diff --git a/parse/train/3SV-ZePhnZM/3SV-ZePhnZM.md b/parse/train/3SV-ZePhnZM/3SV-ZePhnZM.md new file mode 100644 index 0000000000000000000000000000000000000000..356ba510c5c3aff6eecaf587d379e85f9bbb174a --- /dev/null +++ b/parse/train/3SV-ZePhnZM/3SV-ZePhnZM.md @@ -0,0 +1,317 @@ +# INCREMENTAL FEW-SHOT LEARNING VIA VECTOR QUANTIZATION IN DEEP EMBEDDED SPACE + +# Kuilin Chen + +# Chi-Guhn Lee + +Department of Mechanical and Industrial Engineering +University of Toronto +Toronto, Ontario, Canada +kuilin.chen@mail.utoronto.ca +Department of Mechanical and Industrial Engineering +University of Toronto +Toronto, Ontario, Canada +cglee@mie.utoronto.ca + +# ABSTRACT + +The capability of incrementally learning new tasks without forgetting old ones is a challenging problem due to catastrophic forgetting. This challenge becomes greater when novel tasks contain very few labelled training samples. Currently, most methods are dedicated to class-incremental learning and rely on sufficient training data to learn additional weights for newly added classes. Those methods cannot be easily extended to incremental regression tasks and could suffer from severe overfitting when learning few-shot novel tasks. In this study, we propose a nonparametric method in deep embedded space to tackle incremental few-shot learning problems. The knowledge about the learned tasks is compressed into a small number of quantized reference vectors. The proposed method learns new tasks sequentially by adding more reference vectors to the model using few-shot samples in each novel task. For classification problems, we employ the nearest neighbor scheme to make classification on sparsely available data and incorporate intra-class variation, less forgetting regularization and calibration of reference vectors to mitigate catastrophic forgetting. In addition, the proposed learning vector quantization (LVQ) in deep embedded space can be customized as a kernel smoother to handle incremental few-shot regression tasks. Experimental results demonstrate that the proposed method outperforms other state-of-the-art methods in incremental learning. + +# 1 INTRODUCTION + +Incremental learning is a learning paradigm that allows the model to continually learn new tasks on novel data, without forgetting how to perform previously learned tasks (Cauwenberghs & Poggio, 2001; Kuzborskij et al., 2013; Mensink et al., 2013). The capability of incremental learning becomes more important in real-world applications, in which the deployed models are exposed to possible out-of-sample data. Typically, hundreds of thousands of labelled samples in new tasks are required to re-train or fine-tune the model (Rebuffi et al., 2017). Unfortunately, it is impractical to gather sufficient samples of new tasks in real applications. In contrast, humans can learn new concepts from just one or a few examples, without losing old knowledge. Therefore, it is desirable to develop algorithms to support incremental learning from very few samples. + +While a natural approach for incremental few-shot learning is to fine-tune part of the base model using novel training data (Donahue et al., 2014; Girshick et al., 2014), the model could suffer from severe over-fitting on new tasks due to a limited number of training samples. Moreover, simple fine-tuning also leads to significant performance drop on previously learned tasks, termed as catastrophic forgetting (Goodfellow et al., 2014). Recent attempts to mitigate the catastrophic forgetting are generally categorized into two streams: memory relay of old training samples (Rebuffi et al., 2017; Shin et al., 2017; Kemker & Kanan, 2018) and regularization on important model parameters (Kirkpatrick et al., 2017; Zenke et al., 2017). However, those incremental learning approaches are developed and tested on unrealistic scenarios where sufficient training samples are available in novel tasks. They may not work well when the training samples in novel tasks are few (Tao et al., 2020b). + +To the best of our knowledge, the majority of incremental learning methodologies focus on classification problems and they cannot be extended to regression problems easily. In class-incremental learning, the model has to expand output dimensions to learn $N ^ { \prime }$ novel classes while keeping the knowledge of existing $N$ classes. Parametric models estimate additional classification weights for novel classes, while nonparametric methods compute the class centroids for novel classes. In comparison, output dimensions in regression problems do not change in incremental learning as neither additional weights nor class centroids are applicable to regression problems. + +Besides, we find that catastrophic forgetting in incremental few-shot classification can be attributed to three reasons. First, the model is biased towards new classes and forgets old classes because the model is fine-tuned on new data only (Hou et al., 2019; Zhao et al., 2020). Meanwhile, the prediction accuracy on novel classes is not good due to over-fitting on few-shot training samples. Second, features of novel samples could overlap with those of old classes in the feature space, leading to ambiguity among classes in the feature space. Finally, features of old classes and classification weights are no longer compatible after the model is fine-tuned with new data. + +In this paper, we investigate the problem of incremental few-shot learning, where only a few training samples are available in new tasks. A unified model is learned sequentially to jointly recognize all classes or regression targets that have been encountered in previous tasks (Rebuffi et al., 2017; Wu et al., 2019). To tackle aforementioned problems, we propose a nonparametric method to handle incremental few-shot learning based on learning vector quantization (LVQ) (Sato & Yamada, 1996) in deep embedded space. As such, the adverse effects of imbalanced weights in a parametric classifier can be completely avoided (Mensink et al., 2013; Snell et al., 2017; Yu et al., 2020). Our contributions are three fold. First, a unified framework is developed, termed as incremental deep learning vector quantization (IDLVQ), to handle both incremental classification (IDLVQ-C) and regression (IDLVQ-R) problems. Second, we develop intra-class variance regularization, less forgetting constraints and calibration factors to mitigate catastrophic forgetting in class-incremental learning. Finally, the proposed methods achieve state-of-the-art performance on incremental fewshot classification and regression datasets. + +# 2 RELATED WORK + +Incremental learning: Some incremental learning approaches rely on memory replay of old exemplars to prevent forgetting previously learned knowledge. Old exemplars can be saved in memory (Rebuffi et al., 2017; Castro et al., 2018; Prabhu et al., 2020) or sampled from generative models (Shin et al., 2017; Kemker & Kanan, 2018; van de Ven et al., 2020). However, explicit storage of training samples is not scalable if the number of classes is large. Furthermore, it is difficult to train a reliable generative model for all classes from very few training samples. In parallel, regularization approaches do not require old exemplars and impose regularization on network weights or outputs to minimize the change of parameters that are important to old tasks (Kirkpatrick et al., 2017; Zenke et al., 2017). To avoid quick performance deterioration after learning a sequence of novel tasks in regularization approaches, semantic drift compensation (SDC) is developed by learning an embedding network via triplet loss (Schroff et al., 2015) and compensates the drift of class centroids using novel data only (Yu et al., 2020). In comparison, IDLVQ-C saves only one exemplar per class and uses saved exemplars to regularize the change in feature extractor and calibrate the change in reference vectors. + +Few-shot learning: Few-shot learning attempts to obtain models for classification or regression tasks with only a few labelled samples. Few-shot models are trained on widely-varying episodes of fake few-shot tasks with labelled samples drawn from a large-scale meta-training dataset (Vinyals et al., 2016; Finn et al., 2017; Ravi & Larochelle, 2017; Snell et al., 2017; Sung et al., 2018). Meanwhile, recent works attempt to handle novel few-shot tasks while retraining the knowledge of the base task. These methods are referred to as dynamic few-shot learning (Gidaris & Komodakis, 2018; Ren et al., 2019a; Gidaris & Komodakis, 2019). However, dynamic few-shot learning is different from incremental few-shot learning, because they rely on the entire base training dataset and an extra meta-training dataset during meta-training. In addition, dynamic few-shot learning does not accumulate knowledge for multiple novel tasks sequentially. + +Incremental few-shot learning: Prior works on incremental few-shot learning focus on classification problems by computing the weights for novel classes in parametric classifiers, without iterative gradient descent. For instance, the weights of novel classes can be imprinted by normalized prototypes of novel classes, while keeping the feature extractor fixed (Qi et al., 2018). Since novel weights are computed only with the samples of novel classes, the fixed feature extractor may not be compatible with novel classification weights. More recently, neural gas network is employed to construct an undirected graph to represent knowledge of old classes (Tao et al., 2020b;a). The vertices in the graph are constructed in an unsupervised manner using competitive Hebbian learning (Fritzke, 1995), while the feature embedding is fixed. In contrast, IDLVQ learns both feature extractor and reference vectors concurrently in a supervised manner. + +# 3 BACKGROUND + +# 3.1 INCREMENTAL FEW-SHOT LEARNING + +In this paper, incremental few-shot learning is studied for both classification and regression tasks. For classification tasks, we consider the standard class-incremental setup in literature. After the model is trained on a base task $\mathit { t } = 1$ ) with sufficient data, the model learns novel tasks sequentially. Each novel task contains a number of novel classes with only a few training samples per class. Learning a novel task $( t > 1 )$ ) is referred to as an incremental learning session. In task $t$ , we have access only to training data $\mathcal { D } ^ { t }$ in the current task and previously saved exemplars (one exemplar per class in this study). Each task has a set of classes $\hat { C } ^ { t } = \{ c _ { 1 } ^ { t } , . . . , c _ { n ^ { t } } ^ { t } \}$ , where $n _ { t }$ is the number of classes in task $t$ . In addition, it is assumed that there is no overlap between classes in different tasks $C ^ { t } \bigcap C ^ { s } = \emptyset$ for $t \neq s$ . After an incremental learning session, the performance of the model is evaluated on a test set that contains all previously seen classes $C = \cap _ { i } { \bar { C } } ^ { i }$ . Note that our focus is not multi-task scenario, where a task ID is exposed to the model during test phase and the model is only required to perform a given task one time (van de Ven & Tolias, 2019). Our model is evaluated in a task-agnostic setting, where task ID is not exposed to the model at test time. + +For regression tasks, we follow a similar setting with a notable difference that the target is realvalued $y \in \mathbb { R }$ . In addition, the target values in different tasks do not have to be mutually exclusive, unlike the class-incremental setup. + +# 3.2 LEARNING VECTOR QUANTIZATION + +Traditional nonparametric methods, such as nearest neighbors, represent knowledge and make predictions by storing the entire training set. Despite the simplicity and effectiveness, they are not scalable to a large-scale base dataset. Typically, incremental learning methods are only allowed to store a small number of exemplars to preserve the knowledge of previously learned tasks. However, randomly selected exemplars may not well present the knowledge in old tasks. LVQ is a classical data compression method that represents the knowledge through a few learned reference vectors (Sato & Yamada, 1996; Seo & Obermayer, 2003; Biehl et al., 2007). A new sample is classified to the same label as the nearest reference vector in the input space. LVQ has been combined with deep feature extractors as an alternative to standard neural networks for better interpretability (De Vries et al., 2016; Villmann et al., 2017; Saralajew et al., 2018). The combinations of LVQ and deep feature extractors have been applied to natural language processing (NLP), facial recognition and biometrics (Variani et al., 2015; Wang et al., 2016; Ren et al., 2019b; Leng et al., 2015). We notice that LVQ is a nonparametric method which is well suited for incremental few-shot learning because the model capacity grows by incorporating more reference vectors to learn new knowledge. For example, incremental learning vector quantization (ILVQ) has been developed to learn classification models adaptively from raw features (Xu et al., 2012). In this study, we present the knowledge by learning reference vectors in the feature space through LVQ and adapt them in incremental few-shot learning. Compared with ILVQ by Xu et al. (2012), our method does not rely on predefined rules to update reference vectors and can be learned along with deep neural networks in an end-to-end fashion. Besides, our method uses a single reference vector for each class, while ILVQ automatically assigns different numbers of prototypes for different classes. + +# 4 METHODOLOGY + +# 4.1 INCREMENTAL DEEP LEARNING VECTOR QUANTIZATION + +The general framework of IDLVQ for both classification and regression can be derived from a Gaussian mixture perspective (Ghahramani $\&$ Jordan, 1994), with a simplified covariance structure and supervised deep representation learning. In the base dataset $( t = 1 )$ ), a raw input $\mathbf { x }$ is projected into a feature space ${ \bar { \mathcal { F } } } ^ { 1 }$ by a deep neural network $f _ { \theta ^ { 1 } }$ , where $\theta ^ { 1 }$ denotes the parameters in neural networks. In addition, $N ^ { 1 }$ reference vectors $\mathbf { M } ^ { 1 } = \{ \mathbf { m } _ { 1 } ^ { 1 } , . . . , \mathbf { m } _ { N ^ { 1 } } ^ { 1 } \}$ are placed in the feature space ${ \mathcal { F } } ^ { 1 }$ , which can be learned to capture the representation of the base dataset. More reference vectors will be added incrementally while learning novel tasks. The marginal distribution $p ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) )$ of feature vector can be described by a Gaussian mixture model $\begin{array} { r } { p ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) ) = \sum _ { i = 1 } ^ { N ^ { 1 } } p ( i ) p ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) | i ) } \end{array}$ of $N ^ { 1 }$ components, where the prior $p ( i ) = 1 / N ^ { 1 }$ and the component distribution $p ( f _ { \boldsymbol { \theta } ^ { 1 } } ( \mathbf { x } ) | i )$ is Gaussian. By assuming that each component distribution $p ( f _ { \boldsymbol { \theta } ^ { 1 } } ( \mathbf { x } ) | i )$ is isotropic Gaussian centered at $\mathbf { m } _ { i } ^ { 1 }$ with the same covariance, the posterior distribution of a component given the input is + +$$ +p ^ { 1 } ( i | \mathbf { x } ) = \frac { \kappa ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) , \mathbf { m } _ { i } ^ { 1 } ) } { \sum _ { j = 1 } ^ { N ^ { 1 } } \kappa ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) , \mathbf { m } _ { j } ^ { 1 } ) } , +$$ + +where $\kappa ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) , \mathbf { m } _ { i } ^ { 1 } ) = \exp ( - \| f _ { \theta ^ { 1 } } ( \mathbf { x } ) - \mathbf { m } _ { i } ^ { 1 } \| ^ { 2 } / \gamma )$ is a Gaussian kernel and $\gamma$ is a scale factor. The conditional expectation of the output from a Gaussian mixture is $\begin{array} { r } { \hat { y } = \sum _ { i = 1 } ^ { N ^ { 1 } } p ^ { 1 } ( i | \mathbf { x } ) q _ { i } ^ { 1 } } \end{array}$ , where $q _ { i } ^ { 1 }$ is the reference target associated with reference vector $\mathbf { m } _ { i } ^ { 1 }$ . In classification problems, $q _ { i } ^ { 1 }$ is either 0 or 1 indicating whether $\mathbf { m } _ { i } ^ { 1 }$ and $\mathbf { x }$ have the same label. Since each reference vector is assigned to a class at initialization, $q _ { i } ^ { 1 }$ is fixed and does not require learning. Meanwhile, $q _ { i } ^ { 1 }$ in regression problems is real-valued and has to be learned. The weights in neural networks $\theta ^ { \mathrm { { I } } }$ , reference vectors ${ \bf { M } } ^ { 1 }$ , reference targets $q _ { i } ^ { 1 }$ (in regression problems only) and the scale factor $\gamma$ are learned concurrently by minimizing a loss function between the true label $y$ and the predicted label $\hat { y }$ . + +The proposed IDLVQ is a nonparametric method as it makes prediction based on similarity to reference vectors, instead of using any regression or classification weights. The capacity of the model grows naturally by adding more reference vectors to learn novel tasks, while the old knowledge is preserved in existing reference vectors. + +# 4.2 INCREMENTAL DEEP LEARNING VECTOR QUANTIZATION FOR CLASSIFICATION + +For classification problems, one reference vector is assigned to each class in our study. Thus, $\hat { y }$ represents the predicted probability that an input belongs to a class. The model can be trained to classify data correctly by minimizing the cross-entropy loss $\mathcal { L } _ { C E }$ between the predicted probability $\hat { y }$ and the true label $y$ . Although the cross-entropy loss encourages separability of features in base classes, it does not guarantee compact intra-class variation in the feature space. Specifically, in an incremental learning session, features of novel classes could overlap with those of previously learned classes. As a result, the overall classification accuracy could deteriorate after incremental learning sessions. A desirable feature embedding leaves large margin between classes to mitigate overlap in features across old and new classes. Inspired by center loss (Wen et al., 2016) to enhance discriminative capability in facial recognition, a regularization term on intra-class distance to reference vectors is added to get compact intra-class variation. + +$$ +\mathcal { L } _ { i n t r a } = \sum _ { \forall ( \mathbf { x } , y ) , y = i } \left\| f _ { \theta ^ { 1 } } ( \mathbf { x } ) - \mathbf { m } _ { i } ^ { 1 } \right\| ^ { 2 } +$$ + +As such, $f _ { \theta ^ { 1 } } ( \mathbf { x } )$ is forced to stay close to the reference vector with the same label and naturally moves away from other reference vectors. Consequently, features of new classes are more likely to lie in the margin between old classes to mitigate ambiguity in features across different classes. The total loss in training the base task is given by $\mathcal { L } = \mathcal { L } _ { C E } + \lambda _ { i n t r a } \mathcal { L } _ { i n t r a }$ , where $\lambda _ { i n t r a }$ is a hyper-parameter to control the weight for intra-class variation loss. The total loss is differentiable w.r.t. neural network parameters $\theta ^ { \mathrm { { i } } }$ , reference vectors $\mathbf { M } ^ { 1 } = \{ \mathbf { m } _ { 1 } ^ { 1 } , . . . , \mathbf { m } _ { n ^ { 1 } } ^ { 1 } \}$ and scaling factor $\gamma$ . All parameters in the model can be trained jointly in an end-to-end fashion. + +In an incremental session $( t > 1 )$ ), a novel dataset $\mathcal { D } ^ { t }$ contains $n ^ { t }$ classes and $K ^ { t }$ samples per class ( $\cdot \boldsymbol { n } ^ { t }$ -way $K ^ { t }$ -shot). $n ^ { t }$ new reference vectors are added and each reference vector is initialized as the centroid of features in a class mti = 1Kt PKtk=1 fθt (xk). The new reference vectors along with the neural network parameters are fine-tuned on $\mathcal { D } ^ { t }$ to learn new knowledge in task $t$ . To preserve the knowledge from the old tasks during incremental learning, the model should be updated only when necessary. Therefore, cross-entropy loss is not used in incremental learning sessions because it always updates model parameters even if the sample is correctly classified. Let $\mathbf { m } _ { + } ^ { t }$ be the reference vector with the correct label and $\mathbf { m } _ { - } ^ { t }$ be the nearest reference vector with a wrong label. For a training sample $\left( \mathbf { x } , y \right)$ in $\mathcal { D } ^ { t }$ , the sample is classified correctly if $\left\| f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { + } ^ { t } \right\| ^ { 2 } <$ $\left\| f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { - } ^ { t } \right\| ^ { 2 }$ . In this case, the loss should be 0. When $\left\| f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { + } ^ { t } \right\| ^ { 2 } > \left\| f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { - } ^ { t } \right\| ^ { 2 }$ , the sample is misclassified. We adapt the margin based loss function ${ \mathcal { L } } _ { M }$ from De Vries et al. (2016) with a minor modification + +$$ +\mathcal { L } _ { M } = \mathrm { R e L U } \left( \frac { \left. f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { + } ^ { t } \right. ^ { 2 } - \left. f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { - } ^ { t } \right. ^ { 2 } } { \left. f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { + } ^ { t } \right. ^ { 2 } + \left. f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { - } ^ { t } \right. ^ { 2 } } \right) , +$$ + +where $\mathrm { R e L U } ( \cdot )$ stands for the rectified linear unit function. The margin based loss leads to slow training convergence because it only updates two reference vectors one time. However, the adapted margin based loss is well suited in learning from few-shot samples while avoids unnecessary parameter updates. + +Features for an old class could deviate away from the corresponding reference vector due to changes in $\theta ^ { t }$ during incremental learning, leading to catastrophic forgetting. A forgetting loss $\mathcal { L } _ { F }$ is developed to regularize the drift in the feature space + +$$ +\mathcal { L } _ { F } = \sum _ { i = 1 } ^ { N ^ { t - 1 } } \| f _ { \theta ^ { t } } ( \mathbf { x } _ { i } ^ { ' } ) - f _ { \theta ^ { t - 1 } } ( \mathbf { x } _ { i } ^ { ' } ) \| ^ { 2 } , +$$ + +where $\mathbf { x } _ { i } ^ { ' }$ is the selected exemplar for class $i$ and $N ^ { t - 1 }$ denotes the total number of classes in the base task and all previous novel tasks. Note that the exemplar $\mathbf { x } _ { i } ^ { ' }$ for class $i \in [ N ^ { t - 1 } , N ^ { t } ]$ is picked from $\mathcal { D } ^ { t }$ whose feature is nearest to $\mathbf { m } _ { i } ^ { t }$ at the end of each learning session. The total loss in the incremental learning session $t$ is $\mathcal { L } = \mathcal { L } _ { M } + \lambda _ { F } \mathcal { L } _ { F } + \lambda _ { i n t r a } \mathcal { L } _ { i n t r a }$ , where $\lambda _ { F }$ and $\lambda _ { i n t r a }$ are weights for forgetting loss and intra-class variation loss, respectively. The total loss is optimized w.r.t. neural network parameters $\theta ^ { t }$ and new reference vectors $\{ \mathbf { m } _ { N ^ { t - 1 } + 1 } ^ { t } , . . . , \mathbf { m } _ { N ^ { t } } ^ { t } \}$ . + +The reference vectors for previously learned tasks are not updated by novel data to prevent catastrophic forgetting. However, they may not be well suited to represent knowledge and make classification in the new feature space $\mathcal { F } ^ { t }$ as feature embedding is changed with updated $\theta ^ { t }$ . Although the true optimal location of those reference vectors are difficult to estimate without using the entire data from all tasks, they can be calculated approximately using the shift in features of exemplars. Considering that features of an exemplar $\mathbf { x } _ { i } ^ { ' }$ are close to $\mathbf { m } _ { i }$ in the feature space, the shift of a reference vector $\delta _ { i } ^ { t }$ in the new feature space can be approximated by the shift of the exemplar’s features $\delta _ { i } ^ { t } = f _ { \theta ^ { t } } ( \mathbf { x } _ { i } ^ { ' } ) { - } f _ { \theta ^ { t - 1 } } ( \mathbf { x } _ { i } ^ { ' } )$ . Therefore, the reference vectors for previously learned tasks are calibrated $\mathbf { m } _ { i } ^ { t } = \mathbf { m } _ { i } ^ { t - 1 } + \delta _ { i } ^ { t }$ , where $\mathbf { m } _ { i } ^ { t - 1 }$ is the uncalibrated reference vector for class $i \in [ 1 , N ^ { t - 1 } ]$ . A test sample, which could be from any seen classes, is classified according to the distance to reference vectors $\{ \mathbf { m } _ { 1 } ^ { t } , . . . , \mathbf { m } _ { N ^ { t } } ^ { t } \}$ . The pseudo code for IDLVQ-C is presented in the appendix. + +# 4.3 INCREMENTAL DEEP LEARNING VECTOR QUANTIZATION FOR REGRESSION + +For regression problems, the model is trained to recognize regression targets by the minimizing mean squared error (MSE) loss $\mathcal { L } _ { M S E } = ( y - \hat { y } ) ^ { 2 }$ , where $y$ is the real-valued target in training dataset. The MSE loss function is differentiable w.r.t. neural network weights, reference vectors and targets, and scale factor. Therefore, all parameters can be trained jointly in an end-to-end manner. The proposed IDLVQ-R can also be interpreted as a kernel smoother in deep embedded space. Compared with traditional kernel smoother, such as Nadaraya-Watson estimator (Nadaraya, 1964), IDLVQ-R is sparse and hence more scalable as it only relies on a few reference vectors and targets. + +In an incremental learning session $( t > 1 )$ ), we have access to data $\mathcal { D } ^ { t }$ that contains $K ^ { t }$ pairs of training samples $( \mathbf { x } _ { i } ^ { t } , y _ { i } ^ { t } )$ . ${ \bar { n } } ^ { t }$ new reference vectors $( n ^ { t } \leq K ^ { t } )$ along with corresponding targets are added to the model to learn new knowledge in the novel task $t$ . We randomly select $n ^ { t }$ samples from $\mathcal { D } ^ { t }$ to initialize reference vectors and targets as follows + +$$ +\begin{array} { c } { { { \bf m } _ { i + N ^ { t - 1 } } = f _ { \theta } ( { \bf x } _ { i } ^ { t } ) , } } \\ { { { \bf \ q } _ { i + N ^ { t - 1 } } = y _ { i } ^ { t } , } } \end{array} +$$ + +where $N ^ { t - 1 }$ is the total number of reference vectors in all previous tasks. The new reference vectors and targets are fine-tuned by minimizing MSE on $\mathcal { D } ^ { t }$ while keeping other parameters frozen. After new reference vectors and targets are fine-tuned with novel data $\mathcal { D } ^ { t }$ , the model makes prediction by smoothing targets of all reference vectors + +$$ +\hat { y } = \frac { \sum _ { i = 1 } ^ { N ^ { t } } \kappa ( f _ { \theta } ( \mathbf { x } ) , \mathbf { m } _ { i } ) q _ { i } } { \sum _ { i = 1 } ^ { N ^ { t } } \kappa ( f _ { \theta } ( \mathbf { x } ) , \mathbf { m } _ { i } ) } , +$$ + +where $N ^ { t }$ is the current total number of reference vectors. + +# 5 EXPERIMENTS + +We first describe the overall protocols, then we present the results on incremental few-shot classification and regression problems. + +# 5.1 INCREMENTAL FEW-SHOT CLASSIFICATION + +We empirically evaluate the performance of IDLVQ-C on incremental few-shot classification on CUB200-2011 (Welinder et al., 2010) and miniImageNet datasets (Vinyals et al., 2016). The dataset is split into base classes and multiple groups of novel classes. We apply standard data augmentation, including random crop, horizontal flip and color jitter, on all training images. After each training session, the model performance is evaluated on a test set, which contains all classes that the model has been trained on. + +CUB dataset is composed of 200 fine-grained bird species with 11,788 images. We split the dataset into 5894 training images, 2947 validation images and 2947 test images. All images are resized to $2 2 4 \times 2 2 4$ . In addition, the first 100 classes are chosen as base classes, where all training samples in base classes are used to train the base model. The remaining 100 classes are treated as novel categories and split into 10 incremental learning sessions. Each incremental learning session contains 10 novel classes and 5 randomly selected training samples per class (10-way 5-shot). + +miniImageNet dataset is a 100-class subset of the original ImageNet dataset (Deng et al., 2009). Each class contains 500 training images, 50 validation images, and 50 test images. The images are in RGB format of the size $8 4 \times 8 4$ . We choose 60 and 40 classes for base and novel classes, respectfully. The 40 novel classes are divided into 8 sessions and each session contains 5 novel classes with 5 randomly selected training samples per class (5-way 5-shot). + +ResNet18 (He et al., 2016) is used as the feature extractor for incremental classification problems. The learning process for each dataset is repeated 10 times and the average test accuracy is reported. The proposed method is compared with six methods for few-shot class-incremental learning: fineturning using $\mathcal { D } ^ { t }$ , joint training using the entire training set from all encountered classes, iCaRL (Rebuffi et al., 2017), Rebalancing (Hou et al., 2019), ProtoNet (Snell et al., 2017), incremental learning vector quantization (ILVQ) (Xu et al., 2012), SDC (Yu et al., 2020), and Imprint (Qi et al., 2018). Note that ILVQ is applied to the features extracted by neural networks in our experiment. + +The incremental few-shot learning results on CUB and miniImageNet are shown in Table 1 and 2, respectively. Our method outperforms fine-tuning, iCaRL (Rebuffi et al., 2017), and ProtoNet (Snell et al., 2017) by a large margin. Simply fine-tuning the weights in classifier with few-shot training samples for novel classes significantly deteriorates the prediction accuracy. Although iCaRL alleviates catastrophic forgetting by tuning the model with a mix of old exemplars and novel few-shot data, the prediction accuracy still drops quickly because iCaRL requires sufficient samples per class to achieve satisfactory performance. The ProtoNet relies on distance to prototypes (the mean of features within a class) to make classification but the fixed feature extractor may not be able to well separate novel classes. ILVQ is slightly better than ProtoNet because prototypes can be learned adaptively when more classes are available in incremental learning sessions. Some prototypes in + +ILVQ are close to the border of a class, which are more effective than class centroids in ProtoNet. However, ILVQ does not achieve the best performance because the feature extractor is fixed and cannot be learned along with the prototypes. IDLVQ-C has a small gain in the first couple of incremental few-shot learning sessions compared with SDC (Yu et al., 2020) and Imprint (Qi et al., 2018). Similar to ProtoNet, SDC also relies on prototypes to make classification. The performance of SDC is better than that of ProtoNet because SDC fine-tunes the feature extractor with novel dataset and compensates the drift in prototypes. However, the compensation for the drift of old-class prototypes can be less accurate in SDC because it is approximated by samples in novel classes. In parallel, the imprint method directly computes the normalized classification weights from the average of normalized features within a novel class. The imprint method avoids imbalanced classification weights and circumvents the overfitting in few-shot class-incremental learning through weight normalization. Nevertheless, the fixed feature extractor in the imprint method may not be well suited for novel classes. In contrast, IDLVQ-C updates the feature extractor only when necessary and compensates the shift of old reference vectors more accurately using exemplars from old classes. That is why the gain of IDLVQ-C increases with more incremental few-shot learning sessions. The performance of SDC, Imprint and IDLVQ-C is better than offline joint training in early sessions of incremental few-shot learning. Offline joint training may not result in oracle performance due to extremely imbalanced samples between base classes and novel classes. + +Table 1: Prediction accuracy on CUB all classes using the 10-way 5-shot incremental setting. + +
Methodsessions
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Fine-tune77.3046.2334.7125.3523.1620.6516.2113.3211.9811.1710.76
Joint train77.3073.2868.8065.3463.7562.0060.8159.7159.0658.6958.23
iCaRL (Rebuffi et al.,2017)77.3057.1854.6748.1140.7636.8533.1230.4228.2226.8425.23
Rebalancing (Hou et al., 2019)77.3064.5356.1447.2938.9234.3931.0427.9327.1224.4623.61
ProtoNet (Snell et al., 2017)77.3069.7666.0162.2959.5857.1055.1354.0952.4051.6550.36
ILVQ (Xu et al., 2012)77.3071.5066.7962.7160.2057.8455.2755.0652.4251.7250.47
SDC (Yu et al., 2020)77.3474.4569.4565.2761.8158.2656.1455.7153.3152.7951.52
Imprint (Qi et al., 2018)77.0273.3969.5065.6162.8160.7459.3958.6156.8555.9354.82
IDLVQ-C77.3774.7270.2867.1365.3463.5262.1061.5459.0458.6857.81
+ +Table 2: Prediction accuracy on miniImageNet all classes using the 5-way 5-shot incremental setting. + +
Methodsessions
123456789
Fine-tune64.2530.1118.536.312.862.681.871.561.42
Joint train64.2558.8055.2652.3849.7148.3745.9144.6843.38
iCaRL (Rebuffi et al., 2017)64.2548.0443.1338.2830.0124.4621.8519.8417.76
Rebalancing (Hou et al.,2019)64.2549.2144.1737.7130.1122.9219.9917.9616.25
ProtoNet (Snell et al., 2017)64.2555.1251.6748.9146.5244.2541.9140.0738.42
ILVQ (Xu et al., 2012)64.2556.0152.4349.3146.9844.3742.0640.1138.43
SDC (Yu et al., 2020)64.6259.6355.3950.9248.3045.2842.9742.5141.24
Imprint (Qi et al., 2018)64.7159.8555.7152.4749.9047.3144.5742.5741.26
IDLVQ-C64.7759.8755.9352.6249.8847.5544.8343.1441.84
+ +Ablation studies are conducted to analyze how individual components affect the performance of incremental few-shot learning. We study five variants of our methods: (a) new reference vectors are initialized as class centroids and no tuning is done for feature extractor or old reference vectors; (b) $\mathcal { L } _ { i n t r a }$ is not used in incremental learning sessions; (c) $\mathcal { L } _ { F }$ is not used in the incremental learning sessions; (d) shift in old reference vectors are not compensated; (e) replace the margin based loss ${ \mathcal { L } } _ { M }$ with the cross entropy loss $\mathcal { L } _ { C E }$ . Table 3 shows the results of our ablation studies on CUB dataset. Without any fine-tuning, the initial reference vectors for novel classes lead to descent accuracy in incremental few-shot classification. It demonstrates the robustness of nonparametric classifier. $\mathcal { L } _ { i n t r a }$ leads to $0 . 5 7 \%$ gain due to tight intra-class variation. The less forgetting regularization $\mathcal { L } _ { F }$ is proved to prevent forgetting old classes and achieves a performance boost by $2 . 3 5 \%$ . The shift compensation for reference vectors effectively adapts old reference vector in new embedding spaces with a gain round of $1 \%$ . The margin based loss is more effective in preventing forgetting in early sessions of incremental learning. + +The effect of the number of training samples per class. The proposed method is evaluated under different few-shot settings on CUB dataset to investigate the effect of different numbers of training samples in novel classes, including 5-shot, 10-shot and 20-shot settings. One reference vector is assigned to each class in all few-shot settings. As shown in Fig. 2 in the appendix, the performance of incremental learning improves as the number of samples per class increases. When the training samples are scarce, the training samples may not well present the generative distribution of training data. Therefore, the learned reference vectors could be biased and classification accuracy is low. With more training samples, the learned reference vectors could well present the center of the distribution and classification accuracy is improved. The gap in performance becomes more obvious as the number of incremental learning sessions grows. The detailed results are reported in Table 7 and 8. + +Table 3: Ablation study on CUB using the 10-way 5-shot incremental setting. + +
Methodsessions
234567891011
No tuning71.9367.1464.2162.6160.1359.0458.4755.6454.2553.66
W.o.Lintra74.7570.2666.8965.0563.1861.8461.3658.6158.1457.24
w.0. LF73.8569.5466.2164.0262.7460.2859.4956.9756.3855.46
w.o. δ74.6770.0166.7464.8163.9061.4260.7358.1657.6256.79
LM→LCE73.2269.4166.0363.9363.0761.1460.9858.6758.1157.32
IDLVQ-C74.7270.2867.1365.3463.5262.1061.5459.0458.6857.81
+ +# 5.2 INCREMENTAL FEW-SHOT REGRESSION + +IDLVQ-R is tested on two regression datasets: sinusoidal wave and 3D spatial data. Considering that there is no state-of-the-art method for incremental few-shot regression, we compare IDLVQ-R against three alternative methods: fine-tuning using novel task data only, fine-tuning using novel task data along with exemplars and offline training using the entire training dataset from all tasks. + +Sinusoidal wave is defined by a function $y = \sin ( 3 \pi x ) + 0 . 3 \cos ( 9 \pi x ) + 0 . 5 \sin ( 7 \pi x ) + \epsilon ,$ , where $\epsilon$ is white noise with a standard deviation of 0.1. 1000 training samples in the first task (bas task) are generated by sampling $x \in [ - 1 . 0 , 1 . 0 ]$ uniformly. 5-shot training samples in two novel tasks are generated by sampling $x \in [ 1 . 0 , 1 . 5 ]$ and $x \in [ 1 . 5 , 2 . 0 ]$ , respectively. + +As shown in Fig. 1(a), IDLVQ-R achieves comparable performance to offline neural networks in Fig. 1(d) which are trained using the entire training set from all tasks. In comparison, neural networks trained sequentially with few-shot training samples show catastrophic forgetting on old tasks in Fig. 1(b). With the addition of exemplars during training, the networks perform better but still suffer from catastrophic forgetting on the base task in Fig. 1(c). In conclusion, IDLVQ-R preserves old knowledge and adapts to new knowledge quickly using a few reference vectors and achieves satisfactory performance on incremental few-shot regression tasks. The experiment details can be found in the appendix. + +![](images/83a97f4c3e066abb502771d10b9d6938bf3934093e0891e51d8029367598a481.jpg) +Figure 1: Comparison of performance for incremental few-shot regression. Red dots denote test samples for base task, purple dots denotes test sample for novel tasks, grey lines denote model predictions, and black crosses denotes few-shot training samples in novel tasks. (a) IDLVQ-R; (b) neural networks incrementally fine-tuned with novel data only in each session; (c) neural networks incrementally fine-tuned with exemplars and novel training samples; (d) offline neural networks trained with training samples from all tasks. + +3D spatial data1 is collected in North Jutland, Denmark. The inputs are longitude $x _ { 1 }$ and latitude $x _ { 2 }$ , and the output is altitude y. 2482 training samples in the 1st task are collected in the area where $x _ { 1 } \in [ 9 . 9 8 , 9 . 9 9 5 ]$ and $x _ { 2 } \in [ 5 7 . 0 , 5 7 . 0 5 ] .$ 20 training samples in the 2nd task are collected in the area where $x _ { 1 } \ \in \ [ 9 . 9 9 5 , 1 0 . 0 ]$ and $x _ { 2 } ~ \in ~ [ 5 7 . 0 2$ , 57.03]. 20 training samples in the 3rd task are collected in the area where $x _ { 1 } \in [ 9 . 9 9 5 , 1 0 . 0 ]$ and $x _ { 2 } ~ \overset { \cdot } { \in } ~ [ 5 7 . 0 3 , 5 7 . 0 4 ]$ . Approximately 300 test samples are collected for each task. + +Table 4: Normalized RMSE of incremental few-shot regression on 3D spatial data + +
Methodsessions
123
Joint train offline0.02174(2e-4)0.02232(2e-4)0.02296(2e-4)
Fine-tune w.novel data0.02174(2e-4)0.08462(4e-4)0.11870(6e-4)
Fine-tune w. exemplars0.02174(2e-4)0.02988(2e-4)0.03128(2e-4)
IDLVQ-R0.02181(2e-4)0.02641(2e-4)0.02817(2e-4)
+ +The normalized root mean squared errors (RMSE) between actual and predicted altitude in the test set are listed in Table 4. The prediction accuracy drops significantly when the model is finetuned with novel data only. Catastrophic forgetting can be alleviated using exemplars from previous tasks. IDLVQ-R achieves better results than fine tuning with exemplars. The good performance of IDLVQ-R can be attributed to two reasons. First, IDLVQ-R learns a number of reference vectors and targets to preserve the knowledge in encountered tasks. Compared with a linear layer on top of neural networks, a number of reference vectors represent richer information about the training data. Second, IDLVQ-R is nonparametric and can represent local and nonlinear relationship without learning any regression coefficient from few-shot data. + +# 6 CONCLUSIONS + +A new incremental few-shot learning approach is developed to harmonize old knowledge preserving and new knowledge adaptation through quantized vector in deep embedded space. Prediction is made in a nonparametric way using similarity to learned reference vectors, which circumvents biased weights in a parametric classification layer during incremental few-shot learning. For classification problems, additional mechanisms are developed to mitigate the forgetting in old classes and improve representation learning for few-shot novel classes. For regression problems, the proposed approach has been reinterpreted as a kernel smoother to predict real-valued target over novel domain. + +# REFERENCES + +Michael Biehl, Anarta Ghosh, and Barbara Hammer. Dynamics and generalization ability of lvq algorithms. Journal of Machine Learning Research, 8(Feb):323–360, 2007. +Francisco M Castro, Manuel J Mar´ın-Jimenez, Nicol ´ as Guil, Cordelia Schmid, and Karteek Alahari. ´ End-to-end incremental learning. In Proceedings of the European conference on computer vision (ECCV), pp. 233–248, 2018. +Gert Cauwenberghs and Tomaso Poggio. Incremental and decremental support vector machine learning. In Advances in neural information processing systems, pp. 409–415, 2001. +Harm De Vries, Roland Memisevic, and Aaron C Courville. Deep learning vector quantization. In ESANN, 2016. +Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248–255. Ieee, 2009. +Jeff Donahue, Yangqing Jia, Oriol Vinyals, Judy Hoffman, Ning Zhang, Eric Tzeng, and Trevor Darrell. Decaf: A deep convolutional activation feature for generic visual recognition. 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Optimization as a model for few-shot learning. In 5th International Conference on Learning Representations, ICLR 2017, 2017. + +Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, Georg Sperl, and Christoph H Lampert. icarl: Incremental classifier and representation learning. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pp. 2001–2010, 2017. + +Mengye Ren, Renjie Liao, Ethan Fetaya, and Richard Zemel. Incremental few-shot learning with attention attractor networks. In Advances in Neural Information Processing Systems, pp. 5275– 5285, 2019a. + +Zhenwen Ren, Bin Wu, Quansen Sun, and Mingna Wu. Simultaneous learning of reduced prototypes and local metric for image set classification. Expert Systems with Applications, 134:102–111, 2019b. + +Sascha Saralajew, Lars Holdijk, Maike Rees, and Thomas Villmann. Prototype-based neural network layers: incorporating vector quantization. In Advances in neural information processing systems - workshop poster, 2018. + +Atsushi Sato and Keiji Yamada. Generalized learning vector quantization. In Advances in neural information processing systems, pp. 423–429, 1996. + +Florian Schroff, Dmitry Kalenichenko, and James Philbin. Facenet: A unified embedding for face recognition and clustering. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 815–823, 2015. + +Sambu Seo and Klaus Obermayer. Soft learning vector quantization. Neural computation, 15(7): 1589–1604, 2003. + +Hanul Shin, Jung Kwon Lee, Jaehong Kim, and Jiwon Kim. Continual learning with deep generative replay. In Advances in Neural Information Processing Systems, pp. 2990–2999, 2017. + +Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Advances in Neural Information Processing Systems, pp. 4077–4087, 2017. + +Flood Sung, Yongxin Yang, Li Zhang, Tao Xiang, Philip HS Torr, and Timothy M Hospedales. Learning to compare: Relation network for few-shot learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1199–1208, 2018. + +Xiaoyu Tao, Xinyuan Chang, Xiaopeng Hong, Xing Wei, and Yihong Gong. Topology-preserving class-incremental learning. In European Conference on Computer Vision. Springer, 2020a. + +Xiaoyu Tao, Xiaopeng Hong, Xinyuan Chang, Songlin Dong, Xing Wei, and Yihong Gong. Fewshot class-incremental learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12183–12192, 2020b. + +Gido M van de Ven and Andreas S Tolias. Three scenarios for continual learning. In Advances in Neural Information Processing Systems - Continual Learning workshop, 2019. + +Gido M van de Ven, Hava T Siegelmann, and Andreas S Tolias. Brain-inspired replay for continual learning with artificial neural networks. Nature Communications, 11:4069, 2020. + +Ehsan Variani, Erik McDermott, and Georg Heigold. A gaussian mixture model layer jointly optimized with discriminative features within a deep neural network architecture. In 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4270–4274. IEEE, 2015. + +Thomas Villmann, Michael Biehl, Andrea Villmann, and Sascha Saralajew. Fusion of deep learning architectures, multilayer feedforward networks and learning vector quantizers for deep classification learning. In 2017 12th international workshop on self-organizing maps and learning vector quantization, clustering and data visualization (WSOM), pp. 1–8. IEEE, 2017. + +Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In Advances in neural information processing systems, pp. 3630–3638, 2016. + +Wen Wang, Ruiping Wang, Shiguang Shan, and Xilin Chen. Prototype discriminative learning for face image set classification. In Asian Conference on Computer Vision, pp. 344–360. Springer, 2016. + +P. Welinder, S. Branson, T. Mita, C. Wah, F. Schroff, S. Belongie, and P. Perona. Caltech-UCSD Birds 200. Technical Report CNS-TR-2010-001, California Institute of Technology, 2010. + +Yandong Wen, Kaipeng Zhang, Zhifeng Li, and Yu Qiao. A discriminative feature learning approach for deep face recognition. In European conference on computer vision, pp. 499–515. Springer, 2016. + +Yue Wu, Yinpeng Chen, Lijuan Wang, Yuancheng Ye, Zicheng Liu, Yandong Guo, and Yun Fu. Large scale incremental learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 374–382, 2019. + +Ye Xu, Furao Shen, and Jinxi Zhao. An incremental learning vector quantization algorithm for pattern classification. Neural Computing and Applications, 21(6):1205–1215, 2012. + +Lu Yu, Bartlomiej Twardowski, Xialei Liu, Luis Herranz, Kai Wang, Yongmei Cheng, Shangling Jui, and Joost van de Weijer. Semantic drift compensation for class-incremental learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6982–6991, 2020. + +Friedemann Zenke, Ben Poole, and Surya Ganguli. Continual learning through synaptic intelligence. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 3987– 3995, 2017. + +Bowen Zhao, Xi Xiao, Guojun Gan, Bin Zhang, and Shu-Tao Xia. Maintaining discrimination and fairness in class incremental learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13208–13217, 2020. + +# A APPENDIX + +A.1 PSEUDO CODE FOR IDLVQ-C + +# Algorithm 1 IDLVQ-C + +
In the base task(t = 1)
Initialize 01,{m𝑖,.,mN1} and γ
Minimize L = LcE + XintraLintra W.r.t. θ1,{ml,.,mN1} and γ
Pick exemplars from D1 for classes in the base task: xi = arg minxeD1 |lft-1(x) - m1|l²
for novel task t = 2,3,... do
Initialize {mt-1+1.*., mNt}
Minimize L=LM +XFLF + XintraLintra W.r.t. 0t and {mNt-1+1,.,mNt}
t-1 t-1 +
Pick exemplars from Dt for classes in the novel task t: x' = arg minx∈Dt |lfet-1(x) - m ll²
end for
+ +# A.2 PSEUDO CODE FOR IDLVQ-R + +# Algorithm 2 IDLVQ-R + +
In the base task (t =1) Initializeθ,{m1,..,mN1},{q],..,qN1} and γ
Minimize Lmse w.r.t. θ,{ml,..,mN1},{ql,.,qN1} and γ
for novel task t = 2,3,.. do
Initialize {mNt-1+1,, mNt} and {qNt-1+1.…,qNt}
Minimize LmsE w.r.t. {mNt-1+1..*,mNt} and {qNt-1+1,.,qNt}
end for
+ +# A.3 EXPERIMENT DETAILS FOR INCREMENTAL FEW-SHOT CLASSIFICATIO + +The base model is trained by the SGD optimizer (momentum of 0.9 and weight decay of 1e-4) with a mini-batch size of 64. For CUB dataset, the initial learning rate is 0.01 and is decayed by 0.1 after 60 and 120 epochs (200 epochs in total). For miniImageNet, the learning rate also starts from 0.01 and is decayed by 0.1 every 200 epochs (600 epochs in total). In an incremental learning session $( t > 1 )$ ), the model is fine-tuned with with $\mathcal { D } ^ { t }$ with a learning rate of 0.01 for 100 epochs. Since novel data $\mathcal { D } ^ { t }$ $t > 1$ ) contains very few training samples, all training samples in $\mathcal { D } ^ { t }$ are included in one mini-batch. In addition, we use $\lambda _ { i n t r a } = 1 . 0$ and $\lambda _ { F } = 0 . 5$ for both datasets. Empirically, larger $\lambda _ { i n t r a }$ leads to more compact intra-class variation. However, convergence could be slow if $\lambda _ { i n t r a }$ is too large. In addition, larger $\lambda _ { F }$ results in less forgetting in old classes but makes learning novel classes more difficult. + +# A.4 ADDITIONAL RESULTS FOR INCREMENTAL FEW-SHOT CLASSIFICATION + +The accuracies for base and novel classes are reported separately in Table 5 and 6 for CUB amd miniImageNet, respectively. The prediction accuracy of novel classes is calculated upon all novel classes the model has been trained on. Note that the accuracy in Table 1 and 2 is calculated upon all classes (including base and novel classes) that the model has been trained on. The proposed IDLVQC demonstrates strong capability of preserving old knowledge by achieving the best performance on old classes across all learning sessions. In parallel, Imprint method performs slightly better on novel classes than IDLVQ-C in early incremental learning sessions, while IDLVQ-C outperforms Imprint method in longer incremental learning sessions. The advantage of IDLVQ-C can be attributed to the adaptive feature extractor, which is tuned in each learning session. + +Table 5: Prediction accuracy on CUB base and novel classes using the 10-way 5-shot incremental setting. + +
Base classessessions
1234567891011
Fine-tune77.3044.2336.2827.5225.9623.0517.6813.0711.7810.9910.71
Joint train77.3075.8375.2574.5174.5873.7473.9573.2573.1173.2573.18
iCaRL (Rebuffi et al., 2017)77.3059.3858.8154.4348.2743.2839.1734.9132.4329.3625.87
Rebalancing (Hou et al.,2019)77.3066.4360.3255.3646.3941.7637.1232.5831.2627.0324.25
ProtoNet (Snell et al., 2017)77.3072.5572.2172.0671.6471.2971.0270.9470.6770.6070.53
ILVQ (Xu et al., 2012)77.3074.1873.5772.6672.5671.5771.1471.1271.0270.9870.85
SDC (Yu et al., 2020)77.3476.0575.2174.1272.3671.8171.6871.4371.2571.2770.96
Imprint (Qi et al., 2018)77.0274.7674.5773.6972.6970.8870.3470.1270.0769.8469.27
IDLVQ-C77.3776.3275.9075.9175.4974.8674.5874.3774.0273.3973.32
Novel classessessions
1234567891011
Fine-tune=66.2326.8618.1216.1615.8513.7613.6812.2311.3710.81
Joint train=47.7836.5534.7736.6838.5238.9140.3741.5042.5143.28
iCaRL (Rebuffi et al., 2017)-35.1833.9727.0421.9923.9923.0424.0122.9624.0424.59
Rebalancing (Hou et al.,2019)=45.5335.2420.3920.2519.6520.9121.2921.9521.6022.97
ProtoNet (Snell et al., 2017)41.8635.0129.7229.4328.7228.6530.0229.5630.5930.19
ILVQ (Xu et al., 2012)40.7432.8929.5429.3030.3828.8232.1229.1730.3230.09
SDC (Yu et al., 2020)58.4540.6535.7733.2331.1630.2433.2530.8932.2632.08
Imprint (Qi et al.,2018)59.6944.1538.6838.1140.4641.1442.1740.3340.4740.37
IDLVQ-C=58.7242.1837.8639.9740.8441.3043.2140.3242.3442.30
+ +The test accuracy on CUB dataset using 10-way 10-shot and 10-way 20-shot incremental settings are reported in Table 7 and 8, respectively. The prediction accuracy improves in all methods with more training samples per class. The iCaRL and Rebalancing methods show the most significant improvement when the number of training samples increases. The proposed IDLVQ-C is effective in different incremental few-shot scenarios as it achieves the best performance on 5-shot, 10-shot and 20-shot settings. + +# A.5 EXPERIMENT DETAILS FOR INCREMENTAL FEW-SHOT REGRESSION + +Sinusoidal wave: A six-layer feedforward neural network with ReLU nonlinear activation is used as the feature extractor. IDLVQ-R learns 10 reference vectors and targets from the base task. In each incremental learning sessions, 5 pairs of reference vectors and targets are added. After new reference vectors and targets are fine-tuned, the model is capable of make prediction for all seen tasks. 10 exemplars are selected uniformly from the training set in the base task. In the incremental learning session of the 2nd task, the model is fine-tuned on 10 exemplars and 5 novel training samples. After the training converges, 5 novel training samples in the current task are added to the exemplar set. In the incremental learning session of the 3rd task, the model is fine-tuned with 15 exemplars from old tasks and 5 novel training samples. + +Table 6: Prediction accuracy on miniImageNet base and novel classes using the 5-way 5-shot incremental setting. + +
Base classessessions
123456789
Fine-tune64.2532.2820.876.953.173.161.921.531.46
Joint train64.2563.3062.8362.1662.1862.6861.8661.8761.89
iCaRL (Rebuffi et al., 2017)64.2551.6648.9745.6237.3930.8628.6826.8324.47
Rebalancing (Hou et al., 2019)64.2552.8750.1644.7837.4828.7525.5822.9721.57
ProtoNet (Snell et al., 2017)64.2559.2758.8858.6958.2257.6357.0356.8056.47
ILVQ (Xu et al.,2012)64.2560.2459.6259.0258.6157.7157.1656.8356.49
SDC (Yu et al., 2020)64.6263.5862.7861.1260.2959.3759.0559.9759.87
Imprint (Qi et al.,2018)64.7163.5262.9662.1361.1761.2760.6359.8659.64
IDLVQ-C64.77 63.7763.2262.4461.2261.4760.9760.6660.44
Novel classessessions
123456789
Fine-tune4.074.493.751.931.531.771.611.36
Joint train-4.809.8413.2612.3014.0314.0115.2115.62
iCaRL (Rebuffi et al., 2017)=4.608.098.927.879.108.197.867.70
Rebalancing (Hou et al.,2019)5.298.239.438.008.938.818.838.27
ProtoNet (Snell et al., 2017)5.328.419.7911.4212.1411.6711.3911.35
ILVQ (Xu et al.,2012)5.259.2910.4712.0912.3511.8611.4511.34
SDC (Yu et al., 2020)12.2311.0510.1212.3311.4610.8112.5813.30
Imprint (Qi et al., 2018)15.8112.2113.8316.0913.8112.4512.9313.69
IDLVQ-C-13.0712.1913.3415.8614.1412.5513.1113.94
+ +Table 7: Prediction accuracy on CUB using the 10-way 10-shot incremental setting. + +
Methodsessions
1234567891011
Fine-tune77.3047.1636.3426.9224.0821.2417.1914.3112.7311.7511.43
Joint train77.3073.2971.5468.7266.3865.4264.9865.7464.8264.4764.16
iCaRL (Rebuffi et al., 2017)77.3059.6656.2452.2648.7746.3744.5443.1742.3541.1940.92
Rebalancing (Hou et al., 2019)77.3064.5358.3553.8249.2747.1245.1643.0542.3741.0240.86
ProtoNet (Snell et al., 2017)77.3069.8266.1263.1961.1758.8558.0457.7555.8455.8255.60
ILVQ (Xu et al., 2012)77.3071.2666.8463.8262.6659.7158.9258.1156.3156.1456.03
SDC (Yu et al., 2020)77.3474.6769.7366.7166.4962.1461.3359.8458.0157.3956.62
Imprint (Qi et al.,2018)77.0274.0770.2666.8464.4562.4661.8561.0259.2058.9358.43
IDLVQ-C77.3774.7970.9668.0865.9464.1263.5862.9860.8560.5459.72
+ +Table 8: Prediction accuracy on CUB using the 10-way 20-shot incremental setting. + +
Methodsessions
1234567891011
Fine-tune77.3047.6836.7527.0124.5621.7217.8815.2412.8411.8911.52
Joint train77.3074.3472.5970.8570.1470.0169.6969.6769.4869.4169.42
iCaRL (Rebuff et al.,2017)77.3068.8563.5660.3457.7156.2855.9755.0254.6252.2152.23
Rebalancing (Hou et al.,2019)77.3069.3665.4961.3259.3058.6858.7757.4256.2555.3354.17
ProtoNet (Snell et al., 2017)77.3070.8467.8264.8162.9561.7160.9860.7359.5159.3058.95
ILVQ (Xu et al., 2012)77.3071.5068.7766.1264.3162.8962.3162.0061.5259.7959.23
SDC (Yu et al.,2020)77.3474.6271.6368.6666.7565.2464.2163.6261.9761.5461.11
Imprint (Qi et al., 2018)77.0274.1470.7267.7565.8864.6364.2863.8262.0061.8461.40
IDLVQ-C77.3774.8472.0769.0567.2865.5165.1964.8462.7762.5561.96
+ +3D spatial data: We follow the same training and test protocols as the sinusoidal wave dataset. We choose 40, 15 and 15 reference vectors and targets for 1st, 2nd and 3rd tasks, respectively. Adding more reference vectors does not result in obvious improvement in accuracy in our experiments. + +![](images/5e1419f899b33ff87b466bb8184b86bfa4ee63793fee7e949fdeb2a29f2e20af.jpg) +Figure 2: Comparison results of different few-shot settings, evaluated with ResNet18 on CUB dataset + +# A.6 VISUALIZATION OF IDLVQ-C + +We show the visualization of standard neural networks and IDLVQ-C with/without intra-class variation loss in Fig. 3. MNIST dataset is used as a toy example for visualization. Classes 0-7 are old classes with sufficient training samples, and classes 8 and 9 are novel classes with few-shot training samples. It can be observed in Fig. 3(a) and 3(b) that standard neural networks and IDLVQ-C (without intra-class variation loss) trained by cross-entropy loss do not have compact intra-class variation. Consequently, features of novel classes are more likely to overlap with old classes. In this case, the performance of class-incremental learning degrades very quickly because the classifier cannot distinguish between features from different classes. In comparison, the proposed IDLVQ-C makes intra-class variation compact and leaves large margin between classes in Fig 3(c). As a result, the features of novel classes are less likely to overlap existing classes. The compact intra-class variation and large margin between classes make features of novel classes distinguishable so that learning novel classes is easier. In addition, the margin based loss only updates the model parameters when necessary and avoids catastrophic forgetting of old classes. + +![](images/49e99100b8d6c5c5a8ae0c3a024f4b687cf490c746e6964ee89e2468e958123b.jpg) +Figure 3: Visualization of feature spaces in different methods. Dots represent features of samples and crosses denote references vectors of classes. (a) standard neural networks; (b) IDLVQ-C without $\mathcal { L } _ { i n t r a }$ ; (c) IDLVQ-C. \ No newline at end of file diff --git a/parse/train/3SV-ZePhnZM/3SV-ZePhnZM_content_list.json b/parse/train/3SV-ZePhnZM/3SV-ZePhnZM_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..d6bf84cad75fa5666baeb56f54c79f286c4c5d3a --- /dev/null +++ b/parse/train/3SV-ZePhnZM/3SV-ZePhnZM_content_list.json @@ -0,0 +1,1682 @@ +[ + { + "type": "text", + "text": "INCREMENTAL FEW-SHOT LEARNING VIA VECTOR QUANTIZATION IN DEEP EMBEDDED SPACE ", + "text_level": 1, + "bbox": [ + 174, + 101, + 820, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Kuilin Chen ", + "text_level": 1, + "bbox": [ + 184, + 172, + 271, + 186 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Chi-Guhn Lee ", + "text_level": 1, + "bbox": [ + 557, + 172, + 658, + 186 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Mechanical and Industrial Engineering \nUniversity of Toronto \nToronto, Ontario, Canada \nkuilin.chen@mail.utoronto.ca \nDepartment of Mechanical and Industrial Engineering \nUniversity of Toronto \nToronto, Ontario, Canada \ncglee@mie.utoronto.ca ", + "bbox": [ + 184, + 188, + 537, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 555, + 188, + 910, + 242 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 279, + 544, + 294 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The capability of incrementally learning new tasks without forgetting old ones is a challenging problem due to catastrophic forgetting. This challenge becomes greater when novel tasks contain very few labelled training samples. Currently, most methods are dedicated to class-incremental learning and rely on sufficient training data to learn additional weights for newly added classes. Those methods cannot be easily extended to incremental regression tasks and could suffer from severe overfitting when learning few-shot novel tasks. In this study, we propose a nonparametric method in deep embedded space to tackle incremental few-shot learning problems. The knowledge about the learned tasks is compressed into a small number of quantized reference vectors. The proposed method learns new tasks sequentially by adding more reference vectors to the model using few-shot samples in each novel task. For classification problems, we employ the nearest neighbor scheme to make classification on sparsely available data and incorporate intra-class variation, less forgetting regularization and calibration of reference vectors to mitigate catastrophic forgetting. In addition, the proposed learning vector quantization (LVQ) in deep embedded space can be customized as a kernel smoother to handle incremental few-shot regression tasks. Experimental results demonstrate that the proposed method outperforms other state-of-the-art methods in incremental learning. ", + "bbox": [ + 233, + 314, + 764, + 579 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 617, + 336, + 632 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Incremental learning is a learning paradigm that allows the model to continually learn new tasks on novel data, without forgetting how to perform previously learned tasks (Cauwenberghs & Poggio, 2001; Kuzborskij et al., 2013; Mensink et al., 2013). The capability of incremental learning becomes more important in real-world applications, in which the deployed models are exposed to possible out-of-sample data. Typically, hundreds of thousands of labelled samples in new tasks are required to re-train or fine-tune the model (Rebuffi et al., 2017). Unfortunately, it is impractical to gather sufficient samples of new tasks in real applications. In contrast, humans can learn new concepts from just one or a few examples, without losing old knowledge. Therefore, it is desirable to develop algorithms to support incremental learning from very few samples. ", + "bbox": [ + 174, + 652, + 825, + 777 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "While a natural approach for incremental few-shot learning is to fine-tune part of the base model using novel training data (Donahue et al., 2014; Girshick et al., 2014), the model could suffer from severe over-fitting on new tasks due to a limited number of training samples. Moreover, simple fine-tuning also leads to significant performance drop on previously learned tasks, termed as catastrophic forgetting (Goodfellow et al., 2014). Recent attempts to mitigate the catastrophic forgetting are generally categorized into two streams: memory relay of old training samples (Rebuffi et al., 2017; Shin et al., 2017; Kemker & Kanan, 2018) and regularization on important model parameters (Kirkpatrick et al., 2017; Zenke et al., 2017). However, those incremental learning approaches are developed and tested on unrealistic scenarios where sufficient training samples are available in novel tasks. They may not work well when the training samples in novel tasks are few (Tao et al., 2020b). ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "To the best of our knowledge, the majority of incremental learning methodologies focus on classification problems and they cannot be extended to regression problems easily. In class-incremental learning, the model has to expand output dimensions to learn $N ^ { \\prime }$ novel classes while keeping the knowledge of existing $N$ classes. Parametric models estimate additional classification weights for novel classes, while nonparametric methods compute the class centroids for novel classes. In comparison, output dimensions in regression problems do not change in incremental learning as neither additional weights nor class centroids are applicable to regression problems. ", + "bbox": [ + 174, + 103, + 823, + 202 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Besides, we find that catastrophic forgetting in incremental few-shot classification can be attributed to three reasons. First, the model is biased towards new classes and forgets old classes because the model is fine-tuned on new data only (Hou et al., 2019; Zhao et al., 2020). Meanwhile, the prediction accuracy on novel classes is not good due to over-fitting on few-shot training samples. Second, features of novel samples could overlap with those of old classes in the feature space, leading to ambiguity among classes in the feature space. Finally, features of old classes and classification weights are no longer compatible after the model is fine-tuned with new data. ", + "bbox": [ + 174, + 208, + 825, + 305 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, we investigate the problem of incremental few-shot learning, where only a few training samples are available in new tasks. A unified model is learned sequentially to jointly recognize all classes or regression targets that have been encountered in previous tasks (Rebuffi et al., 2017; Wu et al., 2019). To tackle aforementioned problems, we propose a nonparametric method to handle incremental few-shot learning based on learning vector quantization (LVQ) (Sato & Yamada, 1996) in deep embedded space. As such, the adverse effects of imbalanced weights in a parametric classifier can be completely avoided (Mensink et al., 2013; Snell et al., 2017; Yu et al., 2020). Our contributions are three fold. First, a unified framework is developed, termed as incremental deep learning vector quantization (IDLVQ), to handle both incremental classification (IDLVQ-C) and regression (IDLVQ-R) problems. Second, we develop intra-class variance regularization, less forgetting constraints and calibration factors to mitigate catastrophic forgetting in class-incremental learning. Finally, the proposed methods achieve state-of-the-art performance on incremental fewshot classification and regression datasets. ", + "bbox": [ + 173, + 313, + 825, + 492 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 515, + 344, + 531 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Incremental learning: Some incremental learning approaches rely on memory replay of old exemplars to prevent forgetting previously learned knowledge. Old exemplars can be saved in memory (Rebuffi et al., 2017; Castro et al., 2018; Prabhu et al., 2020) or sampled from generative models (Shin et al., 2017; Kemker & Kanan, 2018; van de Ven et al., 2020). However, explicit storage of training samples is not scalable if the number of classes is large. Furthermore, it is difficult to train a reliable generative model for all classes from very few training samples. In parallel, regularization approaches do not require old exemplars and impose regularization on network weights or outputs to minimize the change of parameters that are important to old tasks (Kirkpatrick et al., 2017; Zenke et al., 2017). To avoid quick performance deterioration after learning a sequence of novel tasks in regularization approaches, semantic drift compensation (SDC) is developed by learning an embedding network via triplet loss (Schroff et al., 2015) and compensates the drift of class centroids using novel data only (Yu et al., 2020). In comparison, IDLVQ-C saves only one exemplar per class and uses saved exemplars to regularize the change in feature extractor and calibrate the change in reference vectors. ", + "bbox": [ + 174, + 547, + 825, + 742 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Few-shot learning: Few-shot learning attempts to obtain models for classification or regression tasks with only a few labelled samples. Few-shot models are trained on widely-varying episodes of fake few-shot tasks with labelled samples drawn from a large-scale meta-training dataset (Vinyals et al., 2016; Finn et al., 2017; Ravi & Larochelle, 2017; Snell et al., 2017; Sung et al., 2018). Meanwhile, recent works attempt to handle novel few-shot tasks while retraining the knowledge of the base task. These methods are referred to as dynamic few-shot learning (Gidaris & Komodakis, 2018; Ren et al., 2019a; Gidaris & Komodakis, 2019). However, dynamic few-shot learning is different from incremental few-shot learning, because they rely on the entire base training dataset and an extra meta-training dataset during meta-training. In addition, dynamic few-shot learning does not accumulate knowledge for multiple novel tasks sequentially. ", + "bbox": [ + 173, + 750, + 825, + 888 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Incremental few-shot learning: Prior works on incremental few-shot learning focus on classification problems by computing the weights for novel classes in parametric classifiers, without iterative gradient descent. For instance, the weights of novel classes can be imprinted by normalized prototypes of novel classes, while keeping the feature extractor fixed (Qi et al., 2018). Since novel weights are computed only with the samples of novel classes, the fixed feature extractor may not be compatible with novel classification weights. More recently, neural gas network is employed to construct an undirected graph to represent knowledge of old classes (Tao et al., 2020b;a). The vertices in the graph are constructed in an unsupervised manner using competitive Hebbian learning (Fritzke, 1995), while the feature embedding is fixed. In contrast, IDLVQ learns both feature extractor and reference vectors concurrently in a supervised manner. ", + "bbox": [ + 173, + 895, + 821, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 255, + 326, + 271 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 INCREMENTAL FEW-SHOT LEARNING ", + "text_level": 1, + "bbox": [ + 176, + 299, + 470, + 313 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this paper, incremental few-shot learning is studied for both classification and regression tasks. For classification tasks, we consider the standard class-incremental setup in literature. After the model is trained on a base task $\\mathit { t } = 1$ ) with sufficient data, the model learns novel tasks sequentially. Each novel task contains a number of novel classes with only a few training samples per class. Learning a novel task $( t > 1 )$ ) is referred to as an incremental learning session. In task $t$ , we have access only to training data $\\mathcal { D } ^ { t }$ in the current task and previously saved exemplars (one exemplar per class in this study). Each task has a set of classes $\\hat { C } ^ { t } = \\{ c _ { 1 } ^ { t } , . . . , c _ { n ^ { t } } ^ { t } \\}$ , where $n _ { t }$ is the number of classes in task $t$ . In addition, it is assumed that there is no overlap between classes in different tasks $C ^ { t } \\bigcap C ^ { s } = \\emptyset$ for $t \\neq s$ . After an incremental learning session, the performance of the model is evaluated on a test set that contains all previously seen classes $C = \\cap _ { i } { \\bar { C } } ^ { i }$ . Note that our focus is not multi-task scenario, where a task ID is exposed to the model during test phase and the model is only required to perform a given task one time (van de Ven & Tolias, 2019). Our model is evaluated in a task-agnostic setting, where task ID is not exposed to the model at test time. ", + "bbox": [ + 174, + 332, + 825, + 512 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "For regression tasks, we follow a similar setting with a notable difference that the target is realvalued $y \\in \\mathbb { R }$ . In addition, the target values in different tasks do not have to be mutually exclusive, unlike the class-incremental setup. ", + "bbox": [ + 176, + 520, + 825, + 561 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 LEARNING VECTOR QUANTIZATION ", + "text_level": 1, + "bbox": [ + 178, + 598, + 459, + 612 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Traditional nonparametric methods, such as nearest neighbors, represent knowledge and make predictions by storing the entire training set. Despite the simplicity and effectiveness, they are not scalable to a large-scale base dataset. Typically, incremental learning methods are only allowed to store a small number of exemplars to preserve the knowledge of previously learned tasks. However, randomly selected exemplars may not well present the knowledge in old tasks. LVQ is a classical data compression method that represents the knowledge through a few learned reference vectors (Sato & Yamada, 1996; Seo & Obermayer, 2003; Biehl et al., 2007). A new sample is classified to the same label as the nearest reference vector in the input space. LVQ has been combined with deep feature extractors as an alternative to standard neural networks for better interpretability (De Vries et al., 2016; Villmann et al., 2017; Saralajew et al., 2018). The combinations of LVQ and deep feature extractors have been applied to natural language processing (NLP), facial recognition and biometrics (Variani et al., 2015; Wang et al., 2016; Ren et al., 2019b; Leng et al., 2015). We notice that LVQ is a nonparametric method which is well suited for incremental few-shot learning because the model capacity grows by incorporating more reference vectors to learn new knowledge. For example, incremental learning vector quantization (ILVQ) has been developed to learn classification models adaptively from raw features (Xu et al., 2012). In this study, we present the knowledge by learning reference vectors in the feature space through LVQ and adapt them in incremental few-shot learning. Compared with ILVQ by Xu et al. (2012), our method does not rely on predefined rules to update reference vectors and can be learned along with deep neural networks in an end-to-end fashion. Besides, our method uses a single reference vector for each class, while ILVQ automatically assigns different numbers of prototypes for different classes. ", + "bbox": [ + 174, + 632, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "4 METHODOLOGY ", + "text_level": 1, + "bbox": [ + 176, + 102, + 341, + 118 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 INCREMENTAL DEEP LEARNING VECTOR QUANTIZATION ", + "text_level": 1, + "bbox": [ + 174, + 133, + 604, + 147 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The general framework of IDLVQ for both classification and regression can be derived from a Gaussian mixture perspective (Ghahramani $\\&$ Jordan, 1994), with a simplified covariance structure and supervised deep representation learning. In the base dataset $( t = 1 )$ ), a raw input $\\mathbf { x }$ is projected into a feature space ${ \\bar { \\mathcal { F } } } ^ { 1 }$ by a deep neural network $f _ { \\theta ^ { 1 } }$ , where $\\theta ^ { 1 }$ denotes the parameters in neural networks. In addition, $N ^ { 1 }$ reference vectors $\\mathbf { M } ^ { 1 } = \\{ \\mathbf { m } _ { 1 } ^ { 1 } , . . . , \\mathbf { m } _ { N ^ { 1 } } ^ { 1 } \\}$ are placed in the feature space ${ \\mathcal { F } } ^ { 1 }$ , which can be learned to capture the representation of the base dataset. More reference vectors will be added incrementally while learning novel tasks. The marginal distribution $p ( f _ { \\theta ^ { 1 } } ( \\mathbf { x } ) )$ of feature vector can be described by a Gaussian mixture model $\\begin{array} { r } { p ( f _ { \\theta ^ { 1 } } ( \\mathbf { x } ) ) = \\sum _ { i = 1 } ^ { N ^ { 1 } } p ( i ) p ( f _ { \\theta ^ { 1 } } ( \\mathbf { x } ) | i ) } \\end{array}$ of $N ^ { 1 }$ components, where the prior $p ( i ) = 1 / N ^ { 1 }$ and the component distribution $p ( f _ { \\boldsymbol { \\theta } ^ { 1 } } ( \\mathbf { x } ) | i )$ is Gaussian. By assuming that each component distribution $p ( f _ { \\boldsymbol { \\theta } ^ { 1 } } ( \\mathbf { x } ) | i )$ is isotropic Gaussian centered at $\\mathbf { m } _ { i } ^ { 1 }$ with the same covariance, the posterior distribution of a component given the input is ", + "bbox": [ + 173, + 159, + 825, + 319 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9f3ef359407319e967888b856eee1aad28e196abe3f96b9b130df03282a554c6.jpg", + "text": "$$\np ^ { 1 } ( i | \\mathbf { x } ) = \\frac { \\kappa ( f _ { \\theta ^ { 1 } } ( \\mathbf { x } ) , \\mathbf { m } _ { i } ^ { 1 } ) } { \\sum _ { j = 1 } ^ { N ^ { 1 } } \\kappa ( f _ { \\theta ^ { 1 } } ( \\mathbf { x } ) , \\mathbf { m } _ { j } ^ { 1 } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 387, + 323, + 609, + 363 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\kappa ( f _ { \\theta ^ { 1 } } ( \\mathbf { x } ) , \\mathbf { m } _ { i } ^ { 1 } ) = \\exp ( - \\| f _ { \\theta ^ { 1 } } ( \\mathbf { x } ) - \\mathbf { m } _ { i } ^ { 1 } \\| ^ { 2 } / \\gamma )$ is a Gaussian kernel and $\\gamma$ is a scale factor. The conditional expectation of the output from a Gaussian mixture is $\\begin{array} { r } { \\hat { y } = \\sum _ { i = 1 } ^ { N ^ { 1 } } p ^ { 1 } ( i | \\mathbf { x } ) q _ { i } ^ { 1 } } \\end{array}$ , where $q _ { i } ^ { 1 }$ is the reference target associated with reference vector $\\mathbf { m } _ { i } ^ { 1 }$ . In classification problems, $q _ { i } ^ { 1 }$ is either 0 or 1 indicating whether $\\mathbf { m } _ { i } ^ { 1 }$ and $\\mathbf { x }$ have the same label. Since each reference vector is assigned to a class at initialization, $q _ { i } ^ { 1 }$ is fixed and does not require learning. Meanwhile, $q _ { i } ^ { 1 }$ in regression problems is real-valued and has to be learned. The weights in neural networks $\\theta ^ { \\mathrm { { I } } }$ , reference vectors ${ \\bf { M } } ^ { 1 }$ , reference targets $q _ { i } ^ { 1 }$ (in regression problems only) and the scale factor $\\gamma$ are learned concurrently by minimizing a loss function between the true label $y$ and the predicted label $\\hat { y }$ . ", + "bbox": [ + 173, + 367, + 825, + 487 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The proposed IDLVQ is a nonparametric method as it makes prediction based on similarity to reference vectors, instead of using any regression or classification weights. The capacity of the model grows naturally by adding more reference vectors to learn novel tasks, while the old knowledge is preserved in existing reference vectors. ", + "bbox": [ + 173, + 492, + 825, + 549 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.2 INCREMENTAL DEEP LEARNING VECTOR QUANTIZATION FOR CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 174, + 565, + 754, + 580 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For classification problems, one reference vector is assigned to each class in our study. Thus, $\\hat { y }$ represents the predicted probability that an input belongs to a class. The model can be trained to classify data correctly by minimizing the cross-entropy loss $\\mathcal { L } _ { C E }$ between the predicted probability $\\hat { y }$ and the true label $y$ . Although the cross-entropy loss encourages separability of features in base classes, it does not guarantee compact intra-class variation in the feature space. Specifically, in an incremental learning session, features of novel classes could overlap with those of previously learned classes. As a result, the overall classification accuracy could deteriorate after incremental learning sessions. A desirable feature embedding leaves large margin between classes to mitigate overlap in features across old and new classes. Inspired by center loss (Wen et al., 2016) to enhance discriminative capability in facial recognition, a regularization term on intra-class distance to reference vectors is added to get compact intra-class variation. ", + "bbox": [ + 173, + 590, + 825, + 744 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/136678fbac51893814b09458f21a4efb8214aa053cdbe04a19c1f7dbed2adb21.jpg", + "text": "$$\n\\mathcal { L } _ { i n t r a } = \\sum _ { \\forall ( \\mathbf { x } , y ) , y = i } \\left\\| f _ { \\theta ^ { 1 } } ( \\mathbf { x } ) - \\mathbf { m } _ { i } ^ { 1 } \\right\\| ^ { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 375, + 748, + 622, + 786 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As such, $f _ { \\theta ^ { 1 } } ( \\mathbf { x } )$ is forced to stay close to the reference vector with the same label and naturally moves away from other reference vectors. Consequently, features of new classes are more likely to lie in the margin between old classes to mitigate ambiguity in features across different classes. The total loss in training the base task is given by $\\mathcal { L } = \\mathcal { L } _ { C E } + \\lambda _ { i n t r a } \\mathcal { L } _ { i n t r a }$ , where $\\lambda _ { i n t r a }$ is a hyper-parameter to control the weight for intra-class variation loss. The total loss is differentiable w.r.t. neural network parameters $\\theta ^ { \\mathrm { { i } } }$ , reference vectors $\\mathbf { M } ^ { 1 } = \\{ \\mathbf { m } _ { 1 } ^ { 1 } , . . . , \\mathbf { m } _ { n ^ { 1 } } ^ { 1 } \\}$ and scaling factor $\\gamma$ . All parameters in the model can be trained jointly in an end-to-end fashion. ", + "bbox": [ + 173, + 790, + 825, + 888 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In an incremental session $( t > 1 )$ ), a novel dataset $\\mathcal { D } ^ { t }$ contains $n ^ { t }$ classes and $K ^ { t }$ samples per class ( $\\cdot \\boldsymbol { n } ^ { t }$ -way $K ^ { t }$ -shot). $n ^ { t }$ new reference vectors are added and each reference vector is initialized as the centroid of features in a class mti = 1Kt PKtk=1 fθt (xk). The new reference vectors along with the neural network parameters are fine-tuned on $\\mathcal { D } ^ { t }$ to learn new knowledge in task $t$ . To preserve the knowledge from the old tasks during incremental learning, the model should be updated only when necessary. Therefore, cross-entropy loss is not used in incremental learning sessions because it always updates model parameters even if the sample is correctly classified. Let $\\mathbf { m } _ { + } ^ { t }$ be the reference vector with the correct label and $\\mathbf { m } _ { - } ^ { t }$ be the nearest reference vector with a wrong label. For a training sample $\\left( \\mathbf { x } , y \\right)$ in $\\mathcal { D } ^ { t }$ , the sample is classified correctly if $\\left\\| f _ { \\theta ^ { t } } ( \\mathbf { x } ) - \\mathbf { m } _ { + } ^ { t } \\right\\| ^ { 2 } <$ $\\left\\| f _ { \\theta ^ { t } } ( \\mathbf { x } ) - \\mathbf { m } _ { - } ^ { t } \\right\\| ^ { 2 }$ . In this case, the loss should be 0. When $\\left\\| f _ { \\theta ^ { t } } ( \\mathbf { x } ) - \\mathbf { m } _ { + } ^ { t } \\right\\| ^ { 2 } > \\left\\| f _ { \\theta ^ { t } } ( \\mathbf { x } ) - \\mathbf { m } _ { - } ^ { t } \\right\\| ^ { 2 }$ , the sample is misclassified. We adapt the margin based loss function ${ \\mathcal { L } } _ { M }$ from De Vries et al. (2016) with a minor modification ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 101, + 826, + 257 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/7c9894a4932012ea0e1399acc01026fe480d5c85c5670a86fd3d9c533a34d28d.jpg", + "text": "$$\n\\mathcal { L } _ { M } = \\mathrm { R e L U } \\left( \\frac { \\left. f _ { \\theta ^ { t } } ( \\mathbf { x } ) - \\mathbf { m } _ { + } ^ { t } \\right. ^ { 2 } - \\left. f _ { \\theta ^ { t } } ( \\mathbf { x } ) - \\mathbf { m } _ { - } ^ { t } \\right. ^ { 2 } } { \\left. f _ { \\theta ^ { t } } ( \\mathbf { x } ) - \\mathbf { m } _ { + } ^ { t } \\right. ^ { 2 } + \\left. f _ { \\theta ^ { t } } ( \\mathbf { x } ) - \\mathbf { m } _ { - } ^ { t } \\right. ^ { 2 } } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 307, + 262, + 689, + 306 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathrm { R e L U } ( \\cdot )$ stands for the rectified linear unit function. The margin based loss leads to slow training convergence because it only updates two reference vectors one time. However, the adapted margin based loss is well suited in learning from few-shot samples while avoids unnecessary parameter updates. ", + "bbox": [ + 176, + 313, + 825, + 368 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Features for an old class could deviate away from the corresponding reference vector due to changes in $\\theta ^ { t }$ during incremental learning, leading to catastrophic forgetting. A forgetting loss $\\mathcal { L } _ { F }$ is developed to regularize the drift in the feature space ", + "bbox": [ + 174, + 376, + 823, + 417 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/14d85a4e5a9ffff4439035a551a0eb6485fd94bcd662359ee2ec50f96e77e5e7.jpg", + "text": "$$\n\\mathcal { L } _ { F } = \\sum _ { i = 1 } ^ { N ^ { t - 1 } } \\| f _ { \\theta ^ { t } } ( \\mathbf { x } _ { i } ^ { ' } ) - f _ { \\theta ^ { t - 1 } } ( \\mathbf { x } _ { i } ^ { ' } ) \\| ^ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 377, + 425, + 619, + 472 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\mathbf { x } _ { i } ^ { ' }$ is the selected exemplar for class $i$ and $N ^ { t - 1 }$ denotes the total number of classes in the base task and all previous novel tasks. Note that the exemplar $\\mathbf { x } _ { i } ^ { ' }$ for class $i \\in [ N ^ { t - 1 } , N ^ { t } ]$ is picked from $\\mathcal { D } ^ { t }$ whose feature is nearest to $\\mathbf { m } _ { i } ^ { t }$ at the end of each learning session. The total loss in the incremental learning session $t$ is $\\mathcal { L } = \\mathcal { L } _ { M } + \\lambda _ { F } \\mathcal { L } _ { F } + \\lambda _ { i n t r a } \\mathcal { L } _ { i n t r a }$ , where $\\lambda _ { F }$ and $\\lambda _ { i n t r a }$ are weights for forgetting loss and intra-class variation loss, respectively. The total loss is optimized w.r.t. neural network parameters $\\theta ^ { t }$ and new reference vectors $\\{ \\mathbf { m } _ { N ^ { t - 1 } + 1 } ^ { t } , . . . , \\mathbf { m } _ { N ^ { t } } ^ { t } \\}$ . ", + "bbox": [ + 174, + 478, + 825, + 569 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The reference vectors for previously learned tasks are not updated by novel data to prevent catastrophic forgetting. However, they may not be well suited to represent knowledge and make classification in the new feature space $\\mathcal { F } ^ { t }$ as feature embedding is changed with updated $\\theta ^ { t }$ . Although the true optimal location of those reference vectors are difficult to estimate without using the entire data from all tasks, they can be calculated approximately using the shift in features of exemplars. Considering that features of an exemplar $\\mathbf { x } _ { i } ^ { ' }$ are close to $\\mathbf { m } _ { i }$ in the feature space, the shift of a reference vector $\\delta _ { i } ^ { t }$ in the new feature space can be approximated by the shift of the exemplar’s features $\\delta _ { i } ^ { t } = f _ { \\theta ^ { t } } ( \\mathbf { x } _ { i } ^ { ' } ) { - } f _ { \\theta ^ { t - 1 } } ( \\mathbf { x } _ { i } ^ { ' } )$ . Therefore, the reference vectors for previously learned tasks are calibrated $\\mathbf { m } _ { i } ^ { t } = \\mathbf { m } _ { i } ^ { t - 1 } + \\delta _ { i } ^ { t }$ , where $\\mathbf { m } _ { i } ^ { t - 1 }$ is the uncalibrated reference vector for class $i \\in [ 1 , N ^ { t - 1 } ]$ . A test sample, which could be from any seen classes, is classified according to the distance to reference vectors $\\{ \\mathbf { m } _ { 1 } ^ { t } , . . . , \\mathbf { m } _ { N ^ { t } } ^ { t } \\}$ . The pseudo code for IDLVQ-C is presented in the appendix. ", + "bbox": [ + 173, + 573, + 825, + 734 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.3 INCREMENTAL DEEP LEARNING VECTOR QUANTIZATION FOR REGRESSION ", + "text_level": 1, + "bbox": [ + 176, + 751, + 728, + 765 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For regression problems, the model is trained to recognize regression targets by the minimizing mean squared error (MSE) loss $\\mathcal { L } _ { M S E } = ( y - \\hat { y } ) ^ { 2 }$ , where $y$ is the real-valued target in training dataset. The MSE loss function is differentiable w.r.t. neural network weights, reference vectors and targets, and scale factor. Therefore, all parameters can be trained jointly in an end-to-end manner. The proposed IDLVQ-R can also be interpreted as a kernel smoother in deep embedded space. Compared with traditional kernel smoother, such as Nadaraya-Watson estimator (Nadaraya, 1964), IDLVQ-R is sparse and hence more scalable as it only relies on a few reference vectors and targets. ", + "bbox": [ + 173, + 776, + 825, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In an incremental learning session $( t > 1 )$ ), we have access to data $\\mathcal { D } ^ { t }$ that contains $K ^ { t }$ pairs of training samples $( \\mathbf { x } _ { i } ^ { t } , y _ { i } ^ { t } )$ . ${ \\bar { n } } ^ { t }$ new reference vectors $( n ^ { t } \\leq K ^ { t } )$ along with corresponding targets are added to the model to learn new knowledge in the novel task $t$ . We randomly select $n ^ { t }$ samples from $\\mathcal { D } ^ { t }$ to initialize reference vectors and targets as follows ", + "bbox": [ + 176, + 881, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 531, + 118 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/75e96725b2430e784118277fc55e79aa641727a9a25ca47ba5fed3ed2909d6fc.jpg", + "text": "$$\n\\begin{array} { c } { { { \\bf m } _ { i + N ^ { t - 1 } } = f _ { \\theta } ( { \\bf x } _ { i } ^ { t } ) , } } \\\\ { { { \\bf \\ q } _ { i + N ^ { t - 1 } } = y _ { i } ^ { t } , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 429, + 123, + 566, + 169 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $N ^ { t - 1 }$ is the total number of reference vectors in all previous tasks. The new reference vectors and targets are fine-tuned by minimizing MSE on $\\mathcal { D } ^ { t }$ while keeping other parameters frozen. After new reference vectors and targets are fine-tuned with novel data $\\mathcal { D } ^ { t }$ , the model makes prediction by smoothing targets of all reference vectors ", + "bbox": [ + 174, + 171, + 823, + 227 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/c9b1e01d4cd5216b1589677f4286fe5cc07c1fb553d92ea9d9c8d2e6f1264b95.jpg", + "text": "$$\n\\hat { y } = \\frac { \\sum _ { i = 1 } ^ { N ^ { t } } \\kappa ( f _ { \\theta } ( \\mathbf { x } ) , \\mathbf { m } _ { i } ) q _ { i } } { \\sum _ { i = 1 } ^ { N ^ { t } } \\kappa ( f _ { \\theta } ( \\mathbf { x } ) , \\mathbf { m } _ { i } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 405, + 233, + 591, + 276 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $N ^ { t }$ is the current total number of reference vectors. ", + "bbox": [ + 173, + 284, + 553, + 297 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 319, + 328, + 335 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We first describe the overall protocols, then we present the results on incremental few-shot classification and regression problems. ", + "bbox": [ + 174, + 351, + 823, + 378 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 INCREMENTAL FEW-SHOT CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 174, + 397, + 513, + 410 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We empirically evaluate the performance of IDLVQ-C on incremental few-shot classification on CUB200-2011 (Welinder et al., 2010) and miniImageNet datasets (Vinyals et al., 2016). The dataset is split into base classes and multiple groups of novel classes. We apply standard data augmentation, including random crop, horizontal flip and color jitter, on all training images. After each training session, the model performance is evaluated on a test set, which contains all classes that the model has been trained on. ", + "bbox": [ + 174, + 421, + 825, + 506 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "CUB dataset is composed of 200 fine-grained bird species with 11,788 images. We split the dataset into 5894 training images, 2947 validation images and 2947 test images. All images are resized to $2 2 4 \\times 2 2 4$ . In addition, the first 100 classes are chosen as base classes, where all training samples in base classes are used to train the base model. The remaining 100 classes are treated as novel categories and split into 10 incremental learning sessions. Each incremental learning session contains 10 novel classes and 5 randomly selected training samples per class (10-way 5-shot). ", + "bbox": [ + 173, + 512, + 825, + 597 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "miniImageNet dataset is a 100-class subset of the original ImageNet dataset (Deng et al., 2009). Each class contains 500 training images, 50 validation images, and 50 test images. The images are in RGB format of the size $8 4 \\times 8 4$ . We choose 60 and 40 classes for base and novel classes, respectfully. The 40 novel classes are divided into 8 sessions and each session contains 5 novel classes with 5 randomly selected training samples per class (5-way 5-shot). ", + "bbox": [ + 173, + 603, + 825, + 674 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "ResNet18 (He et al., 2016) is used as the feature extractor for incremental classification problems. The learning process for each dataset is repeated 10 times and the average test accuracy is reported. The proposed method is compared with six methods for few-shot class-incremental learning: fineturning using $\\mathcal { D } ^ { t }$ , joint training using the entire training set from all encountered classes, iCaRL (Rebuffi et al., 2017), Rebalancing (Hou et al., 2019), ProtoNet (Snell et al., 2017), incremental learning vector quantization (ILVQ) (Xu et al., 2012), SDC (Yu et al., 2020), and Imprint (Qi et al., 2018). Note that ILVQ is applied to the features extracted by neural networks in our experiment. ", + "bbox": [ + 173, + 680, + 825, + 777 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The incremental few-shot learning results on CUB and miniImageNet are shown in Table 1 and 2, respectively. Our method outperforms fine-tuning, iCaRL (Rebuffi et al., 2017), and ProtoNet (Snell et al., 2017) by a large margin. Simply fine-tuning the weights in classifier with few-shot training samples for novel classes significantly deteriorates the prediction accuracy. Although iCaRL alleviates catastrophic forgetting by tuning the model with a mix of old exemplars and novel few-shot data, the prediction accuracy still drops quickly because iCaRL requires sufficient samples per class to achieve satisfactory performance. The ProtoNet relies on distance to prototypes (the mean of features within a class) to make classification but the fixed feature extractor may not be able to well separate novel classes. ILVQ is slightly better than ProtoNet because prototypes can be learned adaptively when more classes are available in incremental learning sessions. Some prototypes in ", + "bbox": [ + 174, + 784, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "ILVQ are close to the border of a class, which are more effective than class centroids in ProtoNet. However, ILVQ does not achieve the best performance because the feature extractor is fixed and cannot be learned along with the prototypes. IDLVQ-C has a small gain in the first couple of incremental few-shot learning sessions compared with SDC (Yu et al., 2020) and Imprint (Qi et al., 2018). Similar to ProtoNet, SDC also relies on prototypes to make classification. The performance of SDC is better than that of ProtoNet because SDC fine-tunes the feature extractor with novel dataset and compensates the drift in prototypes. However, the compensation for the drift of old-class prototypes can be less accurate in SDC because it is approximated by samples in novel classes. In parallel, the imprint method directly computes the normalized classification weights from the average of normalized features within a novel class. The imprint method avoids imbalanced classification weights and circumvents the overfitting in few-shot class-incremental learning through weight normalization. Nevertheless, the fixed feature extractor in the imprint method may not be well suited for novel classes. In contrast, IDLVQ-C updates the feature extractor only when necessary and compensates the shift of old reference vectors more accurately using exemplars from old classes. That is why the gain of IDLVQ-C increases with more incremental few-shot learning sessions. The performance of SDC, Imprint and IDLVQ-C is better than offline joint training in early sessions of incremental few-shot learning. Offline joint training may not result in oracle performance due to extremely imbalanced samples between base classes and novel classes. ", + "bbox": [ + 174, + 104, + 825, + 353 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/b55a30f0cbd49cd71d38ca0fdb858eb515d4bcc9f52ebde1c26c02eb74b053a7.jpg", + "table_caption": [ + "Table 1: Prediction accuracy on CUB all classes using the 10-way 5-shot incremental setting. " + ], + "table_footnote": [], + "table_body": "
Methodsessions
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Fine-tune77.3046.2334.7125.3523.1620.6516.2113.3211.9811.1710.76
Joint train77.3073.2868.8065.3463.7562.0060.8159.7159.0658.6958.23
iCaRL (Rebuffi et al.,2017)77.3057.1854.6748.1140.7636.8533.1230.4228.2226.8425.23
Rebalancing (Hou et al., 2019)77.3064.5356.1447.2938.9234.3931.0427.9327.1224.4623.61
ProtoNet (Snell et al., 2017)77.3069.7666.0162.2959.5857.1055.1354.0952.4051.6550.36
ILVQ (Xu et al., 2012)77.3071.5066.7962.7160.2057.8455.2755.0652.4251.7250.47
SDC (Yu et al., 2020)77.3474.4569.4565.2761.8158.2656.1455.7153.3152.7951.52
Imprint (Qi et al., 2018)77.0273.3969.5065.6162.8160.7459.3958.6156.8555.9354.82
IDLVQ-C77.3774.7270.2867.1365.3463.5262.1061.5459.0458.6857.81
", + "bbox": [ + 196, + 392, + 799, + 522 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f3eca11e64a705f943f655bae6af2313d06ff94cfff279020fd0500ace9283f0.jpg", + "table_caption": [ + "Table 2: Prediction accuracy on miniImageNet all classes using the 5-way 5-shot incremental setting. " + ], + "table_footnote": [], + "table_body": "
Methodsessions
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Fine-tune64.2530.1118.536.312.862.681.871.561.42
Joint train64.2558.8055.2652.3849.7148.3745.9144.6843.38
iCaRL (Rebuffi et al., 2017)64.2548.0443.1338.2830.0124.4621.8519.8417.76
Rebalancing (Hou et al.,2019)64.2549.2144.1737.7130.1122.9219.9917.9616.25
ProtoNet (Snell et al., 2017)64.2555.1251.6748.9146.5244.2541.9140.0738.42
ILVQ (Xu et al., 2012)64.2556.0152.4349.3146.9844.3742.0640.1138.43
SDC (Yu et al., 2020)64.6259.6355.3950.9248.3045.2842.9742.5141.24
Imprint (Qi et al., 2018)64.7159.8555.7152.4749.9047.3144.5742.5741.26
IDLVQ-C64.7759.8755.9352.6249.8847.5544.8343.1441.84
", + "bbox": [ + 238, + 566, + 758, + 695 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Ablation studies are conducted to analyze how individual components affect the performance of incremental few-shot learning. We study five variants of our methods: (a) new reference vectors are initialized as class centroids and no tuning is done for feature extractor or old reference vectors; (b) $\\mathcal { L } _ { i n t r a }$ is not used in incremental learning sessions; (c) $\\mathcal { L } _ { F }$ is not used in the incremental learning sessions; (d) shift in old reference vectors are not compensated; (e) replace the margin based loss ${ \\mathcal { L } } _ { M }$ with the cross entropy loss $\\mathcal { L } _ { C E }$ . Table 3 shows the results of our ablation studies on CUB dataset. Without any fine-tuning, the initial reference vectors for novel classes lead to descent accuracy in incremental few-shot classification. It demonstrates the robustness of nonparametric classifier. $\\mathcal { L } _ { i n t r a }$ leads to $0 . 5 7 \\%$ gain due to tight intra-class variation. The less forgetting regularization $\\mathcal { L } _ { F }$ is proved to prevent forgetting old classes and achieves a performance boost by $2 . 3 5 \\%$ . The shift compensation for reference vectors effectively adapts old reference vector in new embedding spaces with a gain round of $1 \\%$ . The margin based loss is more effective in preventing forgetting in early sessions of incremental learning. ", + "bbox": [ + 174, + 708, + 825, + 888 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The effect of the number of training samples per class. The proposed method is evaluated under different few-shot settings on CUB dataset to investigate the effect of different numbers of training samples in novel classes, including 5-shot, 10-shot and 20-shot settings. One reference vector is assigned to each class in all few-shot settings. As shown in Fig. 2 in the appendix, the performance of incremental learning improves as the number of samples per class increases. When the training samples are scarce, the training samples may not well present the generative distribution of training data. Therefore, the learned reference vectors could be biased and classification accuracy is low. With more training samples, the learned reference vectors could well present the center of the distribution and classification accuracy is improved. The gap in performance becomes more obvious as the number of incremental learning sessions grows. The detailed results are reported in Table 7 and 8. ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/7299019371a92f6be79e9273eb52225e4f5133ff3e3f75c9760fc01563d26c34.jpg", + "table_caption": [ + "Table 3: Ablation study on CUB using the 10-way 5-shot incremental setting. " + ], + "table_footnote": [], + "table_body": "
Methodsessions
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No tuning71.9367.1464.2162.6160.1359.0458.4755.6454.2553.66
W.o.Lintra74.7570.2666.8965.0563.1861.8461.3658.6158.1457.24
w.0. LF73.8569.5466.2164.0262.7460.2859.4956.9756.3855.46
w.o. δ74.6770.0166.7464.8163.9061.4260.7358.1657.6256.79
LM→LCE73.2269.4166.0363.9363.0761.1460.9858.6758.1157.32
IDLVQ-C74.7270.2867.1365.3463.5262.1061.5459.0458.6857.81
", + "bbox": [ + 245, + 127, + 750, + 222 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 244, + 825, + 369 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 INCREMENTAL FEW-SHOT REGRESSION ", + "text_level": 1, + "bbox": [ + 178, + 387, + 485, + 401 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "IDLVQ-R is tested on two regression datasets: sinusoidal wave and 3D spatial data. Considering that there is no state-of-the-art method for incremental few-shot regression, we compare IDLVQ-R against three alternative methods: fine-tuning using novel task data only, fine-tuning using novel task data along with exemplars and offline training using the entire training dataset from all tasks. ", + "bbox": [ + 174, + 412, + 825, + 468 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Sinusoidal wave is defined by a function $y = \\sin ( 3 \\pi x ) + 0 . 3 \\cos ( 9 \\pi x ) + 0 . 5 \\sin ( 7 \\pi x ) + \\epsilon ,$ , where $\\epsilon$ is white noise with a standard deviation of 0.1. 1000 training samples in the first task (bas task) are generated by sampling $x \\in [ - 1 . 0 , 1 . 0 ]$ uniformly. 5-shot training samples in two novel tasks are generated by sampling $x \\in [ 1 . 0 , 1 . 5 ]$ and $x \\in [ 1 . 5 , 2 . 0 ]$ , respectively. ", + "bbox": [ + 174, + 476, + 825, + 532 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As shown in Fig. 1(a), IDLVQ-R achieves comparable performance to offline neural networks in Fig. 1(d) which are trained using the entire training set from all tasks. In comparison, neural networks trained sequentially with few-shot training samples show catastrophic forgetting on old tasks in Fig. 1(b). With the addition of exemplars during training, the networks perform better but still suffer from catastrophic forgetting on the base task in Fig. 1(c). In conclusion, IDLVQ-R preserves old knowledge and adapts to new knowledge quickly using a few reference vectors and achieves satisfactory performance on incremental few-shot regression tasks. The experiment details can be found in the appendix. ", + "bbox": [ + 173, + 537, + 825, + 650 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/83a97f4c3e066abb502771d10b9d6938bf3934093e0891e51d8029367598a481.jpg", + "image_caption": [ + "Figure 1: Comparison of performance for incremental few-shot regression. Red dots denote test samples for base task, purple dots denotes test sample for novel tasks, grey lines denote model predictions, and black crosses denotes few-shot training samples in novel tasks. (a) IDLVQ-R; (b) neural networks incrementally fine-tuned with novel data only in each session; (c) neural networks incrementally fine-tuned with exemplars and novel training samples; (d) offline neural networks trained with training samples from all tasks. " + ], + "image_footnote": [], + "bbox": [ + 205, + 669, + 785, + 760 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3D spatial data1 is collected in North Jutland, Denmark. The inputs are longitude $x _ { 1 }$ and latitude $x _ { 2 }$ , and the output is altitude y. 2482 training samples in the 1st task are collected in the area where $x _ { 1 } \\in [ 9 . 9 8 , 9 . 9 9 5 ]$ and $x _ { 2 } \\in [ 5 7 . 0 , 5 7 . 0 5 ] .$ 20 training samples in the 2nd task are collected in the area where $x _ { 1 } \\ \\in \\ [ 9 . 9 9 5 , 1 0 . 0 ]$ and $x _ { 2 } ~ \\in ~ [ 5 7 . 0 2$ , 57.03]. 20 training samples in the 3rd task are collected in the area where $x _ { 1 } \\in [ 9 . 9 9 5 , 1 0 . 0 ]$ and $x _ { 2 } ~ \\overset { \\cdot } { \\in } ~ [ 5 7 . 0 3 , 5 7 . 0 4 ]$ . Approximately 300 test samples are collected for each task. ", + "bbox": [ + 174, + 872, + 825, + 901 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 159 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/a1af85a5e635057cd677d7be3fcf2d4ede7f10920cb3ddbe340a3fcc32589448.jpg", + "table_caption": [ + "Table 4: Normalized RMSE of incremental few-shot regression on 3D spatial data " + ], + "table_footnote": [], + "table_body": "
Methodsessions
123
Joint train offline0.02174(2e-4)0.02232(2e-4)0.02296(2e-4)
Fine-tune w.novel data0.02174(2e-4)0.08462(4e-4)0.11870(6e-4)
Fine-tune w. exemplars0.02174(2e-4)0.02988(2e-4)0.03128(2e-4)
IDLVQ-R0.02181(2e-4)0.02641(2e-4)0.02817(2e-4)
", + "bbox": [ + 272, + 196, + 722, + 299 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The normalized root mean squared errors (RMSE) between actual and predicted altitude in the test set are listed in Table 4. The prediction accuracy drops significantly when the model is finetuned with novel data only. Catastrophic forgetting can be alleviated using exemplars from previous tasks. IDLVQ-R achieves better results than fine tuning with exemplars. The good performance of IDLVQ-R can be attributed to two reasons. First, IDLVQ-R learns a number of reference vectors and targets to preserve the knowledge in encountered tasks. Compared with a linear layer on top of neural networks, a number of reference vectors represent richer information about the training data. Second, IDLVQ-R is nonparametric and can represent local and nonlinear relationship without learning any regression coefficient from few-shot data. ", + "bbox": [ + 173, + 310, + 825, + 435 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 455, + 328, + 472 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "A new incremental few-shot learning approach is developed to harmonize old knowledge preserving and new knowledge adaptation through quantized vector in deep embedded space. Prediction is made in a nonparametric way using similarity to learned reference vectors, which circumvents biased weights in a parametric classification layer during incremental few-shot learning. For classification problems, additional mechanisms are developed to mitigate the forgetting in old classes and improve representation learning for few-shot novel classes. For regression problems, the proposed approach has been reinterpreted as a kernel smoother to predict real-valued target over novel domain. 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In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13208–13217, 2020. ", + "bbox": [ + 176, + 416, + 821, + 459 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 488, + 299, + 505 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1 PSEUDO CODE FOR IDLVQ-C ", + "bbox": [ + 176, + 520, + 426, + 536 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 1 IDLVQ-C ", + "text_level": 1, + "bbox": [ + 174, + 556, + 331, + 571 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/19ee517a1c5d40b03242f97800042cf891fc63b3a8c14d47459c3ffe5b9db0e7.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
In the base task(t = 1)
Initialize 01,{m𝑖,.,mN1} and γ
Minimize L = LcE + XintraLintra W.r.t. θ1,{ml,.,mN1} and γ
Pick exemplars from D1 for classes in the base task: xi = arg minxeD1 |lft-1(x) - m1|l²
for novel task t = 2,3,... do
Initialize {mt-1+1.*., mNt}
Minimize L=LM +XFLF + XintraLintra W.r.t. 0t and {mNt-1+1,.,mNt}
t-1 t-1 +
Pick exemplars from Dt for classes in the novel task t: x' = arg minx∈Dt |lfet-1(x) - m ll²
end for
", + "bbox": [ + 184, + 574, + 818, + 732 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.2 PSEUDO CODE FOR IDLVQ-R ", + "text_level": 1, + "bbox": [ + 176, + 761, + 426, + 776 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Algorithm 2 IDLVQ-R ", + "text_level": 1, + "bbox": [ + 174, + 797, + 331, + 813 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/c1fbd41cb34a9209802fd8942234152a7e50878c86bd9a88a45a41c2044df0df.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
In the base task (t =1) Initializeθ,{m1,..,mN1},{q],..,qN1} and γ
Minimize Lmse w.r.t. θ,{ml,..,mN1},{ql,.,qN1} and γ
for novel task t = 2,3,.. do
Initialize {mNt-1+1,, mNt} and {qNt-1+1.…,qNt}
Minimize LmsE w.r.t. {mNt-1+1..*,mNt} and {qNt-1+1,.,qNt}
end for
", + "bbox": [ + 179, + 815, + 663, + 920 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.3 EXPERIMENT DETAILS FOR INCREMENTAL FEW-SHOT CLASSIFICATIO ", + "text_level": 1, + "bbox": [ + 184, + 104, + 692, + 117 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The base model is trained by the SGD optimizer (momentum of 0.9 and weight decay of 1e-4) with a mini-batch size of 64. For CUB dataset, the initial learning rate is 0.01 and is decayed by 0.1 after 60 and 120 epochs (200 epochs in total). For miniImageNet, the learning rate also starts from 0.01 and is decayed by 0.1 every 200 epochs (600 epochs in total). In an incremental learning session $( t > 1 )$ ), the model is fine-tuned with with $\\mathcal { D } ^ { t }$ with a learning rate of 0.01 for 100 epochs. Since novel data $\\mathcal { D } ^ { t }$ $t > 1$ ) contains very few training samples, all training samples in $\\mathcal { D } ^ { t }$ are included in one mini-batch. In addition, we use $\\lambda _ { i n t r a } = 1 . 0$ and $\\lambda _ { F } = 0 . 5$ for both datasets. Empirically, larger $\\lambda _ { i n t r a }$ leads to more compact intra-class variation. However, convergence could be slow if $\\lambda _ { i n t r a }$ is too large. In addition, larger $\\lambda _ { F }$ results in less forgetting in old classes but makes learning novel classes more difficult. ", + "bbox": [ + 173, + 130, + 825, + 268 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.4 ADDITIONAL RESULTS FOR INCREMENTAL FEW-SHOT CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 176, + 289, + 705, + 301 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The accuracies for base and novel classes are reported separately in Table 5 and 6 for CUB amd miniImageNet, respectively. The prediction accuracy of novel classes is calculated upon all novel classes the model has been trained on. Note that the accuracy in Table 1 and 2 is calculated upon all classes (including base and novel classes) that the model has been trained on. The proposed IDLVQC demonstrates strong capability of preserving old knowledge by achieving the best performance on old classes across all learning sessions. In parallel, Imprint method performs slightly better on novel classes than IDLVQ-C in early incremental learning sessions, while IDLVQ-C outperforms Imprint method in longer incremental learning sessions. The advantage of IDLVQ-C can be attributed to the adaptive feature extractor, which is tuned in each learning session. ", + "bbox": [ + 174, + 314, + 825, + 439 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/05ca6f836c7776d404f47b953b3e02cbae9c589b42216e869c9b9f11999d652e.jpg", + "table_caption": [ + "Table 5: Prediction accuracy on CUB base and novel classes using the 10-way 5-shot incremental setting. " + ], + "table_footnote": [], + "table_body": "
Base classessessions
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Fine-tune77.3044.2336.2827.5225.9623.0517.6813.0711.7810.9910.71
Joint train77.3075.8375.2574.5174.5873.7473.9573.2573.1173.2573.18
iCaRL (Rebuffi et al., 2017)77.3059.3858.8154.4348.2743.2839.1734.9132.4329.3625.87
Rebalancing (Hou et al.,2019)77.3066.4360.3255.3646.3941.7637.1232.5831.2627.0324.25
ProtoNet (Snell et al., 2017)77.3072.5572.2172.0671.6471.2971.0270.9470.6770.6070.53
ILVQ (Xu et al., 2012)77.3074.1873.5772.6672.5671.5771.1471.1271.0270.9870.85
SDC (Yu et al., 2020)77.3476.0575.2174.1272.3671.8171.6871.4371.2571.2770.96
Imprint (Qi et al., 2018)77.0274.7674.5773.6972.6970.8870.3470.1270.0769.8469.27
IDLVQ-C77.3776.3275.9075.9175.4974.8674.5874.3774.0273.3973.32
Novel classessessions
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Fine-tune=66.2326.8618.1216.1615.8513.7613.6812.2311.3710.81
Joint train=47.7836.5534.7736.6838.5238.9140.3741.5042.5143.28
iCaRL (Rebuffi et al., 2017)-35.1833.9727.0421.9923.9923.0424.0122.9624.0424.59
Rebalancing (Hou et al.,2019)=45.5335.2420.3920.2519.6520.9121.2921.9521.6022.97
ProtoNet (Snell et al., 2017)41.8635.0129.7229.4328.7228.6530.0229.5630.5930.19
ILVQ (Xu et al., 2012)40.7432.8929.5429.3030.3828.8232.1229.1730.3230.09
SDC (Yu et al., 2020)58.4540.6535.7733.2331.1630.2433.2530.8932.2632.08
Imprint (Qi et al.,2018)59.6944.1538.6838.1140.4641.1442.1740.3340.4740.37
IDLVQ-C=58.7242.1837.8639.9740.8441.3043.2140.3242.3442.30
", + "bbox": [ + 196, + 494, + 799, + 751 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The test accuracy on CUB dataset using 10-way 10-shot and 10-way 20-shot incremental settings are reported in Table 7 and 8, respectively. The prediction accuracy improves in all methods with more training samples per class. The iCaRL and Rebalancing methods show the most significant improvement when the number of training samples increases. The proposed IDLVQ-C is effective in different incremental few-shot scenarios as it achieves the best performance on 5-shot, 10-shot and 20-shot settings. ", + "bbox": [ + 174, + 767, + 825, + 851 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "A.5 EXPERIMENT DETAILS FOR INCREMENTAL FEW-SHOT REGRESSION ", + "text_level": 1, + "bbox": [ + 176, + 869, + 679, + 882 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Sinusoidal wave: A six-layer feedforward neural network with ReLU nonlinear activation is used as the feature extractor. IDLVQ-R learns 10 reference vectors and targets from the base task. In each incremental learning sessions, 5 pairs of reference vectors and targets are added. After new reference vectors and targets are fine-tuned, the model is capable of make prediction for all seen tasks. 10 exemplars are selected uniformly from the training set in the base task. In the incremental learning session of the 2nd task, the model is fine-tuned on 10 exemplars and 5 novel training samples. After the training converges, 5 novel training samples in the current task are added to the exemplar set. In the incremental learning session of the 3rd task, the model is fine-tuned with 15 exemplars from old tasks and 5 novel training samples. ", + "bbox": [ + 173, + 895, + 821, + 922 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/de44843eba5451a104b1775b37ae299324f4046c003e296000ccb891b5f4cf3b.jpg", + "table_caption": [ + "Table 6: Prediction accuracy on miniImageNet base and novel classes using the 5-way 5-shot incremental setting. " + ], + "table_footnote": [], + "table_body": "
Base classessessions
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Fine-tune64.2532.2820.876.953.173.161.921.531.46
Joint train64.2563.3062.8362.1662.1862.6861.8661.8761.89
iCaRL (Rebuffi et al., 2017)64.2551.6648.9745.6237.3930.8628.6826.8324.47
Rebalancing (Hou et al., 2019)64.2552.8750.1644.7837.4828.7525.5822.9721.57
ProtoNet (Snell et al., 2017)64.2559.2758.8858.6958.2257.6357.0356.8056.47
ILVQ (Xu et al.,2012)64.2560.2459.6259.0258.6157.7157.1656.8356.49
SDC (Yu et al., 2020)64.6263.5862.7861.1260.2959.3759.0559.9759.87
Imprint (Qi et al.,2018)64.7163.5262.9662.1361.1761.2760.6359.8659.64
IDLVQ-C64.77 63.7763.2262.4461.2261.4760.9760.6660.44
Novel classessessions
123456789
Fine-tune4.074.493.751.931.531.771.611.36
Joint train-4.809.8413.2612.3014.0314.0115.2115.62
iCaRL (Rebuffi et al., 2017)=4.608.098.927.879.108.197.867.70
Rebalancing (Hou et al.,2019)5.298.239.438.008.938.818.838.27
ProtoNet (Snell et al., 2017)5.328.419.7911.4212.1411.6711.3911.35
ILVQ (Xu et al.,2012)5.259.2910.4712.0912.3511.8611.4511.34
SDC (Yu et al., 2020)12.2311.0510.1212.3311.4610.8112.5813.30
Imprint (Qi et al., 2018)15.8112.2113.8316.0913.8112.4512.9313.69
IDLVQ-C-13.0712.1913.3415.8614.1412.5513.1113.94
", + "bbox": [ + 236, + 141, + 758, + 398 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/64a01462b4a68f489cdaa3ed37296b10b079cd8662f7acfeb4ce1a30d0db9c31.jpg", + "table_caption": [ + "Table 7: Prediction accuracy on CUB using the 10-way 10-shot incremental setting. " + ], + "table_footnote": [], + "table_body": "
Methodsessions
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Fine-tune77.3047.1636.3426.9224.0821.2417.1914.3112.7311.7511.43
Joint train77.3073.2971.5468.7266.3865.4264.9865.7464.8264.4764.16
iCaRL (Rebuffi et al., 2017)77.3059.6656.2452.2648.7746.3744.5443.1742.3541.1940.92
Rebalancing (Hou et al., 2019)77.3064.5358.3553.8249.2747.1245.1643.0542.3741.0240.86
ProtoNet (Snell et al., 2017)77.3069.8266.1263.1961.1758.8558.0457.7555.8455.8255.60
ILVQ (Xu et al., 2012)77.3071.2666.8463.8262.6659.7158.9258.1156.3156.1456.03
SDC (Yu et al., 2020)77.3474.6769.7366.7166.4962.1461.3359.8458.0157.3956.62
Imprint (Qi et al.,2018)77.0274.0770.2666.8464.4562.4661.8561.0259.2058.9358.43
IDLVQ-C77.3774.7970.9668.0865.9464.1263.5862.9860.8560.5459.72
", + "bbox": [ + 197, + 444, + 799, + 573 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/513bdce2a277ac08ca72d9a042640875d50b176ebd00b04dc3b19e624cd2686e.jpg", + "table_caption": [ + "Table 8: Prediction accuracy on CUB using the 10-way 20-shot incremental setting. " + ], + "table_footnote": [], + "table_body": "
Methodsessions
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Fine-tune77.3047.6836.7527.0124.5621.7217.8815.2412.8411.8911.52
Joint train77.3074.3472.5970.8570.1470.0169.6969.6769.4869.4169.42
iCaRL (Rebuff et al.,2017)77.3068.8563.5660.3457.7156.2855.9755.0254.6252.2152.23
Rebalancing (Hou et al.,2019)77.3069.3665.4961.3259.3058.6858.7757.4256.2555.3354.17
ProtoNet (Snell et al., 2017)77.3070.8467.8264.8162.9561.7160.9860.7359.5159.3058.95
ILVQ (Xu et al., 2012)77.3071.5068.7766.1264.3162.8962.3162.0061.5259.7959.23
SDC (Yu et al.,2020)77.3474.6271.6368.6666.7565.2464.2163.6261.9761.5461.11
Imprint (Qi et al., 2018)77.0274.1470.7267.7565.8864.6364.2863.8262.0061.8461.40
IDLVQ-C77.3774.8472.0769.0567.2865.5165.1964.8462.7762.5561.96
", + "bbox": [ + 197, + 617, + 799, + 747 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 776, + 825, + 875 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "3D spatial data: We follow the same training and test protocols as the sinusoidal wave dataset. We choose 40, 15 and 15 reference vectors and targets for 1st, 2nd and 3rd tasks, respectively. Adding more reference vectors does not result in obvious improvement in accuracy in our experiments. ", + "bbox": [ + 176, + 882, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/5e1419f899b33ff87b466bb8184b86bfa4ee63793fee7e949fdeb2a29f2e20af.jpg", + "image_caption": [ + "Figure 2: Comparison results of different few-shot settings, evaluated with ResNet18 on CUB dataset " + ], + "image_footnote": [], + "bbox": [ + 179, + 106, + 810, + 430 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.6 VISUALIZATION OF IDLVQ-C ", + "text_level": 1, + "bbox": [ + 176, + 513, + 426, + 527 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We show the visualization of standard neural networks and IDLVQ-C with/without intra-class variation loss in Fig. 3. MNIST dataset is used as a toy example for visualization. Classes 0-7 are old classes with sufficient training samples, and classes 8 and 9 are novel classes with few-shot training samples. It can be observed in Fig. 3(a) and 3(b) that standard neural networks and IDLVQ-C (without intra-class variation loss) trained by cross-entropy loss do not have compact intra-class variation. Consequently, features of novel classes are more likely to overlap with old classes. In this case, the performance of class-incremental learning degrades very quickly because the classifier cannot distinguish between features from different classes. In comparison, the proposed IDLVQ-C makes intra-class variation compact and leaves large margin between classes in Fig 3(c). As a result, the features of novel classes are less likely to overlap existing classes. The compact intra-class variation and large margin between classes make features of novel classes distinguishable so that learning novel classes is easier. In addition, the margin based loss only updates the model parameters when necessary and avoids catastrophic forgetting of old classes. ", + "bbox": [ + 173, + 539, + 825, + 720 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/49e99100b8d6c5c5a8ae0c3a024f4b687cf490c746e6964ee89e2468e958123b.jpg", + "image_caption": [ + "Figure 3: Visualization of feature spaces in different methods. Dots represent features of samples and crosses denote references vectors of classes. (a) standard neural networks; (b) IDLVQ-C without $\\mathcal { L } _ { i n t r a }$ ; (c) IDLVQ-C. 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This challenge becomes", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 272, + 469, + 285 + ], + "spans": [ + { + "bbox": [ + 141, + 272, + 469, + 285 + ], + "score": 1.0, + "content": "greater when novel tasks contain very few labelled training samples. Currently,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 284, + 470, + 295 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 470, + 295 + ], + "score": 1.0, + "content": "most methods are dedicated to class-incremental learning and rely on sufficient", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 294, + 469, + 306 + ], + "spans": [ + { + "bbox": [ + 141, + 294, + 469, + 306 + ], + "score": 1.0, + "content": "training data to learn additional weights for newly added classes. 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Moreover, simple", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "fine-tuning also leads to significant performance drop on previously learned tasks, termed as catas-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 680 + ], + "score": 1.0, + "content": "trophic forgetting (Goodfellow et al., 2014). Recent attempts to mitigate the catastrophic forgetting", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "are generally categorized into two streams: memory relay of old training samples (Rebuffi et al.,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "2017; Shin et al., 2017; Kemker & Kanan, 2018) and regularization on important model parameters", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "(Kirkpatrick et al., 2017; Zenke et al., 2017). However, those incremental learning approaches are", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "developed and tested on unrealistic scenarios where sufficient training samples are available in novel", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "tasks. They may not work well when the training samples in novel tasks are few (Tao et al., 2020b).", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 621, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "To the best of our knowledge, the majority of incremental learning methodologies focus on classi-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "fication problems and they cannot be extended to regression problems easily. In class-incremental", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 359, + 117 + ], + "score": 1.0, + "content": "learning, the model has to expand output dimensions to learn", + "type": "text" + }, + { + "bbox": [ + 360, + 105, + 372, + 114 + ], + "score": 0.84, + "content": "N ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "novel classes while keeping the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 198, + 127 + ], + "score": 1.0, + "content": "knowledge of existing", + "type": "text" + }, + { + "bbox": [ + 199, + 116, + 209, + 125 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "classes. Parametric models estimate additional classification weights for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "novel classes, while nonparametric methods compute the class centroids for novel classes. In com-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "parison, output dimensions in regression problems do not change in incremental learning as neither", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 412, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 412, + 161 + ], + "score": 1.0, + "content": "additional weights nor class centroids are applicable to regression problems.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "Besides, we find that catastrophic forgetting in incremental few-shot classification can be attributed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 187 + ], + "score": 1.0, + "content": "to three reasons. First, the model is biased towards new classes and forgets old classes because the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "model is fine-tuned on new data only (Hou et al., 2019; Zhao et al., 2020). Meanwhile, the prediction", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "accuracy on novel classes is not good due to over-fitting on few-shot training samples. Second,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "features of novel samples could overlap with those of old classes in the feature space, leading to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "ambiguity among classes in the feature space. Finally, features of old classes and classification", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 417, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 417, + 243 + ], + "score": 1.0, + "content": "weights are no longer compatible after the model is fine-tuned with new data.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 504, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 504, + 260 + ], + "score": 1.0, + "content": "In this paper, we investigate the problem of incremental few-shot learning, where only a few train-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "ing samples are available in new tasks. A unified model is learned sequentially to jointly recognize", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "all classes or regression targets that have been encountered in previous tasks (Rebuffi et al., 2017;", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "Wu et al., 2019). To tackle aforementioned problems, we propose a nonparametric method to han-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "dle incremental few-shot learning based on learning vector quantization (LVQ) (Sato & Yamada,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "1996) in deep embedded space. As such, the adverse effects of imbalanced weights in a paramet-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "ric classifier can be completely avoided (Mensink et al., 2013; Snell et al., 2017; Yu et al., 2020).", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 336 + ], + "score": 1.0, + "content": "Our contributions are three fold. First, a unified framework is developed, termed as incremental", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 335, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 349 + ], + "score": 1.0, + "content": "deep learning vector quantization (IDLVQ), to handle both incremental classification (IDLVQ-C)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 346, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 359 + ], + "score": 1.0, + "content": "and regression (IDLVQ-R) problems. Second, we develop intra-class variance regularization, less", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "forgetting constraints and calibration factors to mitigate catastrophic forgetting in class-incremental", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "learning. Finally, the proposed methods achieve state-of-the-art performance on incremental few-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 380, + 276, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 276, + 392 + ], + "score": 1.0, + "content": "shot classification and regression datasets.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20 + }, + { + "type": "title", + "bbox": [ + 108, + 408, + 211, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 213, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 213, + 423 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "Incremental learning: Some incremental learning approaches rely on memory replay of old exem-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "plars to prevent forgetting previously learned knowledge. Old exemplars can be saved in memory", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "(Rebuffi et al., 2017; Castro et al., 2018; Prabhu et al., 2020) or sampled from generative models", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "(Shin et al., 2017; Kemker & Kanan, 2018; van de Ven et al., 2020). However, explicit storage of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "training samples is not scalable if the number of classes is large. Furthermore, it is difficult to train", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "a reliable generative model for all classes from very few training samples. In parallel, regularization", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "approaches do not require old exemplars and impose regularization on network weights or outputs", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "to minimize the change of parameters that are important to old tasks (Kirkpatrick et al., 2017; Zenke", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "et al., 2017). To avoid quick performance deterioration after learning a sequence of novel tasks", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "in regularization approaches, semantic drift compensation (SDC) is developed by learning an em-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "bedding network via triplet loss (Schroff et al., 2015) and compensates the drift of class centroids", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "using novel data only (Yu et al., 2020). In comparison, IDLVQ-C saves only one exemplar per class", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "and uses saved exemplars to regularize the change in feature extractor and calibrate the change in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 577, + 180, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 180, + 589 + ], + "score": 1.0, + "content": "reference vectors.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "Few-shot learning: Few-shot learning attempts to obtain models for classification or regression", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "tasks with only a few labelled samples. Few-shot models are trained on widely-varying episodes of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "fake few-shot tasks with labelled samples drawn from a large-scale meta-training dataset (Vinyals", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "et al., 2016; Finn et al., 2017; Ravi & Larochelle, 2017; Snell et al., 2017; Sung et al., 2018).", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "Meanwhile, recent works attempt to handle novel few-shot tasks while retraining the knowledge of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "the base task. These methods are referred to as dynamic few-shot learning (Gidaris & Komodakis,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "2018; Ren et al., 2019a; Gidaris & Komodakis, 2019). However, dynamic few-shot learning is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "score": 1.0, + "content": "different from incremental few-shot learning, because they rely on the entire base training dataset", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "and an extra meta-training dataset during meta-training. In addition, dynamic few-shot learning does", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 693, + 365, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 365, + 705 + ], + "score": 1.0, + "content": "not accumulate knowledge for multiple novel tasks sequentially.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "Incremental few-shot learning: Prior works on incremental few-shot learning focus on classifica-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "tion problems by computing the weights for novel classes in parametric classifiers, without iterative", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 763 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 160 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "To the best of our knowledge, the majority of incremental learning methodologies focus on classi-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "fication problems and they cannot be extended to regression problems easily. In class-incremental", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 359, + 117 + ], + "score": 1.0, + "content": "learning, the model has to expand output dimensions to learn", + "type": "text" + }, + { + "bbox": [ + 360, + 105, + 372, + 114 + ], + "score": 0.84, + "content": "N ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 373, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "novel classes while keeping the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 116, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 116, + 198, + 127 + ], + "score": 1.0, + "content": "knowledge of existing", + "type": "text" + }, + { + "bbox": [ + 199, + 116, + 209, + 125 + ], + "score": 0.77, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 116, + 505, + 127 + ], + "score": 1.0, + "content": "classes. Parametric models estimate additional classification weights for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "novel classes, while nonparametric methods compute the class centroids for novel classes. In com-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "parison, output dimensions in regression problems do not change in incremental learning as neither", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 412, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 412, + 161 + ], + "score": 1.0, + "content": "additional weights nor class centroids are applicable to regression problems.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 105, + 81, + 506, + 161 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 242 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "Besides, we find that catastrophic forgetting in incremental few-shot classification can be attributed", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 177, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 505, + 187 + ], + "score": 1.0, + "content": "to three reasons. First, the model is biased towards new classes and forgets old classes because the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "model is fine-tuned on new data only (Hou et al., 2019; Zhao et al., 2020). Meanwhile, the prediction", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 210 + ], + "score": 1.0, + "content": "accuracy on novel classes is not good due to over-fitting on few-shot training samples. Second,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "features of novel samples could overlap with those of old classes in the feature space, leading to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 221, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 221, + 505, + 232 + ], + "score": 1.0, + "content": "ambiguity among classes in the feature space. Finally, features of old classes and classification", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 417, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 417, + 243 + ], + "score": 1.0, + "content": "weights are no longer compatible after the model is fine-tuned with new data.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 165, + 505, + 243 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 248, + 505, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 247, + 504, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 504, + 260 + ], + "score": 1.0, + "content": "In this paper, we investigate the problem of incremental few-shot learning, where only a few train-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 505, + 271 + ], + "score": 1.0, + "content": "ing samples are available in new tasks. A unified model is learned sequentially to jointly recognize", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 506, + 282 + ], + "score": 1.0, + "content": "all classes or regression targets that have been encountered in previous tasks (Rebuffi et al., 2017;", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "Wu et al., 2019). To tackle aforementioned problems, we propose a nonparametric method to han-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "dle incremental few-shot learning based on learning vector quantization (LVQ) (Sato & Yamada,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "1996) in deep embedded space. As such, the adverse effects of imbalanced weights in a paramet-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "ric classifier can be completely avoided (Mensink et al., 2013; Snell et al., 2017; Yu et al., 2020).", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 336 + ], + "score": 1.0, + "content": "Our contributions are three fold. First, a unified framework is developed, termed as incremental", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 335, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 349 + ], + "score": 1.0, + "content": "deep learning vector quantization (IDLVQ), to handle both incremental classification (IDLVQ-C)", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 346, + 505, + 359 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 359 + ], + "score": 1.0, + "content": "and regression (IDLVQ-R) problems. Second, we develop intra-class variance regularization, less", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "forgetting constraints and calibration factors to mitigate catastrophic forgetting in class-incremental", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 506, + 381 + ], + "score": 1.0, + "content": "learning. Finally, the proposed methods achieve state-of-the-art performance on incremental few-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 380, + 276, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 276, + 392 + ], + "score": 1.0, + "content": "shot classification and regression datasets.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20, + "bbox_fs": [ + 104, + 247, + 506, + 392 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 408, + 211, + 421 + ], + "lines": [ + { + "bbox": [ + 105, + 407, + 213, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 213, + 423 + ], + "score": 1.0, + "content": "2 RELATED WORK", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "Incremental learning: Some incremental learning approaches rely on memory replay of old exem-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "plars to prevent forgetting previously learned knowledge. Old exemplars can be saved in memory", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "(Rebuffi et al., 2017; Castro et al., 2018; Prabhu et al., 2020) or sampled from generative models", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "(Shin et al., 2017; Kemker & Kanan, 2018; van de Ven et al., 2020). However, explicit storage of", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "training samples is not scalable if the number of classes is large. Furthermore, it is difficult to train", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "a reliable generative model for all classes from very few training samples. In parallel, regularization", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "approaches do not require old exemplars and impose regularization on network weights or outputs", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "to minimize the change of parameters that are important to old tasks (Kirkpatrick et al., 2017; Zenke", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "et al., 2017). To avoid quick performance deterioration after learning a sequence of novel tasks", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 546 + ], + "score": 1.0, + "content": "in regularization approaches, semantic drift compensation (SDC) is developed by learning an em-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 556 + ], + "score": 1.0, + "content": "bedding network via triplet loss (Schroff et al., 2015) and compensates the drift of class centroids", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "using novel data only (Yu et al., 2020). In comparison, IDLVQ-C saves only one exemplar per class", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "and uses saved exemplars to regularize the change in feature extractor and calibrate the change in", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 577, + 180, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 180, + 589 + ], + "score": 1.0, + "content": "reference vectors.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 435, + 506, + 589 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 594, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "Few-shot learning: Few-shot learning attempts to obtain models for classification or regression", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 617 + ], + "score": 1.0, + "content": "tasks with only a few labelled samples. Few-shot models are trained on widely-varying episodes of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "fake few-shot tasks with labelled samples drawn from a large-scale meta-training dataset (Vinyals", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "et al., 2016; Finn et al., 2017; Ravi & Larochelle, 2017; Snell et al., 2017; Sung et al., 2018).", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 506, + 650 + ], + "score": 1.0, + "content": "Meanwhile, recent works attempt to handle novel few-shot tasks while retraining the knowledge of", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 661 + ], + "score": 1.0, + "content": "the base task. These methods are referred to as dynamic few-shot learning (Gidaris & Komodakis,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 506, + 673 + ], + "score": 1.0, + "content": "2018; Ren et al., 2019a; Gidaris & Komodakis, 2019). However, dynamic few-shot learning is", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 506, + 683 + ], + "score": 1.0, + "content": "different from incremental few-shot learning, because they rely on the entire base training dataset", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 694 + ], + "score": 1.0, + "content": "and an extra meta-training dataset during meta-training. In addition, dynamic few-shot learning does", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 693, + 365, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 365, + 705 + ], + "score": 1.0, + "content": "not accumulate knowledge for multiple novel tasks sequentially.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 46.5, + "bbox_fs": [ + 104, + 593, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 504, + 722 + ], + "score": 1.0, + "content": "Incremental few-shot learning: Prior works on incremental few-shot learning focus on classifica-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "tion problems by computing the weights for novel classes in parametric classifiers, without iterative", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "gradient descent. For instance, the weights of novel classes can be imprinted by normalized pro-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "totypes of novel classes, while keeping the feature extractor fixed (Qi et al., 2018). Since novel", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "weights are computed only with the samples of novel classes, the fixed feature extractor may not be", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "compatible with novel classification weights. More recently, neural gas network is employed to con-", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "struct an undirected graph to represent knowledge of old classes (Tao et al., 2020b;a). The vertices", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "in the graph are constructed in an unsupervised manner using competitive Hebbian learning (Fritzke,", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "1995), while the feature embedding is fixed. In contrast, IDLVQ learns both feature extractor and", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 326, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 326, + 172 + ], + "score": 1.0, + "content": "reference vectors concurrently in a supervised manner.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 52.5, + "bbox_fs": [ + 106, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "gradient descent. For instance, the weights of novel classes can be imprinted by normalized pro-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "totypes of novel classes, while keeping the feature extractor fixed (Qi et al., 2018). Since novel", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 117 + ], + "score": 1.0, + "content": "weights are computed only with the samples of novel classes, the fixed feature extractor may not be", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "compatible with novel classification weights. More recently, neural gas network is employed to con-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 505, + 139 + ], + "score": 1.0, + "content": "struct an undirected graph to represent knowledge of old classes (Tao et al., 2020b;a). The vertices", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "in the graph are constructed in an unsupervised manner using competitive Hebbian learning (Fritzke,", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "1995), while the feature embedding is fixed. 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In addition, it is assumed that there is no overlap between classes in different", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 350, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 129, + 364 + ], + "score": 1.0, + "content": "tasks", + "type": "text" + }, + { + "bbox": [ + 129, + 351, + 187, + 363 + ], + "score": 0.92, + "content": "C ^ { t } \\bigcap C ^ { s } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 350, + 202, + 364 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 202, + 352, + 225, + 363 + ], + "score": 0.91, + "content": "t \\neq s", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 350, + 505, + 364 + ], + "score": 1.0, + "content": ". 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Note that our focus is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "not multi-task scenario, where a task ID is exposed to the model during test phase and the model is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "only required to perform a given task one time (van de Ven & Tolias, 2019). Our model is evaluated", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 395, + 429, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 429, + 408 + ], + "score": 1.0, + "content": "in a task-agnostic setting, where task ID is not exposed to the model at test time.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 108, + 412, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "For regression tasks, we follow a similar setting with a notable difference that the target is real-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 135, + 435 + ], + "score": 1.0, + "content": "valued", + "type": "text" + }, + { + "bbox": [ + 136, + 423, + 162, + 434 + ], + "score": 0.91, + "content": "y \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 423, + 505, + 435 + ], + "score": 1.0, + "content": ". In addition, the target values in different tasks do not have to be mutually exclusive,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 433, + 246, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 246, + 447 + ], + "score": 1.0, + "content": "unlike the class-incremental setup.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 109, + 474, + 281, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 284, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 284, + 487 + ], + "score": 1.0, + "content": "3.2 LEARNING VECTOR QUANTIZATION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "Traditional nonparametric methods, such as nearest neighbors, represent knowledge and make pre-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "dictions by storing the entire training set. 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The combinations of LVQ and deep", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "feature extractors have been applied to natural language processing (NLP), facial recognition and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "biometrics (Variani et al., 2015; Wang et al., 2016; Ren et al., 2019b; Leng et al., 2015). We notice", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "that LVQ is a nonparametric method which is well suited for incremental few-shot learning because", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "the model capacity grows by incorporating more reference vectors to learn new knowledge. For", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "example, incremental learning vector quantization (ILVQ) has been developed to learn classification", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "models adaptively from raw features (Xu et al., 2012). 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After the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 284, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 231, + 299 + ], + "score": 1.0, + "content": "model is trained on a base task", + "type": "text" + }, + { + "bbox": [ + 231, + 286, + 255, + 296 + ], + "score": 0.86, + "content": "\\mathit { t } = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 284, + 505, + 299 + ], + "score": 1.0, + "content": ") with sufficient data, the model learns novel tasks sequentially.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 309 + ], + "score": 1.0, + "content": "Each novel task contains a number of novel classes with only a few training samples per class.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 199, + 320 + ], + "score": 1.0, + "content": "Learning a novel task", + "type": "text" + }, + { + "bbox": [ + 199, + 308, + 226, + 318 + ], + "score": 0.83, + "content": "( t > 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 307, + 460, + 320 + ], + "score": 1.0, + "content": ") is referred to as an incremental learning session. 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In addition, it is assumed that there is no overlap between classes in different", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 350, + 505, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 350, + 129, + 364 + ], + "score": 1.0, + "content": "tasks", + "type": "text" + }, + { + "bbox": [ + 129, + 351, + 187, + 363 + ], + "score": 0.92, + "content": "C ^ { t } \\bigcap C ^ { s } = \\emptyset", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 350, + 202, + 364 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 202, + 352, + 225, + 363 + ], + "score": 0.91, + "content": "t \\neq s", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 350, + 505, + 364 + ], + "score": 1.0, + "content": ". After an incremental learning session, the performance of the model", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 361, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 365, + 375 + ], + "score": 1.0, + "content": "is evaluated on a test set that contains all previously seen classes", + "type": "text" + }, + { + "bbox": [ + 366, + 362, + 412, + 374 + ], + "score": 0.93, + "content": "C = \\cap _ { i } { \\bar { C } } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 361, + 506, + 375 + ], + "score": 1.0, + "content": ". Note that our focus is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 386 + ], + "score": 1.0, + "content": "not multi-task scenario, where a task ID is exposed to the model during test phase and the model is", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 396 + ], + "score": 1.0, + "content": "only required to perform a given task one time (van de Ven & Tolias, 2019). Our model is evaluated", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 395, + 429, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 429, + 408 + ], + "score": 1.0, + "content": "in a task-agnostic setting, where task ID is not exposed to the model at test time.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 263, + 507, + 408 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 412, + 505, + 445 + ], + "lines": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "For regression tasks, we follow a similar setting with a notable difference that the target is real-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 135, + 435 + ], + "score": 1.0, + "content": "valued", + "type": "text" + }, + { + "bbox": [ + 136, + 423, + 162, + 434 + ], + "score": 0.91, + "content": "y \\in \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 423, + 505, + 435 + ], + "score": 1.0, + "content": ". In addition, the target values in different tasks do not have to be mutually exclusive,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 433, + 246, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 246, + 447 + ], + "score": 1.0, + "content": "unlike the class-incremental setup.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 106, + 411, + 505, + 447 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 474, + 281, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 284, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 284, + 487 + ], + "score": 1.0, + "content": "3.2 LEARNING VECTOR QUANTIZATION", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 501, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "Traditional nonparametric methods, such as nearest neighbors, represent knowledge and make pre-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "dictions by storing the entire training set. Despite the simplicity and effectiveness, they are not", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 536 + ], + "score": 1.0, + "content": "scalable to a large-scale base dataset. Typically, incremental learning methods are only allowed to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "store a small number of exemplars to preserve the knowledge of previously learned tasks. However,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "randomly selected exemplars may not well present the knowledge in old tasks. LVQ is a classical", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "data compression method that represents the knowledge through a few learned reference vectors", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "(Sato & Yamada, 1996; Seo & Obermayer, 2003; Biehl et al., 2007). A new sample is classified to", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 576, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 104, + 576, + 506, + 592 + ], + "score": 1.0, + "content": "the same label as the nearest reference vector in the input space. LVQ has been combined with deep", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 506, + 602 + ], + "score": 1.0, + "content": "feature extractors as an alternative to standard neural networks for better interpretability (De Vries", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 598, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 598, + 506, + 614 + ], + "score": 1.0, + "content": "et al., 2016; Villmann et al., 2017; Saralajew et al., 2018). The combinations of LVQ and deep", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "feature extractors have been applied to natural language processing (NLP), facial recognition and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "biometrics (Variani et al., 2015; Wang et al., 2016; Ren et al., 2019b; Leng et al., 2015). We notice", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "that LVQ is a nonparametric method which is well suited for incremental few-shot learning because", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "the model capacity grows by incorporating more reference vectors to learn new knowledge. For", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "example, incremental learning vector quantization (ILVQ) has been developed to learn classification", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "models adaptively from raw features (Xu et al., 2012). In this study, we present the knowledge by", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "learning reference vectors in the feature space through LVQ and adapt them in incremental few-shot", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "learning. Compared with ILVQ by Xu et al. (2012), our method does not rely on predefined rules to", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "update reference vectors and can be learned along with deep neural networks in an end-to-end fash-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "ion. 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The new reference vectors along", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 95, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 330, + 108 + ], + "score": 1.0, + "content": "with the neural network parameters are fine-tuned on", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 330, + 96, + 343, + 106 + ], + "score": 0.86, + "content": "\\mathcal { D } ^ { t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 343, + 95, + 479, + 108 + ], + "score": 1.0, + "content": "to learn new knowledge in task", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 479, + 97, + 484, + 106 + ], + "score": 0.66, + "content": "t", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 485, + 95, + 506, + 108 + ], + "score": 1.0, + "content": ". To", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 120 + ], + "score": 1.0, + "content": "preserve the knowledge from the old tasks during incremental learning, the model should be updated", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 118, + 505, + 131 + ], + "spans": [ + { + "bbox": [ + 106, + 118, + 505, + 131 + ], + "score": 1.0, + "content": "only when necessary. Therefore, cross-entropy loss is not used in incremental learning sessions", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 129, + 505, + 142 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 474, + 141 + ], + "score": 1.0, + "content": "because it always updates model parameters even if the sample is correctly classified. Let", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 474, + 129, + 491, + 142 + ], + "score": 0.91, + "content": "\\mathbf { m } _ { + } ^ { t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 492, + 129, + 505, + 141 + ], + "score": 1.0, + "content": "be", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 138, + 506, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 299, + 155 + ], + "score": 1.0, + "content": "the reference vector with the correct label and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 299, + 141, + 317, + 153 + ], + "score": 0.89, + "content": "\\mathbf { m } _ { - } ^ { t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 317, + 138, + 506, + 155 + ], + "score": 1.0, + "content": "be the nearest reference vector with a wrong", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 153, + 505, + 169 + ], + "spans": [ + { + "bbox": [ + 104, + 153, + 223, + 169 + ], + "score": 1.0, + "content": "label. 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The dataset", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "is split into base classes and multiple groups of novel classes. We apply standard data augmentation,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "score": 1.0, + "content": "including random crop, horizontal flip and color jitter, on all training images. 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The remaining 100 classes are treated as novel cat-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "egories and split into 10 incremental learning sessions. Each incremental learning session contains", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 461, + 447, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 447, + 474 + ], + "score": 1.0, + "content": "10 novel classes and 5 randomly selected training samples per class (10-way 5-shot).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "miniImageNet dataset is a 100-class subset of the original ImageNet dataset (Deng et al., 2009).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "Each class contains 500 training images, 50 validation images, and 50 test images. The images", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 231, + 513 + ], + "score": 1.0, + "content": "are in RGB format of the size", + "type": "text" + }, + { + "bbox": [ + 231, + 501, + 267, + 511 + ], + "score": 0.89, + "content": "8 4 \\times 8 4", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 500, + 506, + 513 + ], + "score": 1.0, + "content": ". We choose 60 and 40 classes for base and novel classes,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "respectfully. The 40 novel classes are divided into 8 sessions and each session contains 5 novel", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 523, + 407, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 407, + 534 + ], + "score": 1.0, + "content": "classes with 5 randomly selected training samples per class (5-way 5-shot).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 504, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 504, + 552 + ], + "score": 1.0, + "content": "ResNet18 (He et al., 2016) is used as the feature extractor for incremental classification problems.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "The learning process for each dataset is repeated 10 times and the average test accuracy is reported.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "The proposed method is compared with six methods for few-shot class-incremental learning: fine-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 163, + 585 + ], + "score": 1.0, + "content": "turning using", + "type": "text" + }, + { + "bbox": [ + 164, + 572, + 176, + 582 + ], + "score": 0.87, + "content": "\\mathcal { D } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 572, + 505, + 585 + ], + "score": 1.0, + "content": ", joint training using the entire training set from all encountered classes, iCaRL", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "(Rebuffi et al., 2017), Rebalancing (Hou et al., 2019), ProtoNet (Snell et al., 2017), incremental", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "learning vector quantization (ILVQ) (Xu et al., 2012), SDC (Yu et al., 2020), and Imprint (Qi et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 493, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 493, + 618 + ], + "score": 1.0, + "content": "2018). Note that ILVQ is applied to the features extracted by neural networks in our experiment.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 504, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 504, + 633 + ], + "score": 1.0, + "content": "The incremental few-shot learning results on CUB and miniImageNet are shown in Table 1 and 2,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "respectively. Our method outperforms fine-tuning, iCaRL (Rebuffi et al., 2017), and ProtoNet (Snell", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 642, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 642, + 505, + 658 + ], + "score": 1.0, + "content": "et al., 2017) by a large margin. Simply fine-tuning the weights in classifier with few-shot training", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "samples for novel classes significantly deteriorates the prediction accuracy. 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The dataset", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "is split into base classes and multiple groups of novel classes. We apply standard data augmentation,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 382 + ], + "score": 1.0, + "content": "including random crop, horizontal flip and color jitter, on all training images. After each training", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "session, the model performance is evaluated on a test set, which contains all classes that the model", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 390, + 188, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 188, + 402 + ], + "score": 1.0, + "content": "has been trained on.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 334, + 506, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 505, + 473 + ], + "lines": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "CUB dataset is composed of 200 fine-grained bird species with 11,788 images. 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The remaining 100 classes are treated as novel cat-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "egories and split into 10 incremental learning sessions. Each incremental learning session contains", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 461, + 447, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 447, + 474 + ], + "score": 1.0, + "content": "10 novel classes and 5 randomly selected training samples per class (10-way 5-shot).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 407, + 506, + 474 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 478, + 505, + 534 + ], + "lines": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 490 + ], + "score": 1.0, + "content": "miniImageNet dataset is a 100-class subset of the original ImageNet dataset (Deng et al., 2009).", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "Each class contains 500 training images, 50 validation images, and 50 test images. The images", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 500, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 231, + 513 + ], + "score": 1.0, + "content": "are in RGB format of the size", + "type": "text" + }, + { + "bbox": [ + 231, + 501, + 267, + 511 + ], + "score": 0.89, + "content": "8 4 \\times 8 4", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 500, + 506, + 513 + ], + "score": 1.0, + "content": ". We choose 60 and 40 classes for base and novel classes,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "respectfully. The 40 novel classes are divided into 8 sessions and each session contains 5 novel", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 523, + 407, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 407, + 534 + ], + "score": 1.0, + "content": "classes with 5 randomly selected training samples per class (5-way 5-shot).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 478, + 506, + 534 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 539, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 538, + 504, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 504, + 552 + ], + "score": 1.0, + "content": "ResNet18 (He et al., 2016) is used as the feature extractor for incremental classification problems.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "The learning process for each dataset is repeated 10 times and the average test accuracy is reported.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "The proposed method is compared with six methods for few-shot class-incremental learning: fine-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 163, + 585 + ], + "score": 1.0, + "content": "turning using", + "type": "text" + }, + { + "bbox": [ + 164, + 572, + 176, + 582 + ], + "score": 0.87, + "content": "\\mathcal { D } ^ { t }", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 572, + 505, + 585 + ], + "score": 1.0, + "content": ", joint training using the entire training set from all encountered classes, iCaRL", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 596 + ], + "score": 1.0, + "content": "(Rebuffi et al., 2017), Rebalancing (Hou et al., 2019), ProtoNet (Snell et al., 2017), incremental", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "learning vector quantization (ILVQ) (Xu et al., 2012), SDC (Yu et al., 2020), and Imprint (Qi et al.,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 604, + 493, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 493, + 618 + ], + "score": 1.0, + "content": "2018). Note that ILVQ is applied to the features extracted by neural networks in our experiment.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 538, + 506, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 504, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 504, + 633 + ], + "score": 1.0, + "content": "The incremental few-shot learning results on CUB and miniImageNet are shown in Table 1 and 2,", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "respectively. Our method outperforms fine-tuning, iCaRL (Rebuffi et al., 2017), and ProtoNet (Snell", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 642, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 642, + 505, + 658 + ], + "score": 1.0, + "content": "et al., 2017) by a large margin. Simply fine-tuning the weights in classifier with few-shot training", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "samples for novel classes significantly deteriorates the prediction accuracy. Although iCaRL alle-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "viates catastrophic forgetting by tuning the model with a mix of old exemplars and novel few-shot", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "data, the prediction accuracy still drops quickly because iCaRL requires sufficient samples per class", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "to achieve satisfactory performance. The ProtoNet relies on distance to prototypes (the mean of", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "features within a class) to make classification but the fixed feature extractor may not be able to well", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "score": 1.0, + "content": "separate novel classes. ILVQ is slightly better than ProtoNet because prototypes can be learned", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "adaptively when more classes are available in incremental learning sessions. Some prototypes in", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 42.5, + "bbox_fs": [ + 104, + 622, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "ILVQ are close to the border of a class, which are more effective than class centroids in ProtoNet.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "However, ILVQ does not achieve the best performance because the feature extractor is fixed and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "cannot be learned along with the prototypes. IDLVQ-C has a small gain in the first couple of incre-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 504, + 127 + ], + "score": 1.0, + "content": "mental few-shot learning sessions compared with SDC (Yu et al., 2020) and Imprint (Qi et al., 2018).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "Similar to ProtoNet, SDC also relies on prototypes to make classification. The performance of SDC", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "is better than that of ProtoNet because SDC fine-tunes the feature extractor with novel dataset and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "compensates the drift in prototypes. However, the compensation for the drift of old-class prototypes", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "can be less accurate in SDC because it is approximated by samples in novel classes. In parallel,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "the imprint method directly computes the normalized classification weights from the average of nor-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "malized features within a novel class. The imprint method avoids imbalanced classification weights", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "and circumvents the overfitting in few-shot class-incremental learning through weight normaliza-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "tion. Nevertheless, the fixed feature extractor in the imprint method may not be well suited for novel", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "score": 1.0, + "content": "classes. 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The performance", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "score": 1.0, + "content": "of SDC, Imprint and IDLVQ-C is better than offline joint training in early sessions of incremen-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "tal few-shot learning. Offline joint training may not result in oracle performance due to extremely", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 269, + 349, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 349, + 281 + ], + "score": 1.0, + "content": "imbalanced samples between base classes and novel classes.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 8.5 + }, + { + "type": "table", + "bbox": [ + 120, + 311, + 489, + 414 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 118, + 299, + 491, + 310 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 296, + 492, + 313 + ], + "spans": [ + { + "bbox": [ + 117, + 296, + 492, + 313 + ], + "score": 1.0, + "content": "Table 1: Prediction accuracy on CUB all classes using the 10-way 5-shot incremental setting.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "table_body", + "bbox": [ + 120, + 311, + 489, + 414 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 311, + 489, + 414 + ], + "spans": [ + { + "bbox": [ + 120, + 311, + 489, + 414 + ], + "score": 0.983, + "html": "
Methodsessions
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Fine-tune77.3046.2334.7125.3523.1620.6516.2113.3211.9811.1710.76
Joint train77.3073.2868.8065.3463.7562.0060.8159.7159.0658.6958.23
iCaRL (Rebuffi et al.,2017)77.3057.1854.6748.1140.7636.8533.1230.4228.2226.8425.23
Rebalancing (Hou et al., 2019)77.3064.5356.1447.2938.9234.3931.0427.9327.1224.4623.61
ProtoNet (Snell et al., 2017)77.3069.7666.0162.2959.5857.1055.1354.0952.4051.6550.36
ILVQ (Xu et al., 2012)77.3071.5066.7962.7160.2057.8455.2755.0652.4251.7250.47
SDC (Yu et al., 2020)77.3474.4569.4565.2761.8158.2656.1455.7153.3152.7951.52
Imprint (Qi et al., 2018)77.0273.3969.5065.6162.8160.7459.3958.6156.8555.9354.82
IDLVQ-C77.3774.7270.2867.1365.3463.5262.1061.5459.0458.6857.81
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Methodsessions
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Fine-tune64.2530.1118.536.312.862.681.871.561.42
Joint train64.2558.8055.2652.3849.7148.3745.9144.6843.38
iCaRL (Rebuffi et al., 2017)64.2548.0443.1338.2830.0124.4621.8519.8417.76
Rebalancing (Hou et al.,2019)64.2549.2144.1737.7130.1122.9219.9917.9616.25
ProtoNet (Snell et al., 2017)64.2555.1251.6748.9146.5244.2541.9140.0738.42
ILVQ (Xu et al., 2012)64.2556.0152.4349.3146.9844.3742.0640.1138.43
SDC (Yu et al., 2020)64.6259.6355.3950.9248.3045.2842.9742.5141.24
Imprint (Qi et al., 2018)64.7159.8555.7152.4749.9047.3144.5742.5741.26
IDLVQ-C64.7759.8755.9352.6249.8847.5544.8343.1441.84
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We study five variants of our methods: (a) new reference vectors are", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "initialized as class centroids and no tuning is done for feature extractor or old reference vectors; (b)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 134, + 605 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { i n t r a }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 593, + 331, + 608 + ], + "score": 1.0, + "content": "is not used in incremental learning sessions; (c)", + "type": "text" + }, + { + "bbox": [ + 331, + 594, + 346, + 605 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "is not used in the incremental learning", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 603, + 504, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 487, + 619 + ], + "score": 1.0, + "content": "sessions; (d) shift in old reference vectors are not compensated; (e) replace the margin based loss", + "type": "text" + }, + { + "bbox": [ + 487, + 605, + 504, + 616 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { M }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 216, + 628 + ], + "score": 1.0, + "content": "with the cross entropy loss", + "type": "text" + }, + { + "bbox": [ + 216, + 616, + 237, + 627 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { C E }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 617, + 505, + 628 + ], + "score": 1.0, + "content": ". Table 3 shows the results of our ablation studies on CUB dataset.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 625, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 641 + ], + "score": 1.0, + "content": "Without any fine-tuning, the initial reference vectors for novel classes lead to descent accuracy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "in incremental few-shot classification. It demonstrates the robustness of nonparametric classifier.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 648, + 504, + 663 + ], + "spans": [ + { + "bbox": [ + 107, + 649, + 134, + 660 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { i n t r a }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 648, + 169, + 663 + ], + "score": 1.0, + "content": "leads to", + "type": "text" + }, + { + "bbox": [ + 169, + 649, + 197, + 660 + ], + "score": 0.89, + "content": "0 . 5 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 648, + 489, + 663 + ], + "score": 1.0, + "content": "gain due to tight intra-class variation. The less forgetting regularization", + "type": "text" + }, + { + "bbox": [ + 489, + 649, + 504, + 660 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { F }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 432, + 672 + ], + "score": 1.0, + "content": "is proved to prevent forgetting old classes and achieves a performance boost by", + "type": "text" + }, + { + "bbox": [ + 433, + 660, + 460, + 671 + ], + "score": 0.88, + "content": "2 . 3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 660, + 505, + 672 + ], + "score": 1.0, + "content": ". The shift", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "compensation for reference vectors effectively adapts old reference vector in new embedding spaces", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 191, + 695 + ], + "score": 1.0, + "content": "with a gain round of", + "type": "text" + }, + { + "bbox": [ + 191, + 681, + 206, + 692 + ], + "score": 0.87, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 681, + 505, + 695 + ], + "score": 1.0, + "content": ". The margin based loss is more effective in preventing forgetting in early", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 693, + 239, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 239, + 705 + ], + "score": 1.0, + "content": "sessions of incremental learning.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "The effect of the number of training samples per class. The proposed method is evaluated under", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "score": 1.0, + "content": "different few-shot settings on CUB dataset to investigate the effect of different numbers of training", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 505, + 280 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "ILVQ are close to the border of a class, which are more effective than class centroids in ProtoNet.", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 105 + ], + "score": 1.0, + "content": "However, ILVQ does not achieve the best performance because the feature extractor is fixed and", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "cannot be learned along with the prototypes. IDLVQ-C has a small gain in the first couple of incre-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 116, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 504, + 127 + ], + "score": 1.0, + "content": "mental few-shot learning sessions compared with SDC (Yu et al., 2020) and Imprint (Qi et al., 2018).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "Similar to ProtoNet, SDC also relies on prototypes to make classification. The performance of SDC", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "is better than that of ProtoNet because SDC fine-tunes the feature extractor with novel dataset and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 506, + 162 + ], + "score": 1.0, + "content": "compensates the drift in prototypes. However, the compensation for the drift of old-class prototypes", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "can be less accurate in SDC because it is approximated by samples in novel classes. In parallel,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 505, + 183 + ], + "score": 1.0, + "content": "the imprint method directly computes the normalized classification weights from the average of nor-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 194 + ], + "score": 1.0, + "content": "malized features within a novel class. The imprint method avoids imbalanced classification weights", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 205 + ], + "score": 1.0, + "content": "and circumvents the overfitting in few-shot class-incremental learning through weight normaliza-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 202, + 505, + 216 + ], + "score": 1.0, + "content": "tion. Nevertheless, the fixed feature extractor in the imprint method may not be well suited for novel", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 213, + 506, + 227 + ], + "score": 1.0, + "content": "classes. In contrast, IDLVQ-C updates the feature extractor only when necessary and compensates", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "the shift of old reference vectors more accurately using exemplars from old classes. That is why", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 235, + 505, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 235, + 505, + 249 + ], + "score": 1.0, + "content": "the gain of IDLVQ-C increases with more incremental few-shot learning sessions. The performance", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 505, + 260 + ], + "score": 1.0, + "content": "of SDC, Imprint and IDLVQ-C is better than offline joint training in early sessions of incremen-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 257, + 505, + 271 + ], + "score": 1.0, + "content": "tal few-shot learning. Offline joint training may not result in oracle performance due to extremely", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 269, + 349, + 281 + ], + "spans": [ + { + "bbox": [ + 106, + 269, + 349, + 281 + ], + "score": 1.0, + "content": "imbalanced samples between base classes and novel classes.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 82, + 506, + 281 + ] + }, + { + "type": "table", + "bbox": [ + 120, + 311, + 489, + 414 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 118, + 299, + 491, + 310 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 117, + 296, + 492, + 313 + ], + "spans": [ + { + "bbox": [ + 117, + 296, + 492, + 313 + ], + "score": 1.0, + "content": "Table 1: Prediction accuracy on CUB all classes using the 10-way 5-shot incremental setting.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "table_body", + "bbox": [ + 120, + 311, + 489, + 414 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 311, + 489, + 414 + ], + "spans": [ + { + "bbox": [ + 120, + 311, + 489, + 414 + ], + "score": 0.983, + "html": "
Methodsessions
1234567891011
Fine-tune77.3046.2334.7125.3523.1620.6516.2113.3211.9811.1710.76
Joint train77.3073.2868.8065.3463.7562.0060.8159.7159.0658.6958.23
iCaRL (Rebuffi et al.,2017)77.3057.1854.6748.1140.7636.8533.1230.4228.2226.8425.23
Rebalancing (Hou et al., 2019)77.3064.5356.1447.2938.9234.3931.0427.9327.1224.4623.61
ProtoNet (Snell et al., 2017)77.3069.7666.0162.2959.5857.1055.1354.0952.4051.6550.36
ILVQ (Xu et al., 2012)77.3071.5066.7962.7160.2057.8455.2755.0652.4251.7250.47
SDC (Yu et al., 2020)77.3474.4569.4565.2761.8158.2656.1455.7153.3152.7951.52
Imprint (Qi et al., 2018)77.0273.3969.5065.6162.8160.7459.3958.6156.8555.9354.82
IDLVQ-C77.3774.7270.2867.1365.3463.5262.1061.5459.0458.6857.81
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Methodsessions
123456789
Fine-tune64.2530.1118.536.312.862.681.871.561.42
Joint train64.2558.8055.2652.3849.7148.3745.9144.6843.38
iCaRL (Rebuffi et al., 2017)64.2548.0443.1338.2830.0124.4621.8519.8417.76
Rebalancing (Hou et al.,2019)64.2549.2144.1737.7130.1122.9219.9917.9616.25
ProtoNet (Snell et al., 2017)64.2555.1251.6748.9146.5244.2541.9140.0738.42
ILVQ (Xu et al., 2012)64.2556.0152.4349.3146.9844.3742.0640.1138.43
SDC (Yu et al., 2020)64.6259.6355.3950.9248.3045.2842.9742.5141.24
Imprint (Qi et al., 2018)64.7159.8555.7152.4749.9047.3144.5742.5741.26
IDLVQ-C64.7759.8755.9352.6249.8847.5544.8343.1441.84
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We study five variants of our methods: (a) new reference vectors are", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "initialized as class centroids and no tuning is done for feature extractor or old reference vectors; (b)", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 593, + 506, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 134, + 605 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { i n t r a }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 593, + 331, + 608 + ], + "score": 1.0, + "content": "is not used in incremental learning sessions; (c)", + "type": "text" + }, + { + "bbox": [ + 331, + 594, + 346, + 605 + ], + "score": 0.89, + "content": "\\mathcal { L } _ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 593, + 506, + 608 + ], + "score": 1.0, + "content": "is not used in the incremental learning", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 603, + 504, + 619 + ], + "spans": [ + { + "bbox": [ + 104, + 603, + 487, + 619 + ], + "score": 1.0, + "content": "sessions; (d) shift in old reference vectors are not compensated; (e) replace the margin based loss", + "type": "text" + }, + { + "bbox": [ + 487, + 605, + 504, + 616 + ], + "score": 0.89, + "content": "{ \\mathcal { L } } _ { M }", + "type": "inline_equation" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 617, + 216, + 628 + ], + "score": 1.0, + "content": "with the cross entropy loss", + "type": "text" + }, + { + "bbox": [ + 216, + 616, + 237, + 627 + ], + "score": 0.9, + "content": "\\mathcal { L } _ { C E }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 617, + 505, + 628 + ], + "score": 1.0, + "content": ". Table 3 shows the results of our ablation studies on CUB dataset.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 625, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 505, + 641 + ], + "score": 1.0, + "content": "Without any fine-tuning, the initial reference vectors for novel classes lead to descent accuracy", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "in incremental few-shot classification. It demonstrates the robustness of nonparametric classifier.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 107, + 648, + 504, + 663 + ], + "spans": [ + { + "bbox": [ + 107, + 649, + 134, + 660 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { i n t r a }", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 648, + 169, + 663 + ], + "score": 1.0, + "content": "leads to", + "type": "text" + }, + { + "bbox": [ + 169, + 649, + 197, + 660 + ], + "score": 0.89, + "content": "0 . 5 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 648, + 489, + 663 + ], + "score": 1.0, + "content": "gain due to tight intra-class variation. The less forgetting regularization", + "type": "text" + }, + { + "bbox": [ + 489, + 649, + 504, + 660 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { F }", + "type": "inline_equation" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 432, + 672 + ], + "score": 1.0, + "content": "is proved to prevent forgetting old classes and achieves a performance boost by", + "type": "text" + }, + { + "bbox": [ + 433, + 660, + 460, + 671 + ], + "score": 0.88, + "content": "2 . 3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 660, + 505, + 672 + ], + "score": 1.0, + "content": ". The shift", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 684 + ], + "score": 1.0, + "content": "compensation for reference vectors effectively adapts old reference vector in new embedding spaces", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 191, + 695 + ], + "score": 1.0, + "content": "with a gain round of", + "type": "text" + }, + { + "bbox": [ + 191, + 681, + 206, + 692 + ], + "score": 0.87, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 681, + 505, + 695 + ], + "score": 1.0, + "content": ". The margin based loss is more effective in preventing forgetting in early", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 693, + 239, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 239, + 705 + ], + "score": 1.0, + "content": "sessions of incremental learning.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 562, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 708, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 708, + 505, + 722 + ], + "score": 1.0, + "content": "The effect of the number of training samples per class. The proposed method is evaluated under", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 733 + ], + "score": 1.0, + "content": "different few-shot settings on CUB dataset to investigate the effect of different numbers of training", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "samples in novel classes, including 5-shot, 10-shot and 20-shot settings. One reference vector is", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 506, + 219 + ], + "score": 1.0, + "content": "assigned to each class in all few-shot settings. As shown in Fig. 2 in the appendix, the performance", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 215, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 231 + ], + "score": 1.0, + "content": "of incremental learning improves as the number of samples per class increases. When the training", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 226, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 104, + 226, + 506, + 241 + ], + "score": 1.0, + "content": "samples are scarce, the training samples may not well present the generative distribution of training", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "score": 1.0, + "content": "data. Therefore, the learned reference vectors could be biased and classification accuracy is low.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "score": 1.0, + "content": "With more training samples, the learned reference vectors could well present the center of the distri-", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "score": 1.0, + "content": "bution and classification accuracy is improved. The gap in performance becomes more obvious as", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 270, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 284 + ], + "score": 1.0, + "content": "the number of incremental learning sessions grows. The detailed results are reported in Table 7 and", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 282, + 117, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 117, + 294 + ], + "score": 1.0, + "content": "8.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 708, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 150, + 101, + 459, + 176 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 149, + 89, + 460, + 101 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 149, + 87, + 462, + 104 + ], + "spans": [ + { + "bbox": [ + 149, + 87, + 462, + 104 + ], + "score": 1.0, + "content": "Table 3: Ablation study on CUB using the 10-way 5-shot incremental setting.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 150, + 101, + 459, + 176 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 150, + 101, + 459, + 176 + ], + "spans": [ + { + "bbox": [ + 150, + 101, + 459, + 176 + ], + "score": 0.981, + "html": "
Methodsessions
234567891011
No tuning71.9367.1464.2162.6160.1359.0458.4755.6454.2553.66
W.o.Lintra74.7570.2666.8965.0563.1861.8461.3658.6158.1457.24
w.0. LF73.8569.5466.2164.0262.7460.2859.4956.9756.3855.46
w.o. δ74.6770.0166.7464.8163.9061.4260.7358.1657.6256.79
LM→LCE73.2269.4166.0363.9363.0761.1460.9858.6758.1157.32
IDLVQ-C74.7270.2867.1365.3463.5262.1061.5459.0458.6857.81
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As shown in Fig. 2 in the appendix, the performance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 215, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 506, + 231 + ], + "score": 1.0, + "content": "of incremental learning improves as the number of samples per class increases. When the training", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 226, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 104, + 226, + 506, + 241 + ], + "score": 1.0, + "content": "samples are scarce, the training samples may not well present the generative distribution of training", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "score": 1.0, + "content": "data. Therefore, the learned reference vectors could be biased and classification accuracy is low.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 262 + ], + "score": 1.0, + "content": "With more training samples, the learned reference vectors could well present the center of the distri-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 273 + ], + "score": 1.0, + "content": "bution and classification accuracy is improved. The gap in performance becomes more obvious as", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 270, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 284 + ], + "score": 1.0, + "content": "the number of incremental learning sessions grows. The detailed results are reported in Table 7 and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 282, + 117, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 117, + 294 + ], + "score": 1.0, + "content": "8.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 109, + 307, + 297, + 318 + ], + "lines": [ + { + "bbox": [ + 106, + 306, + 299, + 319 + ], + "spans": [ + { + "bbox": [ + 106, + 306, + 299, + 319 + ], + "score": 1.0, + "content": "5.2 INCREMENTAL FEW-SHOT REGRESSION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 327, + 505, + 371 + ], + "lines": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 340 + ], + "score": 1.0, + "content": "IDLVQ-R is tested on two regression datasets: sinusoidal wave and 3D spatial data. 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Methodsessions
234567891011
No tuning71.9367.1464.2162.6160.1359.0458.4755.6454.2553.66
W.o.Lintra74.7570.2666.8965.0563.1861.8461.3658.6158.1457.24
w.0. LF73.8569.5466.2164.0262.7460.2859.4956.9756.3855.46
w.o. δ74.6770.0166.7464.8163.9061.4260.7358.1657.6256.79
LM→LCE73.2269.4166.0363.9363.0761.1460.9858.6758.1157.32
IDLVQ-C74.7270.2867.1365.3463.5262.1061.5459.0458.6857.81
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Methodsessions
123
Joint train offline0.02174(2e-4)0.02232(2e-4)0.02296(2e-4)
Fine-tune w.novel data0.02174(2e-4)0.08462(4e-4)0.11870(6e-4)
Fine-tune w. exemplars0.02174(2e-4)0.02988(2e-4)0.03128(2e-4)
IDLVQ-R0.02181(2e-4)0.02641(2e-4)0.02817(2e-4)
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Methodsessions
123
Joint train offline0.02174(2e-4)0.02232(2e-4)0.02296(2e-4)
Fine-tune w.novel data0.02174(2e-4)0.08462(4e-4)0.11870(6e-4)
Fine-tune w. exemplars0.02174(2e-4)0.02988(2e-4)0.03128(2e-4)
IDLVQ-R0.02181(2e-4)0.02641(2e-4)0.02817(2e-4)
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In the base task(t = 1)
Initialize 01,{m𝑖,.,mN1} and γ
Minimize L = LcE + XintraLintra W.r.t. θ1,{ml,.,mN1} and γ
Pick exemplars from D1 for classes in the base task: xi = arg minxeD1 |lft-1(x) - m1|l²
for novel task t = 2,3,... do
Initialize {mt-1+1.*., mNt}
Minimize L=LM +XFLF + XintraLintra W.r.t. 0t and {mNt-1+1,.,mNt}
t-1 t-1 +
Pick exemplars from Dt for classes in the novel task t: x' = arg minx∈Dt |lfet-1(x) - m ll²
end for
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In the base task (t =1) Initializeθ,{m1,..,mN1},{q],..,qN1} and γ
Minimize Lmse w.r.t. θ,{ml,..,mN1},{ql,.,qN1} and γ
for novel task t = 2,3,.. do
Initialize {mNt-1+1,, mNt} and {qNt-1+1.…,qNt}
Minimize LmsE w.r.t. {mNt-1+1..*,mNt} and {qNt-1+1,.,qNt}
end for
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In the base task(t = 1)
Initialize 01,{m𝑖,.,mN1} and γ
Minimize L = LcE + XintraLintra W.r.t. θ1,{ml,.,mN1} and γ
Pick exemplars from D1 for classes in the base task: xi = arg minxeD1 |lft-1(x) - m1|l²
for novel task t = 2,3,... do
Initialize {mt-1+1.*., mNt}
Minimize L=LM +XFLF + XintraLintra W.r.t. 0t and {mNt-1+1,.,mNt}
t-1 t-1 +
Pick exemplars from Dt for classes in the novel task t: x' = arg minx∈Dt |lfet-1(x) - m ll²
end for
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In the base task (t =1) Initializeθ,{m1,..,mN1},{q],..,qN1} and γ
Minimize Lmse w.r.t. θ,{ml,..,mN1},{ql,.,qN1} and γ
for novel task t = 2,3,.. do
Initialize {mNt-1+1,, mNt} and {qNt-1+1.…,qNt}
Minimize LmsE w.r.t. {mNt-1+1..*,mNt} and {qNt-1+1,.,qNt}
end for
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Base classessessions
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Fine-tune77.3044.2336.2827.5225.9623.0517.6813.0711.7810.9910.71
Joint train77.3075.8375.2574.5174.5873.7473.9573.2573.1173.2573.18
iCaRL (Rebuffi et al., 2017)77.3059.3858.8154.4348.2743.2839.1734.9132.4329.3625.87
Rebalancing (Hou et al.,2019)77.3066.4360.3255.3646.3941.7637.1232.5831.2627.0324.25
ProtoNet (Snell et al., 2017)77.3072.5572.2172.0671.6471.2971.0270.9470.6770.6070.53
ILVQ (Xu et al., 2012)77.3074.1873.5772.6672.5671.5771.1471.1271.0270.9870.85
SDC (Yu et al., 2020)77.3476.0575.2174.1272.3671.8171.6871.4371.2571.2770.96
Imprint (Qi et al., 2018)77.0274.7674.5773.6972.6970.8870.3470.1270.0769.8469.27
IDLVQ-C77.3776.3275.9075.9175.4974.8674.5874.3774.0273.3973.32
Novel classessessions
1234567891011
Fine-tune=66.2326.8618.1216.1615.8513.7613.6812.2311.3710.81
Joint train=47.7836.5534.7736.6838.5238.9140.3741.5042.5143.28
iCaRL (Rebuffi et al., 2017)-35.1833.9727.0421.9923.9923.0424.0122.9624.0424.59
Rebalancing (Hou et al.,2019)=45.5335.2420.3920.2519.6520.9121.2921.9521.6022.97
ProtoNet (Snell et al., 2017)41.8635.0129.7229.4328.7228.6530.0229.5630.5930.19
ILVQ (Xu et al., 2012)40.7432.8929.5429.3030.3828.8232.1229.1730.3230.09
SDC (Yu et al., 2020)58.4540.6535.7733.2331.1630.2433.2530.8932.2632.08
Imprint (Qi et al.,2018)59.6944.1538.6838.1140.4641.1442.1740.3340.4740.37
IDLVQ-C=58.7242.1837.8639.9740.8441.3043.2140.3242.3442.30
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In addition, larger", + "type": "text" + }, + { + "bbox": [ + 258, + 192, + 272, + 202 + ], + "score": 0.88, + "content": "\\lambda _ { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 190, + 506, + 205 + ], + "score": 1.0, + "content": "results in less forgetting in old classes but makes learning", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 202, + 220, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 202, + 220, + 214 + ], + "score": 1.0, + "content": "novel classes more difficult.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 103, + 506, + 214 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 229, + 432, + 239 + ], + "lines": [ + { + "bbox": [ + 106, + 228, + 433, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 433, + 240 + ], + "score": 1.0, + "content": "A.4 ADDITIONAL RESULTS FOR INCREMENTAL FEW-SHOT CLASSIFICATION", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 348 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "The accuracies for base and novel classes are reported separately in Table 5 and 6 for CUB amd", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "miniImageNet, respectively. The prediction accuracy of novel classes is calculated upon all novel", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 505, + 284 + ], + "score": 1.0, + "content": "classes the model has been trained on. Note that the accuracy in Table 1 and 2 is calculated upon all", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 505, + 295 + ], + "score": 1.0, + "content": "classes (including base and novel classes) that the model has been trained on. The proposed IDLVQ-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 292, + 505, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 505, + 307 + ], + "score": 1.0, + "content": "C demonstrates strong capability of preserving old knowledge by achieving the best performance on", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "old classes across all learning sessions. In parallel, Imprint method performs slightly better on novel", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 506, + 328 + ], + "score": 1.0, + "content": "classes than IDLVQ-C in early incremental learning sessions, while IDLVQ-C outperforms Imprint", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "method in longer incremental learning sessions. The advantage of IDLVQ-C can be attributed to the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 338, + 372, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 372, + 349 + ], + "score": 1.0, + "content": "adaptive feature extractor, which is tuned in each learning session.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 249, + 506, + 349 + ] + }, + { + "type": "table", + "bbox": [ + 120, + 392, + 489, + 595 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 369, + 503, + 390 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 381 + ], + "score": 1.0, + "content": "Table 5: Prediction accuracy on CUB base and novel classes using the 10-way 5-shot incremental", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 378, + 140, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 140, + 394 + ], + "score": 1.0, + "content": "setting.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "table_body", + "bbox": [ + 120, + 392, + 489, + 595 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 392, + 489, + 595 + ], + "spans": [ + { + "bbox": [ + 120, + 392, + 489, + 595 + ], + "score": 0.982, + "html": "
Base classessessions
1234567891011
Fine-tune77.3044.2336.2827.5225.9623.0517.6813.0711.7810.9910.71
Joint train77.3075.8375.2574.5174.5873.7473.9573.2573.1173.2573.18
iCaRL (Rebuffi et al., 2017)77.3059.3858.8154.4348.2743.2839.1734.9132.4329.3625.87
Rebalancing (Hou et al.,2019)77.3066.4360.3255.3646.3941.7637.1232.5831.2627.0324.25
ProtoNet (Snell et al., 2017)77.3072.5572.2172.0671.6471.2971.0270.9470.6770.6070.53
ILVQ (Xu et al., 2012)77.3074.1873.5772.6672.5671.5771.1471.1271.0270.9870.85
SDC (Yu et al., 2020)77.3476.0575.2174.1272.3671.8171.6871.4371.2571.2770.96
Imprint (Qi et al., 2018)77.0274.7674.5773.6972.6970.8870.3470.1270.0769.8469.27
IDLVQ-C77.3776.3275.9075.9175.4974.8674.5874.3774.0273.3973.32
Novel classessessions
1234567891011
Fine-tune=66.2326.8618.1216.1615.8513.7613.6812.2311.3710.81
Joint train=47.7836.5534.7736.6838.5238.9140.3741.5042.5143.28
iCaRL (Rebuffi et al., 2017)-35.1833.9727.0421.9923.9923.0424.0122.9624.0424.59
Rebalancing (Hou et al.,2019)=45.5335.2420.3920.2519.6520.9121.2921.9521.6022.97
ProtoNet (Snell et al., 2017)41.8635.0129.7229.4328.7228.6530.0229.5630.5930.19
ILVQ (Xu et al., 2012)40.7432.8929.5429.3030.3828.8232.1229.1730.3230.09
SDC (Yu et al., 2020)58.4540.6535.7733.2331.1630.2433.2530.8932.2632.08
Imprint (Qi et al.,2018)59.6944.1538.6838.1140.4641.1442.1740.3340.4740.37
IDLVQ-C=58.7242.1837.8639.9740.8441.3043.2140.3242.3442.30
", + "type": "table", + "image_path": "05ca6f836c7776d404f47b953b3e02cbae9c589b42216e869c9b9f11999d652e.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 120, + 392, + 489, + 459.6666666666667 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 120, + 459.6666666666667, + 489, + 527.3333333333334 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 120, + 527.3333333333334, + 489, + 595.0 + ], + "spans": [], + "index": 25 + } + ] + } + ], + "index": 22.75 + }, + { + "type": "text", + "bbox": [ + 107, + 608, + 505, + 674 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 621 + ], + "score": 1.0, + "content": "The test accuracy on CUB dataset using 10-way 10-shot and 10-way 20-shot incremental settings", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 505, + 631 + ], + "score": 1.0, + "content": "are reported in Table 7 and 8, respectively. The prediction accuracy improves in all methods with", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "more training samples per class. The iCaRL and Rebalancing methods show the most significant", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 505, + 653 + ], + "score": 1.0, + "content": "improvement when the number of training samples increases. The proposed IDLVQ-C is effective", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 652, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 106, + 652, + 505, + 663 + ], + "score": 1.0, + "content": "in different incremental few-shot scenarios as it achieves the best performance on 5-shot, 10-shot", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 661, + 191, + 676 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 191, + 676 + ], + "score": 1.0, + "content": "and 20-shot settings.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 606, + 505, + 676 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 689, + 416, + 699 + ], + "lines": [ + { + "bbox": [ + 106, + 689, + 417, + 701 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 417, + 701 + ], + "score": 1.0, + "content": "A.5 EXPERIMENT DETAILS FOR INCREMENTAL FEW-SHOT REGRESSION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 503, + 731 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Sinusoidal wave: A six-layer feedforward neural network with ReLU nonlinear activation is used", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "as the feature extractor. IDLVQ-R learns 10 reference vectors and targets from the base task. In", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "each incremental learning sessions, 5 pairs of reference vectors and targets are added. After new", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "reference vectors and targets are fine-tuned, the model is capable of make prediction for all seen", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 639, + 504, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 504, + 649 + ], + "score": 1.0, + "content": "tasks. 10 exemplars are selected uniformly from the training set in the base task. In the incremental", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "learning session of the 2nd task, the model is fine-tuned on 10 exemplars and 5 novel training", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "samples. After the training converges, 5 novel training samples in the current task are added to the", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "exemplar set. In the incremental learning session of the 3rd task, the model is fine-tuned with 15", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 681, + 329, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 329, + 694 + ], + "score": 1.0, + "content": "exemplars from old tasks and 5 novel training samples.", + "type": "text", + "cross_page": true + } + ], + "index": 19 + } + ], + "index": 33.5, + "bbox_fs": [ + 106, + 709, + 505, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 145, + 112, + 464, + 316 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 502, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 87, + 504, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 504, + 103 + ], + "score": 1.0, + "content": "Table 6: Prediction accuracy on miniImageNet base and novel classes using the 5-way 5-shot incre-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 168, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 168, + 114 + ], + "score": 1.0, + "content": "mental setting.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 145, + 112, + 464, + 316 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 112, + 464, + 316 + ], + "spans": [ + { + "bbox": [ + 145, + 112, + 464, + 316 + ], + "score": 0.98, + "html": "
Base classessessions
123456789
Fine-tune64.2532.2820.876.953.173.161.921.531.46
Joint train64.2563.3062.8362.1662.1862.6861.8661.8761.89
iCaRL (Rebuffi et al., 2017)64.2551.6648.9745.6237.3930.8628.6826.8324.47
Rebalancing (Hou et al., 2019)64.2552.8750.1644.7837.4828.7525.5822.9721.57
ProtoNet (Snell et al., 2017)64.2559.2758.8858.6958.2257.6357.0356.8056.47
ILVQ (Xu et al.,2012)64.2560.2459.6259.0258.6157.7157.1656.8356.49
SDC (Yu et al., 2020)64.6263.5862.7861.1260.2959.3759.0559.9759.87
Imprint (Qi et al.,2018)64.7163.5262.9662.1361.1761.2760.6359.8659.64
IDLVQ-C64.77 63.7763.2262.4461.2261.4760.9760.6660.44
Novel classessessions
123456789
Fine-tune4.074.493.751.931.531.771.611.36
Joint train-4.809.8413.2612.3014.0314.0115.2115.62
iCaRL (Rebuffi et al., 2017)=4.608.098.927.879.108.197.867.70
Rebalancing (Hou et al.,2019)5.298.239.438.008.938.818.838.27
ProtoNet (Snell et al., 2017)5.328.419.7911.4212.1411.6711.3911.35
ILVQ (Xu et al.,2012)5.259.2910.4712.0912.3511.8611.4511.34
SDC (Yu et al., 2020)12.2311.0510.1212.3311.4610.8112.5813.30
Imprint (Qi et al., 2018)15.8112.2113.8316.0913.8112.4512.9313.69
IDLVQ-C-13.0712.1913.3415.8614.1412.5513.1113.94
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Methodsessions
1234567891011
Fine-tune77.3047.1636.3426.9224.0821.2417.1914.3112.7311.7511.43
Joint train77.3073.2971.5468.7266.3865.4264.9865.7464.8264.4764.16
iCaRL (Rebuffi et al., 2017)77.3059.6656.2452.2648.7746.3744.5443.1742.3541.1940.92
Rebalancing (Hou et al., 2019)77.3064.5358.3553.8249.2747.1245.1643.0542.3741.0240.86
ProtoNet (Snell et al., 2017)77.3069.8266.1263.1961.1758.8558.0457.7555.8455.8255.60
ILVQ (Xu et al., 2012)77.3071.2666.8463.8262.6659.7158.9258.1156.3156.1456.03
SDC (Yu et al., 2020)77.3474.6769.7366.7166.4962.1461.3359.8458.0157.3956.62
Imprint (Qi et al.,2018)77.0274.0770.2666.8464.4562.4661.8561.0259.2058.9358.43
IDLVQ-C77.3774.7970.9668.0865.9464.1263.5862.9860.8560.5459.72
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Methodsessions
1234567891011
Fine-tune77.3047.6836.7527.0124.5621.7217.8815.2412.8411.8911.52
Joint train77.3074.3472.5970.8570.1470.0169.6969.6769.4869.4169.42
iCaRL (Rebuff et al.,2017)77.3068.8563.5660.3457.7156.2855.9755.0254.6252.2152.23
Rebalancing (Hou et al.,2019)77.3069.3665.4961.3259.3058.6858.7757.4256.2555.3354.17
ProtoNet (Snell et al., 2017)77.3070.8467.8264.8162.9561.7160.9860.7359.5159.3058.95
ILVQ (Xu et al., 2012)77.3071.5068.7766.1264.3162.8962.3162.0061.5259.7959.23
SDC (Yu et al.,2020)77.3474.6271.6368.6666.7565.2464.2163.6261.9761.5461.11
Imprint (Qi et al., 2018)77.0274.1470.7267.7565.8864.6364.2863.8262.0061.8461.40
IDLVQ-C77.3774.8472.0769.0567.2865.5165.1964.8462.7762.5561.96
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After new", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "reference vectors and targets are fine-tuned, the model is capable of make prediction for all seen", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 639, + 504, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 504, + 649 + ], + "score": 1.0, + "content": "tasks. 10 exemplars are selected uniformly from the training set in the base task. In the incremental", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "learning session of the 2nd task, the model is fine-tuned on 10 exemplars and 5 novel training", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "samples. After the training converges, 5 novel training samples in the current task are added to the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "exemplar set. In the incremental learning session of the 3rd task, the model is fine-tuned with 15", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 681, + 329, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 329, + 694 + ], + "score": 1.0, + "content": "exemplars from old tasks and 5 novel training samples.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "3D spatial data: We follow the same training and test protocols as the sinusoidal wave dataset. We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "choose 40, 15 and 15 reference vectors and targets for 1st, 2nd and 3rd tasks, respectively. Adding", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 720, + 488, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 488, + 733 + ], + "score": 1.0, + "content": "more reference vectors does not result in obvious improvement in accuracy in our experiments.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 145, + 112, + 464, + 316 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 89, + 502, + 111 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 87, + 504, + 103 + ], + "spans": [ + { + "bbox": [ + 105, + 87, + 504, + 103 + ], + "score": 1.0, + "content": "Table 6: Prediction accuracy on miniImageNet base and novel classes using the 5-way 5-shot incre-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 99, + 168, + 114 + ], + "spans": [ + { + "bbox": [ + 105, + 99, + 168, + 114 + ], + "score": 1.0, + "content": "mental setting.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 145, + 112, + 464, + 316 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 112, + 464, + 316 + ], + "spans": [ + { + "bbox": [ + 145, + 112, + 464, + 316 + ], + "score": 0.98, + "html": "
Base classessessions
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Fine-tune64.2532.2820.876.953.173.161.921.531.46
Joint train64.2563.3062.8362.1662.1862.6861.8661.8761.89
iCaRL (Rebuffi et al., 2017)64.2551.6648.9745.6237.3930.8628.6826.8324.47
Rebalancing (Hou et al., 2019)64.2552.8750.1644.7837.4828.7525.5822.9721.57
ProtoNet (Snell et al., 2017)64.2559.2758.8858.6958.2257.6357.0356.8056.47
ILVQ (Xu et al.,2012)64.2560.2459.6259.0258.6157.7157.1656.8356.49
SDC (Yu et al., 2020)64.6263.5862.7861.1260.2959.3759.0559.9759.87
Imprint (Qi et al.,2018)64.7163.5262.9662.1361.1761.2760.6359.8659.64
IDLVQ-C64.77 63.7763.2262.4461.2261.4760.9760.6660.44
Novel classessessions
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Fine-tune4.074.493.751.931.531.771.611.36
Joint train-4.809.8413.2612.3014.0314.0115.2115.62
iCaRL (Rebuffi et al., 2017)=4.608.098.927.879.108.197.867.70
Rebalancing (Hou et al.,2019)5.298.239.438.008.938.818.838.27
ProtoNet (Snell et al., 2017)5.328.419.7911.4212.1411.6711.3911.35
ILVQ (Xu et al.,2012)5.259.2910.4712.0912.3511.8611.4511.34
SDC (Yu et al., 2020)12.2311.0510.1212.3311.4610.8112.5813.30
Imprint (Qi et al., 2018)15.8112.2113.8316.0913.8112.4512.9313.69
IDLVQ-C-13.0712.1913.3415.8614.1412.5513.1113.94
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Methodsessions
1234567891011
Fine-tune77.3047.1636.3426.9224.0821.2417.1914.3112.7311.7511.43
Joint train77.3073.2971.5468.7266.3865.4264.9865.7464.8264.4764.16
iCaRL (Rebuffi et al., 2017)77.3059.6656.2452.2648.7746.3744.5443.1742.3541.1940.92
Rebalancing (Hou et al., 2019)77.3064.5358.3553.8249.2747.1245.1643.0542.3741.0240.86
ProtoNet (Snell et al., 2017)77.3069.8266.1263.1961.1758.8558.0457.7555.8455.8255.60
ILVQ (Xu et al., 2012)77.3071.2666.8463.8262.6659.7158.9258.1156.3156.1456.03
SDC (Yu et al., 2020)77.3474.6769.7366.7166.4962.1461.3359.8458.0157.3956.62
Imprint (Qi et al.,2018)77.0274.0770.2666.8464.4562.4661.8561.0259.2058.9358.43
IDLVQ-C77.3774.7970.9668.0865.9464.1263.5862.9860.8560.5459.72
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Methodsessions
1234567891011
Fine-tune77.3047.6836.7527.0124.5621.7217.8815.2412.8411.8911.52
Joint train77.3074.3472.5970.8570.1470.0169.6969.6769.4869.4169.42
iCaRL (Rebuff et al.,2017)77.3068.8563.5660.3457.7156.2855.9755.0254.6252.2152.23
Rebalancing (Hou et al.,2019)77.3069.3665.4961.3259.3058.6858.7757.4256.2555.3354.17
ProtoNet (Snell et al., 2017)77.3070.8467.8264.8162.9561.7160.9860.7359.5159.3058.95
ILVQ (Xu et al., 2012)77.3071.5068.7766.1264.3162.8962.3162.0061.5259.7959.23
SDC (Yu et al.,2020)77.3474.6271.6368.6666.7565.2464.2163.6261.9761.5461.11
Imprint (Qi et al., 2018)77.0274.1470.7267.7565.8864.6364.2863.8262.0061.8461.40
IDLVQ-C77.3774.8472.0769.0567.2865.5165.1964.8462.7762.5561.96
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We", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 505, + 723 + ], + "score": 1.0, + "content": "choose 40, 15 and 15 reference vectors and targets for 1st, 2nd and 3rd tasks, respectively. Adding", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 720, + 488, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 488, + 733 + ], + "score": 1.0, + "content": "more reference vectors does not result in obvious improvement in accuracy in our experiments.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 699, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 110, + 84, + 496, + 341 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 110, + 84, + 496, + 341 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 110, + 84, + 496, + 341 + ], + "spans": [ + { + "bbox": [ + 110, + 84, + 496, + 341 + ], + "score": 0.971, + "type": "image", + "image_path": "5e1419f899b33ff87b466bb8184b86bfa4ee63793fee7e949fdeb2a29f2e20af.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 110, + 84, + 496, + 169.66666666666669 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 110, + 169.66666666666669, + 496, + 255.33333333333337 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 110, + 255.33333333333337, + 496, + 341.00000000000006 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 366, + 503, + 388 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 379 + ], + "score": 1.0, + "content": "Figure 2: Comparison results of different few-shot settings, evaluated with ResNet18 on CUB", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 376, + 138, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 138, + 390 + ], + "score": 1.0, + "content": "dataset", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "title", + "bbox": [ + 108, + 407, + 261, + 418 + ], + "lines": [ + { + "bbox": [ + 105, + 405, + 263, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 263, + 421 + ], + "score": 1.0, + "content": "A.6 VISUALIZATION OF IDLVQ-C", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 427, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 106, + 428, + 504, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 504, + 440 + ], + "score": 1.0, + "content": "We show the visualization of standard neural networks and IDLVQ-C with/without intra-class vari-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "ation loss in Fig. 3. MNIST dataset is used as a toy example for visualization. Classes 0-7 are old", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 462 + ], + "score": 1.0, + "content": "classes with sufficient training samples, and classes 8 and 9 are novel classes with few-shot train-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 461, + 504, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 504, + 473 + ], + "score": 1.0, + "content": "ing samples. It can be observed in Fig. 3(a) and 3(b) that standard neural networks and IDLVQ-C", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 505, + 484 + ], + "score": 1.0, + "content": "(without intra-class variation loss) trained by cross-entropy loss do not have compact intra-class", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 106, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "variation. Consequently, features of novel classes are more likely to overlap with old classes. 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Methodsessions
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Fine-tune77.3046.2334.7125.3523.1620.6516.2113.3211.9811.1710.76
Joint train77.3073.2868.8065.3463.7562.0060.8159.7159.0658.6958.23
iCaRL (Rebuffi et al.,2017)77.3057.1854.6748.1140.7636.8533.1230.4228.2226.8425.23
Rebalancing (Hou et al., 2019)77.3064.5356.1447.2938.9234.3931.0427.9327.1224.4623.61
ProtoNet (Snell et al., 2017)77.3069.7666.0162.2959.5857.1055.1354.0952.4051.6550.36
ILVQ (Xu et al., 2012)77.3071.5066.7962.7160.2057.8455.2755.0652.4251.7250.47
SDC (Yu et al., 2020)77.3474.4569.4565.2761.8158.2656.1455.7153.3152.7951.52
Imprint (Qi et al., 2018)77.0273.3969.5065.6162.8160.7459.3958.6156.8555.9354.82
IDLVQ-C77.3774.7270.2867.1365.3463.5262.1061.5459.0458.6857.81
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Methodsessions
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Fine-tune64.2530.1118.536.312.862.681.871.561.42
Joint train64.2558.8055.2652.3849.7148.3745.9144.6843.38
iCaRL (Rebuffi et al., 2017)64.2548.0443.1338.2830.0124.4621.8519.8417.76
Rebalancing (Hou et al.,2019)64.2549.2144.1737.7130.1122.9219.9917.9616.25
ProtoNet (Snell et al., 2017)64.2555.1251.6748.9146.5244.2541.9140.0738.42
ILVQ (Xu et al., 2012)64.2556.0152.4349.3146.9844.3742.0640.1138.43
SDC (Yu et al., 2020)64.6259.6355.3950.9248.3045.2842.9742.5141.24
Imprint (Qi et al., 2018)64.7159.8555.7152.4749.9047.3144.5742.5741.26
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Methodsessions
234567891011
No tuning71.9367.1464.2162.6160.1359.0458.4755.6454.2553.66
W.o.Lintra74.7570.2666.8965.0563.1861.8461.3658.6158.1457.24
w.0. LF73.8569.5466.2164.0262.7460.2859.4956.9756.3855.46
w.o. δ74.6770.0166.7464.8163.9061.4260.7358.1657.6256.79
LM→LCE73.2269.4166.0363.9363.0761.1460.9858.6758.1157.32
IDLVQ-C74.7270.2867.1365.3463.5262.1061.5459.0458.6857.81
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Methodsessions
123
Joint train offline0.02174(2e-4)0.02232(2e-4)0.02296(2e-4)
Fine-tune w.novel data0.02174(2e-4)0.08462(4e-4)0.11870(6e-4)
Fine-tune w. exemplars0.02174(2e-4)0.02988(2e-4)0.03128(2e-4)
IDLVQ-R0.02181(2e-4)0.02641(2e-4)0.02817(2e-4)
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In the base task(t = 1)
Initialize 01,{m𝑖,.,mN1} and γ
Minimize L = LcE + XintraLintra W.r.t. θ1,{ml,.,mN1} and γ
Pick exemplars from D1 for classes in the base task: xi = arg minxeD1 |lft-1(x) - m1|l²
for novel task t = 2,3,... do
Initialize {mt-1+1.*., mNt}
Minimize L=LM +XFLF + XintraLintra W.r.t. 0t and {mNt-1+1,.,mNt}
t-1 t-1 +
Pick exemplars from Dt for classes in the novel task t: x' = arg minx∈Dt |lfet-1(x) - m ll²
end for
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In the base task (t =1) Initializeθ,{m1,..,mN1},{q],..,qN1} and γ
Minimize Lmse w.r.t. θ,{ml,..,mN1},{ql,.,qN1} and γ
for novel task t = 2,3,.. do
Initialize {mNt-1+1,, mNt} and {qNt-1+1.…,qNt}
Minimize LmsE w.r.t. {mNt-1+1..*,mNt} and {qNt-1+1,.,qNt}
end for
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Base classessessions
1234567891011
Fine-tune77.3044.2336.2827.5225.9623.0517.6813.0711.7810.9910.71
Joint train77.3075.8375.2574.5174.5873.7473.9573.2573.1173.2573.18
iCaRL (Rebuffi et al., 2017)77.3059.3858.8154.4348.2743.2839.1734.9132.4329.3625.87
Rebalancing (Hou et al.,2019)77.3066.4360.3255.3646.3941.7637.1232.5831.2627.0324.25
ProtoNet (Snell et al., 2017)77.3072.5572.2172.0671.6471.2971.0270.9470.6770.6070.53
ILVQ (Xu et al., 2012)77.3074.1873.5772.6672.5671.5771.1471.1271.0270.9870.85
SDC (Yu et al., 2020)77.3476.0575.2174.1272.3671.8171.6871.4371.2571.2770.96
Imprint (Qi et al., 2018)77.0274.7674.5773.6972.6970.8870.3470.1270.0769.8469.27
IDLVQ-C77.3776.3275.9075.9175.4974.8674.5874.3774.0273.3973.32
Novel classessessions
1234567891011
Fine-tune=66.2326.8618.1216.1615.8513.7613.6812.2311.3710.81
Joint train=47.7836.5534.7736.6838.5238.9140.3741.5042.5143.28
iCaRL (Rebuffi et al., 2017)-35.1833.9727.0421.9923.9923.0424.0122.9624.0424.59
Rebalancing (Hou et al.,2019)=45.5335.2420.3920.2519.6520.9121.2921.9521.6022.97
ProtoNet (Snell et al., 2017)41.8635.0129.7229.4328.7228.6530.0229.5630.5930.19
ILVQ (Xu et al., 2012)40.7432.8929.5429.3030.3828.8232.1229.1730.3230.09
SDC (Yu et al., 2020)58.4540.6535.7733.2331.1630.2433.2530.8932.2632.08
Imprint (Qi et al.,2018)59.6944.1538.6838.1140.4641.1442.1740.3340.4740.37
IDLVQ-C=58.7242.1837.8639.9740.8441.3043.2140.3242.3442.30
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Methodsessions
1234567891011
Fine-tune77.3047.6836.7527.0124.5621.7217.8815.2412.8411.8911.52
Joint train77.3074.3472.5970.8570.1470.0169.6969.6769.4869.4169.42
iCaRL (Rebuff et al.,2017)77.3068.8563.5660.3457.7156.2855.9755.0254.6252.2152.23
Rebalancing (Hou et al.,2019)77.3069.3665.4961.3259.3058.6858.7757.4256.2555.3354.17
ProtoNet (Snell et al., 2017)77.3070.8467.8264.8162.9561.7160.9860.7359.5159.3058.95
ILVQ (Xu et al., 2012)77.3071.5068.7766.1264.3162.8962.3162.0061.5259.7959.23
SDC (Yu et al.,2020)77.3474.6271.6368.6666.7565.2464.2163.6261.9761.5461.11
Imprint (Qi et al., 2018)77.0274.1470.7267.7565.8864.6364.2863.8262.0061.8461.40
IDLVQ-C77.3774.8472.0769.0567.2865.5165.1964.8462.7762.5561.96
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Methodsessions
1234567891011
Fine-tune77.3047.1636.3426.9224.0821.2417.1914.3112.7311.7511.43
Joint train77.3073.2971.5468.7266.3865.4264.9865.7464.8264.4764.16
iCaRL (Rebuffi et al., 2017)77.3059.6656.2452.2648.7746.3744.5443.1742.3541.1940.92
Rebalancing (Hou et al., 2019)77.3064.5358.3553.8249.2747.1245.1643.0542.3741.0240.86
ProtoNet (Snell et al., 2017)77.3069.8266.1263.1961.1758.8558.0457.7555.8455.8255.60
ILVQ (Xu et al., 2012)77.3071.2666.8463.8262.6659.7158.9258.1156.3156.1456.03
SDC (Yu et al., 2020)77.3474.6769.7366.7166.4962.1461.3359.8458.0157.3956.62
Imprint (Qi et al.,2018)77.0274.0770.2666.8464.4562.4661.8561.0259.2058.9358.43
IDLVQ-C77.3774.7970.9668.0865.9464.1263.5862.9860.8560.5459.72
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Base classessessions
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Fine-tune64.2532.2820.876.953.173.161.921.531.46
Joint train64.2563.3062.8362.1662.1862.6861.8661.8761.89
iCaRL (Rebuffi et al., 2017)64.2551.6648.9745.6237.3930.8628.6826.8324.47
Rebalancing (Hou et al., 2019)64.2552.8750.1644.7837.4828.7525.5822.9721.57
ProtoNet (Snell et al., 2017)64.2559.2758.8858.6958.2257.6357.0356.8056.47
ILVQ (Xu et al.,2012)64.2560.2459.6259.0258.6157.7157.1656.8356.49
SDC (Yu et al., 2020)64.6263.5862.7861.1260.2959.3759.0559.9759.87
Imprint (Qi et al.,2018)64.7163.5262.9662.1361.1761.2760.6359.8659.64
IDLVQ-C64.77 63.7763.2262.4461.2261.4760.9760.6660.44
Novel classessessions
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Fine-tune4.074.493.751.931.531.771.611.36
Joint train-4.809.8413.2612.3014.0314.0115.2115.62
iCaRL (Rebuffi et al., 2017)=4.608.098.927.879.108.197.867.70
Rebalancing (Hou et al.,2019)5.298.239.438.008.938.818.838.27
ProtoNet (Snell et al., 2017)5.328.419.7911.4212.1411.6711.3911.35
ILVQ (Xu et al.,2012)5.259.2910.4712.0912.3511.8611.4511.34
SDC (Yu et al., 2020)12.2311.0510.1212.3311.4610.8112.5813.30
Imprint (Qi et al., 2018)15.8112.2113.8316.0913.8112.4512.9313.69
IDLVQ-C-13.0712.1913.3415.8614.1412.5513.1113.94
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+Kaiyang Zhou1, Yongxin Yang1, Yu Qiao2, Tao Xiang1 + +1University of Surrey, UK +2Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China +k.zhou.vision@gmail.com +{yongxin.yang, t.xiang}@surrey.ac.uk +yu.qiao@siat.ac.cn + +# ABSTRACT + +Though convolutional neural networks (CNNs) have demonstrated remarkable ability in learning discriminative features, they often generalize poorly to unseen domains. Domain generalization aims to address this problem by learning from a set of source domains a model that is generalizable to any unseen domain. In this paper, a novel approach is proposed based on probabilistically mixing instancelevel feature statistics of training samples across source domains. Our method, termed MixStyle, is motivated by the observation that visual domain is closely related to image style (e.g., photo vs. sketch images). Such style information is captured by the bottom layers of a CNN where our proposed style-mixing takes place. Mixing styles of training instances results in novel domains being synthesized implicitly, which increase the domain diversity of the source domains, and hence the generalizability of the trained model. MixStyle fits into mini-batch training perfectly and is extremely easy to implement. The effectiveness of MixStyle is demonstrated on a wide range of tasks including category classification, instance retrieval and reinforcement learning. + +# 1 INTRODUCTION + +Key to automated understanding of digital images is to compute a compact and informative feature representation. Deep convolutional neural networks (CNNs) have demonstrated remarkable ability in representation learning, proven to be effective in many visual recognition tasks, such as classifying photo images into 1,000 categories from ImageNet (Krizhevsky et al., 2012) and playing Atari games with reinforcement learning (Mnih et al., 2013). However, it has long been discovered that the success of CNNs heavily relies on the i.i.d. assumption, i.e. training and test data should be drawn from the same distribution; when such an assumption is violated even just slightly, as in most realworld application scenarios, severe performance degradation is expected (Hendrycks & Dietterich, 2019; Recht et al., 2019). + +Domain generalization (DG) aims to address such a problem (Zhou et al., 2021; Blanchard et al., 2011; Muandet et al., 2013; Li et al., 2018a; Zhou et al., 2020b; Balaji et al., 2018; Dou et al., 2019; Carlucci et al., 2019). In particular, assuming that multiple source domains containing the same visual classes are available for model training, the goal of DG is to learn models that are robust against data distribution changes across domains, known as domain shift, so that the trained model can generalize well to any unseen domains. Compared to the closely related and more widely studied domain adaptation (DA) problem, DG is much harder in that no target domain data is available for the model to analyze the distribution shift in order to overcome the negative effects. Instead, a DG model must rely on the source domains and focus on learning domain-invariant feature representation in the hope that it would remain discriminative given target domain data. + +A straightforward solution to DG is to expose a model with a large variety of source domains. Specifically, the task of learning domain-invariant and thus generalizable feature representation becomes easier when data from more diverse source domains are available for the model. This would reduce the burden on designing special models or learning algorithms for DG. Indeed, model training with large-scale data of diverse domains is behind the success of existing commercial face recognition or vision-based autonomous driving systems. A recent work by $\mathrm { X u }$ et al. (2021) also emphasizes the importance of diverse training distributions for out-of-distribution generalization. However, collecting data of a large variety of domains is often costly or even impossible. It thus cannot be a general solution to DG. + +![](images/f1cd7469df889b614caa015810343d6c004d2bbf757f1fbaacd0388b9ffe208e.jpg) +Figure 1: 2-D t-SNE (Maaten & Hinton, 2008) visualization of the style statistics (concatenation of mean and standard deviation) computed from the first residual block’s feature maps of a ResNet-18 (He et al., 2016) trained on four distinct domains (Li et al., 2017). It is clear that different domains are well separated. + +In this paper, a novel approach is proposed based on probabilistically mixing instance-level feature statistics of training samples across source domains. Our model, termed MixStyle, is motivated by the observation that visual domain is closely related to image style. An example is shown in Fig. 1: the four images from four different domains depict the same semantic concept, i.e. dog, but with distinctive styles (e.g., characteristics in color and texture). When these images are fed into a deep CNN, which maps the raw pixel values into category labels, such style information is removed at the output. However, recent style transfer studies (Huang & Belongie, 2017; Dumoulin et al., 2017) suggest that such style information is preserved at the bottom layers of the CNN through the instance-level feature statistics, as shown clearly in Fig. 4. Importantly, since replacing such statistics would lead to replaced style while preserving the semantic content of the image, it is reasonable to assume that mixing styles from images of different domains would result in images of (mixed) new styles. That is, more diverse domains/styles can be made available for training a more domain-generalizable model. + +Concretely, our MixStyle randomly selects two instances of different domains and adopts a probabilistic convex combination between instance-level feature statistics of bottom CNN layers. In contrast to style transfer work (Huang & Belongie, 2017; Dumoulin et al., 2017), no explicit image synthesis is necessary meaning much simpler model design. Moreover, MixStyle perfectly fits into modern mini-batch training. Overall, it is very easy to implement with only few lines of code. To evaluate the effectiveness as well as the general applicability of MixStyle, we conduct extensive experiments on a wide spectrum of datasets covering category classification (Sec. 3.1), instance retrieval (Sec. 3.2), and reinforcement learning (Sec. 3.3). The results demonstrate that MixStyle can significantly improve CNNs’ cross-domain generalization performance.1 + +# 2 METHODOLOGY + +# 2.1 BACKGROUND + +Normalizing feature tensors with instance-specific mean and standard deviation has been found effective for removing image style in style transfer models (Ulyanov et al., 2016; Huang & Belongie, 2017; Dumoulin et al., 2017). Such an operation is widely known as instance normalization (IN, Ulyanov et al. (2016)). Let $\boldsymbol { x } ^ { \prime } \in \mathbb { R } ^ { B \times C \times \dot { H } \times W }$ be a batch of tensors, with $B , C , H$ and $W$ denoting the dimension of batch, channel, height and width, respectively, IN is formulated as + +$$ +\mathrm { I N } ( x ) = \gamma \frac { x - \mu ( x ) } { \sigma ( x ) } + \beta , +$$ + +where $\gamma , \beta \in \mathbb { R } ^ { C }$ are learnable affine transformation parameters, and $\mu ( \boldsymbol { x } ) , \sigma ( \boldsymbol { x } ) \in \mathbb { R } ^ { B \times C }$ are mean and standard deviation computed across the spatial dimension within each channel of each instance (tensor), i.e. + +$$ +\mu ( x ) _ { b , c } = \frac { 1 } { H W } \sum _ { h = 1 } ^ { H } \sum _ { w = 1 } ^ { W } x _ { b , c , h , w } , +$$ + +and + +$$ +\sigma ( x ) _ { b , c } = \sqrt { \frac { 1 } { H W } \sum _ { h = 1 } ^ { H } \sum _ { w = 1 } ^ { W } ( x _ { b , c , h , w } - \mu ( x ) _ { b , c } ) ^ { 2 } } . +$$ + +Huang & Belongie (2017) introduced adaptive instance normalization (AdaIN), which simply replaces the scale and shift parameters in Eq. (1) with the feature statistics of style input $y$ to achieve arbitrary style transfer: + +$$ +\mathrm { A d a I N } ( x ) = \sigma ( y ) \frac { x - \mu ( x ) } { \sigma ( x ) } + \mu ( y ) . +$$ + +# 2.2 MIXSTYLE + +Our method, MixStyle, draws inspiration from AdaIN. However, rather than attaching a decoder for image generation, MixStyle is designed for the purpose of regularizing CNN training by perturbing the style information of source domain training instances. It can be implemented as a plug-andplay module inserted between CNN layers of, e.g., a supervised CNN classifier, without the need to explicitly generate an image of new style. + +More specifically, MixStyle mixes the feature statistics of two instances with a random convex weight to simulate new styles. In terms of implementation, MixStyle can be easily integrated into mini-batch training. Given an input batch $x$ , MixStyle first generates a reference batch $\tilde { x }$ from $x$ . When domain labels are given, $x$ is sampled from two different domains $i$ and $j$ , e.g., $\bar { \boldsymbol { x } } = [ x ^ { i } , x ^ { j } ]$ $\bar { x ^ { i } }$ and $x ^ { j }$ have the same batch size). Then, $\tilde { x }$ is obtained by swapping the position of $x ^ { i }$ and $x ^ { j }$ , followed by a shuffling operation along the batch dimension applied to each batch, i.e. $\mathbf { \widetilde { \boldsymbol { x } } } = [ \mathrm { S h u f f e } ( \boldsymbol { x } ^ { j } ) , \mathrm { S h u f f e } ( \boldsymbol { x } ^ { i } ) ]$ . See Fig. 2(a) for an illustration. In cases where domain labels are unknown, $x$ is randomly sampled from the training data, and $\tilde { x }$ is simply obtained by $\tilde { x } = \mathrm { S h u f f e } ( x )$ (see Fig. 2(b)). Fig. 4 shows that sub-domains exist within each domain, so even if two instances of the same domain are sampled, new domain could be synthesized. After shuffling, MixStyle computes the mixed feature statistics by + +![](images/328a14b30cdf68b6ebf3e522fa5e10ef00672d004ed56b5521dcee7294f42db8.jpg) +Figure 2: A graphical illustration of how a reference batch is generated. Domain label is denoted by color. + +$$ +\begin{array} { r } { \gamma _ { m i x } = \lambda \sigma ( x ) + ( 1 - \lambda ) \sigma ( \tilde { x } ) , } \\ { \beta _ { m i x } = \lambda \mu ( x ) + ( 1 - \lambda ) \mu ( \tilde { x } ) , } \end{array} +$$ + +where $\lambda \in \mathbb { R } ^ { B }$ are instance-wise weights sampled from the Beta distribution, $\lambda \sim B e t a ( \alpha , \alpha )$ with $\alpha \in ( 0 , \infty )$ being a hyper-parameter. Unless specified otherwise, we set $\alpha$ to 0.1 throughout this paper. Finally, the mixed feature statistics are applied to the style-normalized $x$ , + +$$ +{ \mathrm { M i x S t y l e } } ( x ) = \gamma _ { m i x } { \frac { x - \mu ( x ) } { \sigma ( x ) } } + \beta _ { m i x } . +$$ + +In practice, we use a probability of 0.5 to decide if MixStyle is activated or not in the forward pass. At test time, no MixStyle is applied. Note that gradients are blocked in the computational graph of $\mu ( \cdot )$ and $\sigma ( \cdot )$ . MixStyle can be implemented with only few lines of code. See Algorithm 1 in Appendix A.1 for the PyTorch-like pseudo-code. + +Table 1: Leave-one-domain-out generalization results on PACS. + +
MethodArtCartoonPhotoSketchAvg
MMD-AAE CCSA JiGen75.2 80.5 79.4 79.8 82.172.7 76.9 75.3 76.896.0 93.6 96.0 96.064.2 66.8 71.6 70.277.0 79.4 80.5 80.7
Epi-FCR Metareg L2A-OT ResNet-1883.7 83.3 77.0±0.677.0 77.2 78.2 75.9±0.693.9 95.5 96.2 96.0±0.173.0 70.3 73.6 69.2±0.681.5 81.7 82.8 79.5
+ Manifold Mixup75.6±0.770.1±0.993.5±0.765.4±0.676.2
+ Cutout74.9±0.474.9±0.695.9±0.367.7±0.978.3
+ CutMix74.6±0.771.8±0.695.6±0.465.3±0.876.8
74.7±1.072.3±0.9
+ Mixup (w/o label interpolation)93.0±0.469.2±0.277.3
+ Mixup76.8±0.774.9±0.795.8±0.366.6±0.778.5
+ DropBlock76.4±0.775.4±0.795.9±0.369.0±0.379.2
+ MixStyle w/ random shuffle82.3±0.279.0±0.396.3±0.373.8±0.982.8
+ MixStyle w/ domain label84.1±0.478.8±0.496.1±0.375.9±0.9
83.7
+ +# 3 EXPERIMENTS + +# 3.1 GENERALIZATION IN CATEGORY CLASSIFICATION + +Dataset and implementation details. We choose the PACS dataset (Li et al., 2017), a commonly used domain generalization (DG) benchmark concerned with domain shift in image classification. PACS consists of four domains, i.e. Art Painting, Cartoon, Photo and Sketch, with totally 9,991 images of 7 classes. As shown in Fig. 1, the domain shift mainly corresponds to image style changes. For evaluation, a model is trained on three domains and tested on the remaining one. Following prior work (Li et al., 2019; Zhou et al., 2020a), we use ResNet-18 (He et al., 2016) as the classifier where MixStyle is inserted after the 1st, 2nd and 3rd residual blocks. Our code is based on Dassl.pytorch (Zhou et al., 2020c).2 + +Baselines. Our main baselines are general-purpose regularization methods including Mixup (Zhang et al., 2018b), Manifold Mixup (Verma et al., 2019), DropBlock (Ghiasi et al., 2018), CutMix (Yun et al., 2019) and Cutout (DeVries & Taylor, 2017), which are trained using the same training parameters as MixStyle and the optimal hyper-parameter setup as reported in their papers. We also compare with the existing DG methods which reported state-of-the-art performance on PACS. These include domain alignment-based CCSA (Motiian et al., 2017) and MMD-AAE (Li et al., 2018b), Jigsaw puzzle-based JiGen (Carlucci et al., 2019), adversarial gradient-based CrossGrad (Shankar et al., 2018), meta-learning-based Metareg (Balaji et al., 2018) and Epi-FCR (Li et al., 2019), and data augmentation-based L2A-OT (Zhou et al., 2020a). + +Comparison with general-purpose regularization methods. The results are shown in Table 1. Overall, we observe that the general-purpose regularization methods do not offer any clear advantage over the vanilla ResNet-18 in this DG task, while MixStyle improves upon the vanilla ResNet18 with a significant margin. Compared with Mixup, MixStyle is $5 . 2 \%$ better on average. Recall that Mixup also interpolates the output space, we further compare with a variant of Mixup in order to demonstrate the advantage of mixing style statistics at the feature level over mixing images at the pixel level for DG—following Sohn et al. (2020), we remove the label interpolation in Mixup and sample the mixing weights from a uniform distribution of $[ 0 , 1 ]$ . Still, MixStyle outperforms this new baseline with a large margin, which justifies our claim. MixStyle and DropBlock share some commonalities in that they are both applied to feature maps at multiple layers, but MixStyle significantly outperforms DropBlock in all test domains. The reason why DropBlock is ineffective here is because dropping out activations mainly encourages a network to mine discriminative patterns, but does not reinforce the ability to cope with unseen styles, which is exactly what MixStyle aims to achieve: by synthesizing “new” styles (domains) MixStyle regularizes the network to become more robust to domain shift. In addition, it is interesting to see that on Cartoon and Photo, MixStyle w/ random shuffle obtains slightly better results. The reason might be because there exist sub-domains in a source domain (see Fig. 4(a-c)), which allow random shuffling to produce more diverse “new” domains that lead to a more domain-generalizable model. + +Comparison with state-of-the-art DG methods. Overall, MixStyle outperforms most DG methods by a clear margin, despite being a much simpler method. The performance of MixStyle w/ domain label is nearly $1 \%$ better on average than the recently introduced L2A-OT. From a data augmentation perspective, MixStyle and L2A-OT share a similar goal—to synthesize data from pseudo-novel domains. MixStyle accomplishes this goal through mixing style statistics at the feature level. Whereas L2A-OT works at the pixel level: it trains an image generator by maximizing the domain difference (measured by optimal transport) between the original and the generated images, which introduces much heavier computational overhead than MixStyle in terms of GPU memory and training time. It is worth noting that MixStyle’s domain label-free version is highly competitive: its $8 2 . 8 \%$ accuracy is on par with L2A-OT’s. + +# 3.2 GENERALIZATION IN INSTANCE RETRIEVAL + +Dataset and implementation details. We evaluate MixStyle on the person re-identification (re-ID) problem, which aims to match people across disjoint camera views. As each camera view is itself a distinct domain, person re-ID is essentially a cross-domain image matching problem. Instead of using the standard protocol where training and test data come from the same camera views, we adopt the cross-dataset setting so test camera views are never seen during training. Specifically, we train a model on one dataset and then test its performance on the other dataset. Two commonly used re-ID datasets are adopted: Market1501 (Zheng et al., 2015) and Duke (Ristani et al., 2016; Zheng et al., 2017). Ranking accuracy and mean average precision (mAP) are used as the performance measures (displayed in percentage). We test MixStyle on two CNN architectures: ResNet-50 (He et al., 2016) and OSNet (Zhou et al., 2019). The latter was designed specifically for re-ID. In both architectures, MixStyle is inserted after the 1st and 2nd residual blocks. Our code is based on Torchreid (Zhou & Xiang, 2019).3 + +Baselines. We compare with three baseline methods: 1) The vanilla model, which serves as a strong baseline; 2) DropBlock, which was the top-performing competitor in Table 1; 3) RandomErase (Zhong et al., 2020), a widely used regularization method in the re-ID literature (similar to Cutout). + +Results. The results are reported in Table 2. It is clear that only MixStyle consistently outperforms the strong vanilla model under both settings with considerable margins, while DropBlock and RandomErase are unable to show any benefit. Notably, RandomErase, which simulates occlusion by erasing pixels in random rectangular regions with random values, has been used as a default trick when training re-ID CNNs. However, RandomErase shows a detrimental effect in the cross-dataset re-ID setting. Indeed, similar to DropBlock, randomly erasing pixels offers no guarantee to improve the robustness when it comes to domain shift. + +# 3.3 GENERALIZATION IN REINFORCEMENT LEARNING + +Though RL has been greatly advanced by using CNNs for feature learning in raw pixels (Mnih et al., 2013), it has been widely acknowledged that RL agents often overfit training environments while generalize poorly to unseen environments (Cobbe et al., 2019; Igl et al., 2019). + +Dataset and implementation details. We conduct experiments on Coinrun (Cobbe et al., 2019), a recently introduced RL benchmark for evaluating the generalization performance of RL agents. As shown in Fig. 3(a), the goal in Coinrun is to control a character to collect golden coins while avoiding both stationary and non-stationary obstacles. We follow Igl et al. (2019) to construct and train our RL agent: the CNN architecture used in IMPALA (Espeholt et al., 2018) is adopted as the policy network, and is trained by the Proximal Policy Optimization (PPO) algorithm (Schulman et al., 2017). Please refer to Igl et al. (2019) for further implementation details. MixStyle is inserted after the 1st and 2nd convolutional sequences. Training data are sampled from 500 levels while test data are drawn from new levels of only the highest difficulty. As domain labels are difficult to define, we use the random shuffle version of MixStyle. Our code is built on top of Igl et al. (2019).4 + +Table 2: Generalization results on the cross-dataset person re-ID task. + +
ModelMarket1501-→DukeDuke-→Market1501
mAPR1R5R10mAPR1R5R10
ResNet-5019.335.450.356.420.445.263.670.9
+ RandomErase14.327.842.649.116.138.556.864.5
+ DropBlock18.233.249.156.319.745.362.169.1
+ MixStyle w/ random shuffle23.842.258.864.824.151.569.476.2
+ MixStyle w/ domain label23.443.358.964.724.753.070.977.8
OSNet25.944.759.665.424.052.267.574.7
+ RandomErase20.536.252.359.322.449.166.173.0
+ DropBlock23.141.556.562.521.748.265.471.3
+ MixStyle w/ random shuffle27.248.262.768.427.858.174.081.0
+ MixStyle w/ domain label27.347.562.067.129.058.274.980.9
+ +![](images/83f0338beadfd5bc519830499778de6828846e5597dc084ac87299494b1148a1.jpg) +Figure 3: (a) Coinrun benchmark. (b) Test performance in unseen environments. (c) Difference between training and test performance. + +Baselines. Following Igl et al. (2019), we train strong baseline models and add MixStyle on top of them to see whether MixStyle can bring further improvements. To this end, we train two baseline models: 1) Baseline, which combines weight decay and data augmentation;5 2) IBAC-SNI (the $\lambda = 0 . 5$ version), the best-performing model in Igl et al. (2019) which is based on selective noise injection. + +Results. The test performance is shown in Fig. 3(b). Comparing Baseline (blue) with Baseline+MixStyle (orange), we can see that MixStyle brings a significant improvement. Interestingly, the variance is also significantly reduced by using MixStyle, as indicated by the smaller shaded areas (for both orange and red lines). These results strongly demonstrate the effectiveness of MixStyle in enhancing generalization for RL agents. When it comes to the stronger baseline IBAC-SNI (green), MixStyle (red) is able to bring further performance gain, suggesting that MixStyle is complementary to IBAC-SNI. This result also shows the potential of MixStyle as a plug-and-play component to be combined with other advanced RL methods. It is worth noting that Baseline+MixStyle itself is already highly competitive with IBAC-SNI. Fig. 3(c) shows the generalization gap from which it can been seen that the models trained with MixStyle (orange & red) clearly generalize faster and better than those without using MixStyle (blue & green). + +![](images/4d2d1d46181970a60036d74149364fc1b7c14abba9cfcdf85a1601e0d521acda.jpg) +Figure 4: 2-D visualization of flattened feature maps (top) and the corresponding style statistics (bottom). res1-4 denote the four residual blocks in order in a ResNet architecture. We observe that res1 to res3 contain domain-related information while res4 encodes label-related information. + +Table 3: Ablation study on where to apply MixStyle in the ResNet architecture. +(a) Category classification on PACS. + +
ModelAccuracy
ResNet-1879.5
+ MixStyle (res1)80.1
+ MixStyle (res12)81.6
+ MixStyle (res123)82.8
+ MixStyle (res1234)75.6
+ MixStyle (res14)76.3
+ MixStyle (res23)81.7
+ +(b) Cross-dataset person re-ID. + +
ModelmAP
ResNet-5019.3
+ MixStyle (res1)22.6
+ MixStyle (res12) + MixStyle (res123)23.8
+MixStyle (res1234)22.0
+ MixStyle (res14)10.2
+MixStyle (res23)11.1 20.6
+ +# 3.4 ANALYSIS + +Where to apply MixStyle? We repeat the experiments on PACS (category classification) and the re-ID datasets (instance retrieval) using the ResNet architecture. Given that a standard ResNet model has four residual blocks denoted by $\mathtt { r e s l - 4 }$ , we train different models with MixStyle applied to different layers. For notation, res1 means MixStyle is applied after the first residual block; res12 means MixStyle is applied after both the first and second residual blocks; and so forth. The results are shown in Table 3. We have the following observations. 1) Applying MixStyle to multiple lowerlevel layers generally achieves a better performance—for instance, res12 is better than res1 on both tasks. 2) Different tasks favor different combinations—res123 achieves the best performance on PACS, while on the re-ID datasets $\mathtt { r e s 1 2 }$ is the best. 3) On both tasks, the performance plunges when applying MixStyle to the last residual block. This makes sense because res4 is the closest to the prediction layer and tends to capture semantic content (i.e. label-sensitive) information rather than style. In particular, res4 is followed by an average-pooling layer, which essentially forwards the mean vector to the prediction layer and thus forces the mean vector to capture label-related information. As a consequence, mixing the statistics at res4 breaks the inherent label space. This is clearer in Fig. 4: the features and style statistics in $\mathtt { r e s l - 3 }$ exhibit clustering patterns based on domains while those in res4 have a high correlation with class labels. + +![](images/c7e15cbb6312d6d6e57a51169db1f4a01f9832c478ec13254c9df6c7db06a565.jpg) +Figure 5: Evaluation on the hyper-parameter $\alpha$ on (a) PACS, (b) person re-ID datasets and (c) Coinrun. In (b), M and D denote Market1501 and Duke respectively. + +Mixing vs. replacing. Unlike the AdaIN formulation, which completely replaces one style with the other, MixStyle mixes two styles via a convex combination. Table 4 shows that mixing is better than replacing. This is easy to understand: mixing diversifies the styles (imagine an interpolation between two data points). + +Random vs. fixed shuffle at multiple layers. Applying MixStyle to multiple layers, which has been shown advantageous in Table 3, raises another question of whether to shuffle the mini-batch at different layers or use the same shuffled order for all layers. Table 5 suggests that using random shuffle at different layers gives a better performance, which may be attributed to the increased noise level that gives a better regularization effect. + +Table 4: Mixing vs. replacing. + +
Accuracy (%)
Mixing82.8±0.4
Replacing82.1±0.5
+ +Table 5: Random vs. fixed shuffle at multiple layers. + +
Accuracy (%)
Random82.8±0.4
Fixed82.4±0.5
+ +Sensitivity of hyper-parameter. Recall that $\alpha$ is used to control the shape of Beta distribution, which has a direct effect on how the convex weights $\lambda$ are sampled. The smaller $\alpha$ is, the more likely the value in $\lambda$ is close to the extreme value of 0 or 1. In other words, a smaller $\alpha$ favors the style statistics in Eqs. (5) & (6) to be dominated by one side. We first evaluate $\alpha$ on PACS. Fig. 5(a) shows that with $\alpha$ increasing from 0.1 to 0.4, the accuracy slides from $8 2 . 8 \%$ to $8 1 . 7 \%$ . However, further increasing $\alpha$ does not impact on the accuracy. Therefore, the results suggest that the performance is not too sensitive to $\alpha$ ; and selecting $\alpha$ from $\lbrace 0 . 1 , 0 . 2 , 0 . 3 \rbrace$ seems to be a good starting point. We further experiment with $\alpha \in \{ 0 . 1 , 0 . 2 , 0 . 3 \}$ on the re-ID datasets and the Coinrun benchmark. Figs. 5(b) & (c) show that in general the variance for the results of different values is small. Therefore, we suggest practitioners to choose $\alpha$ from $\lbrace 0 . 1 , 0 . 2 , 0 . 3 \rbrace$ , with $\alpha = 0 . 1$ being a good default setting. + +For more analyses and discussions, please see Appendix A.2. + +# 4 RELATED WORK + +Domain generalization, or DG, studies out-of-distribution (OOD) generalization given only source data typically composed of multiple related but distinct domains. We refer readers to Zhou et al. (2021) for a comprehensive survey in this topic. Many DG methods are based on the idea of aligning features between different sources, with a hope that the model can be invariant to domain shift given unseen data. For instance, Li et al. (2018b) achieved distribution alignment in the hidden representation of an autoencoder using maximum mean discrepancy; Li et al. (2018c) resorted to adversarial learning with auxiliary domain classifiers to learn features that are domain-agnostic. Some works explored domain-specific parameterization, such as domain-specific weight matrices (Li et al., 2017) and domain-specific BN (Seo et al., 2020). Recently, meta-learning has drawn increasing attention from the DG community (Li et al., 2018a; Balaji et al., 2018; Dou et al., 2019). The main idea is to expose a model to domain shift during training by using pseudo-train and pseudo-test domains, both drawn from source domains. Data augmentation has also been investigated for learning domain-invariant models. Shankar et al. (2018) introduced a cross-gradient training method (CrossGrad) where source data are augmented by adversarial gradients obtained from a domain classifier. + +Gong et al. (2019) proposed DLOW (for the DA problem), which models intermediate domains between source and target via a domainness factor and learns an image translation model to generate intermediate-domain images. Very recently, Zhou et al. (2020a) introduced L2A-OT to learn a neural network to map source data to pseudo-novel domains by maximizing an optimal transport-based distance measure. Our MixStyle is related to DLOW and L2A-OT in its efforts to synthesizing novel domains. However, MixStyle differs in the fact that it is done implicitly with a much simpler formulation leveraging the feature-level style statistics and only few lines of extra code on top of a standard supervised classifier while being more effective. Essentially, MixStyle can be seen as feature-level augmentation, which is clearly different from the image-level augmentation-based DLOW and L2A-OT. + +Generalization in deep RL has been a challenging problem where RL agents often overfit training environments, and as a result, perform poorly in unseen environments with different visual patterns or levels (Zhang et al., 2018a). A natural way to improve generalization, which has been shown effective in (Cobbe et al., 2019; Farebrother et al., 2018), is to use regularization, e.g., weight decay. However, Igl et al. (2019) suggested that stochastic regularization methods like dropout and batch normalization (which uses estimated population statistics) have adverse effect as the training data in RL are essentially model-dependent. As such, they proposed selective noise injection (SNI), which basically combines a stochastic regularization technique with its deterministic counterpart. They further integrated SNI with information bottleneck actor critic (IBAC-SNI) to reduce the variance in gradients. Curriculum learning has been investigated in (Justesen et al., 2018) where the level of training episodes progresses from easy to difficult over the course of training. Gamrian & Goldberg (2019) leveraged the advances in GAN-based image-to-image translation (Liu et al., 2017) to map target data to the source domain which the agent was trained on. Tobin et al. (2017) introduced domain randomization, which diversifies training data by rendering images with different visual effects via a programmable simulator. With a similar goal of data augmentation, Lee et al. (2020) pre-processed input images with a randomly initialized network. Very recent studies (Laskin et al., 2020; Kostrikov et al., 2020) have shown that it is useful to combine a diverse set of label-preserving transformations, such as rotation, shifting and Cutout. Different from the aforementioned methods, our MixStyle works at the feature level and is orthogonal to most existing methods. For instance, we have shown in Sec. 3.3 that MixStyle significantly improves upon IBAC-SNI. + +# 5 CONCLUSION + +We presented a simple yet effective domain generalization method, termed MixStyle. MixStyle mixes the feature statistics of two instances to synthesize novel domains, which is inspired by the observation in style transfer work that the feature statistics encode style/domain-related information. Extensive experiments covering a wide range of tasks were conducted to demonstrate that MixStyle yields new state-of-the-art on three different tasks. + +# ACKNOWLEDGMENTS + +This work was supported in part by the Shanghai Committee of Science and Technology, China (Grant No. 20DZ1100800), in part by the National Natural Science Foundation of China under Grant (61876176, U1713208), the National Key Research and Development Program of China (No. 2020YFC2004800), and in part by the Guangzhou Research Program (201803010066). + +# REFERENCES + +Yogesh Balaji, Swami Sankaranarayanan, and Rama Chellappa. Metareg: Towards domain generalization using meta-regularization. In NeurIPS, 2018. + +Gilles Blanchard, Gyemin Lee, and Clayton Scott. Generalizing from several related classification tasks to a new unlabeled sample. In NeurIPS, 2011. + +Fabio M. Carlucci, Antonio D’Innocente, Silvia Bucci, Barbara Caputo, and Tatiana Tommasi. Domain generalization by solving jigsaw puzzles. 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Domain adaptive ensemble learning. arXiv preprint arXiv:2003.07325, 2020c. +Kaiyang Zhou, Ziwei Liu, Yu Qiao, Tao Xiang, and Chen Change Loy. Domain generalization: A survey. arXiv preprint arXiv:2103.02503, 2021. + +# A APPENDIX + +# A.1 PSEUDO-CODE OF MIXSTYLE + +Algorithm 1 provides a PyTorch-like pseudo-code. + +# Algorithm 1 PyTorch-like pseudo-code for MixStyle. + +# x: input features of shape (B, C, H, W) +# p: probabillity to apply MixStyle (default: 0.5) +# alpha: hyper-parameter for the Beta distribution (default: 0.1) +# eps: a small value added before square root for numerical stability (default: 1e-6) + +if not in training mode: return x + +if random probability $>$ p: return x + +$\mathrm { ~ \textit ~ { ~ B ~ } ~ } = \mathrm { ~ \textit ~ { ~ x ~ } ~ }$ .size(0) # batch size + +mu $= \times$ .mean(dim=[2, 3], keepdim $\cdot ^ { = }$ True) # compute instance mean var $=$ x.var(dim=[2, 3], keepdim $\cdot ^ { = }$ True) # compute instance variance sig $=$ (var $^ +$ eps).sqrt() # compute instance standard deviation mu, sig $\qquad = \quad \mathtt { m u }$ .detach(), sig.detach() # block gradients x_normed $=$ (x - mu) / sig # normalize input + +lmda $=$ Beta(alpha, alpha).sample((B, 1, 1, 1)) # sample instance-wise convex weights + +if domain label is given: # in this case, input $\begin{array} { r } { \mathrm { ~ ~ \underline { ~ } { ~ \sf ~ } { ~ \sf ~ } { ~ \sf ~ } { ~ \sf ~ } = ~ \mathrm { ~ ~ \tau ~ } [ \times \hat { \mathrm { ~ ~ \underline { ~ } { ~ \bf ~ } { ~ \underline { ~ } { ~ \bf ~ } { ~ \tau ~ } ~ } } } } } \end{array}$ x perm $=$ torch.arange(B-1, -1, -1) # inverse index perm_j, perm_i $=$ perm.chunk(2) # separate indices perm_j $=$ perm_j[torch.randperm(B // 2)] # shuffling perm_i $=$ perm_i[torch.randperm(B // 2)] # shuffling perm $=$ torch.cat([perm_j, perm_i], 0) # concatenation +else: perm $=$ torch.randperm(B) # generate shuffling indices + +mu2, sig2 $=$ mu[perm], sig[perm] # shuffling mu_mix $=$ mu $\star$ lmda $^ +$ mu2 $\star$ (1 - lmda) # generate mixed mean sig_mix $=$ sig $\star$ lmda $^ +$ sig2 $\star$ (1 - lmda) # generate mixed standard deviation return x_normed $\star$ sig_mix $^ +$ mu_mix # denormalize input using the mixed statistics + +Table 7: Test results on the source domains on PACS. A: Art. C: Cartoon. P: Photo. S: Sketch. + +
MethodC,P,SA,P,SA,C,SA,C,PAvg
Vanilla99.49±0.0399.47±0.0499.38±0.0299.65±0.0399.50
MixStyle99.55±0.0299.54±0.0199.47±0.0399.68±0.0399.56
+ +Table 8: Leave-one-domain-out generalization results on Digits-DG. + +
MethodMNIST MNIST-M SVHN SYNAvg
JiGen96.5 61.4 63.7 74.073.9
CCSA95.2 58.2 65.5 79.174.5
MMD-AAECrossGradL2A-OT96.5 58.4 65.0 78.474.675.878.1
96.7 61.1 65.3 80.2
96.7 63.9 68.6 83.2
CNN+ Mixup w/o label interpolation+ Manifold Mixup+ CutMix+ Mixup+ Cutout+ DropBlock+ MixStyle (ours)95.8±0.3 58.8±0.5 61.7±0.5 78.6±0.673.7
93.7±0.6 55.2±1.0 61.6±0.9 74.4±0.871.271.771.8
92.7±0.4 53.1±0.8 64.4±0.2 76.8±0.5
94.9±0.2 50.1±0.5 64.1±0.9 78.1±0.7
94.2±0.5 56.5±0.8 63.3±0.7 76.7±0.672.774.175.376.5
95.8±0.4 58.4±0.6 61.9±0.9 80.6±0.5
96.2±0.1 60.5±0.6 64.1±0.8 80.2±0.6
96.5±0.3 63.5±0.8 64.7±0.7 81.2±0.8
+ +# A.2 FURTHER ANALYSIS + +# MixStyle between same-domain instances. + +We are interested in knowing if mixing styles between same-domain instances helps performance. To this end, we sample each mini-batch from a single domain during training when using MixStyle. The results are shown in Table 6 where we observe that mixing styles between same-domain instances is about $1 \%$ better than the baseline model. This suggests that instancespecific style exists. Nonetheless, the performance is clearly worse than mixing styles between instances of different domains. + +Table 6: Investigation on the effect of mixing styles between same-domain instances. + +
Accuracy (%)
ResNet18 79.5
+MixStyle w/ same-domain 80.4
+MixStyle w/random shuffle 82.8
+MixStyle w/ domain label 83.7
+ +Performance on source domains. To prove that MixStyle does not sacrifice the performance on seen domains in exchange for gains on unseen domains, we report the test accuracy on the held-out validation set of the source domains on PACS in Table 7. + +Results on Digits-DG and Office-Home. In addition to the experiments on PACS (in Sec. 3.1), we further evaluate MixStyle’s effectiveness on two DG datasets, namely Digits-DG (Zhou et al., 2020a) and Office-Home (Venkateswara et al., 2017). Digits-DG contains four digit datasets (domains) including MNIST (LeCun et al., 1998), MNIST-M (Ganin & Lempitsky, 2015), SVHN (Netzer et al., 2011) and SYN (Ganin & Lempitsky, 2015). Images from different digit datasets differ drastically in font style, stroke color and background. Office-Home is composed of four domains (Artistic, Clipart, Product and Real World) with around 15,500 images of 65 classes for home and office object recognition. The results are shown in Tables 8 and 9 where no domain labels are used in MixStyle. Similar to the results on PACS, here we observe that MixStyle also brings clear improvements to the baseline CNN model and outperforms all general-purpose regularization methods on both Digits-DG and Office-Home. Compared with more sophisticated DG methods like L2A-OT, MixStyle’s performance is comparable, despite being much simpler to train and consuming much less computing resources. + +Table 9: Leave-one-domain-out generalization results on Office-Home. + +
MethodArtisticClipartProductReal WorldAvg
JiGen CCSA53.047.571.572.861.2
MMD-AAE59.9 56.549.9 47.374.1 72.175.7 74.864.9 62.7
CrossGrad75.864.4
L2A-OT58.449.473.9
ResNet1860.6 58.9±0.350.1 49.4±0.174.8 74.3±0.177.0 76.2±0.265.6 64.7
+ Manifold Mixup56.2±0.446.3±0.373.6±0.175.2±0.262.8
+ Mixup w/o label interpolation + Cutout57.0±0.2 57.8±0.248.7±0.271.4±0.674.5±0.462.9
48.1±0.373.9±0.275.8±0.363.9
+ CutMix57.9±0.148.3±0.374.5±0.175.6±0.464.1
+ DropBlock58.0±0.148.1±0.174.3±0.375.9±0.464.1
+ Mixup + MixStyle (ours)58.2±0.1 58.7±0.349.3±0.2 53.4±0.274.7±0.1 74.2±0.176.1±0.1 75.9±0.164.6 65.5
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Domain generalization aims to address this problem by learning from a set of source domains a model that is generalizable to any unseen domain. In this paper, a novel approach is proposed based on probabilistically mixing instancelevel feature statistics of training samples across source domains. Our method, termed MixStyle, is motivated by the observation that visual domain is closely related to image style (e.g., photo vs. sketch images). Such style information is captured by the bottom layers of a CNN where our proposed style-mixing takes place. Mixing styles of training instances results in novel domains being synthesized implicitly, which increase the domain diversity of the source domains, and hence the generalizability of the trained model. MixStyle fits into mini-batch training perfectly and is extremely easy to implement. The effectiveness of MixStyle is demonstrated on a wide range of tasks including category classification, instance retrieval and reinforcement learning. ", + "bbox": [ + 233, + 297, + 764, + 506 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 531, + 336, + 546 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Key to automated understanding of digital images is to compute a compact and informative feature representation. Deep convolutional neural networks (CNNs) have demonstrated remarkable ability in representation learning, proven to be effective in many visual recognition tasks, such as classifying photo images into 1,000 categories from ImageNet (Krizhevsky et al., 2012) and playing Atari games with reinforcement learning (Mnih et al., 2013). However, it has long been discovered that the success of CNNs heavily relies on the i.i.d. assumption, i.e. training and test data should be drawn from the same distribution; when such an assumption is violated even just slightly, as in most realworld application scenarios, severe performance degradation is expected (Hendrycks & Dietterich, 2019; Recht et al., 2019). ", + "bbox": [ + 174, + 563, + 825, + 686 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Domain generalization (DG) aims to address such a problem (Zhou et al., 2021; Blanchard et al., 2011; Muandet et al., 2013; Li et al., 2018a; Zhou et al., 2020b; Balaji et al., 2018; Dou et al., 2019; Carlucci et al., 2019). In particular, assuming that multiple source domains containing the same visual classes are available for model training, the goal of DG is to learn models that are robust against data distribution changes across domains, known as domain shift, so that the trained model can generalize well to any unseen domains. Compared to the closely related and more widely studied domain adaptation (DA) problem, DG is much harder in that no target domain data is available for the model to analyze the distribution shift in order to overcome the negative effects. Instead, a DG model must rely on the source domains and focus on learning domain-invariant feature representation in the hope that it would remain discriminative given target domain data. ", + "bbox": [ + 174, + 694, + 825, + 833 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "A straightforward solution to DG is to expose a model with a large variety of source domains. Specifically, the task of learning domain-invariant and thus generalizable feature representation becomes easier when data from more diverse source domains are available for the model. This would reduce the burden on designing special models or learning algorithms for DG. Indeed, model training with large-scale data of diverse domains is behind the success of existing commercial face recognition or vision-based autonomous driving systems. A recent work by $\\mathrm { X u }$ et al. (2021) also emphasizes the importance of diverse training distributions for out-of-distribution generalization. However, collecting data of a large variety of domains is often costly or even impossible. It thus cannot be a general solution to DG. ", + "bbox": [ + 174, + 840, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/f1cd7469df889b614caa015810343d6c004d2bbf757f1fbaacd0388b9ffe208e.jpg", + "image_caption": [ + "Figure 1: 2-D t-SNE (Maaten & Hinton, 2008) visualization of the style statistics (concatenation of mean and standard deviation) computed from the first residual block’s feature maps of a ResNet-18 (He et al., 2016) trained on four distinct domains (Li et al., 2017). It is clear that different domains are well separated. " + ], + "image_footnote": [], + "bbox": [ + 240, + 99, + 758, + 290 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 390, + 821, + 431 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this paper, a novel approach is proposed based on probabilistically mixing instance-level feature statistics of training samples across source domains. Our model, termed MixStyle, is motivated by the observation that visual domain is closely related to image style. An example is shown in Fig. 1: the four images from four different domains depict the same semantic concept, i.e. dog, but with distinctive styles (e.g., characteristics in color and texture). When these images are fed into a deep CNN, which maps the raw pixel values into category labels, such style information is removed at the output. However, recent style transfer studies (Huang & Belongie, 2017; Dumoulin et al., 2017) suggest that such style information is preserved at the bottom layers of the CNN through the instance-level feature statistics, as shown clearly in Fig. 4. Importantly, since replacing such statistics would lead to replaced style while preserving the semantic content of the image, it is reasonable to assume that mixing styles from images of different domains would result in images of (mixed) new styles. That is, more diverse domains/styles can be made available for training a more domain-generalizable model. ", + "bbox": [ + 173, + 439, + 825, + 618 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Concretely, our MixStyle randomly selects two instances of different domains and adopts a probabilistic convex combination between instance-level feature statistics of bottom CNN layers. In contrast to style transfer work (Huang & Belongie, 2017; Dumoulin et al., 2017), no explicit image synthesis is necessary meaning much simpler model design. Moreover, MixStyle perfectly fits into modern mini-batch training. Overall, it is very easy to implement with only few lines of code. To evaluate the effectiveness as well as the general applicability of MixStyle, we conduct extensive experiments on a wide spectrum of datasets covering category classification (Sec. 3.1), instance retrieval (Sec. 3.2), and reinforcement learning (Sec. 3.3). The results demonstrate that MixStyle can significantly improve CNNs’ cross-domain generalization performance.1 ", + "bbox": [ + 174, + 626, + 825, + 751 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 METHODOLOGY ", + "text_level": 1, + "bbox": [ + 176, + 775, + 339, + 791 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 BACKGROUND ", + "text_level": 1, + "bbox": [ + 176, + 809, + 315, + 824 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Normalizing feature tensors with instance-specific mean and standard deviation has been found effective for removing image style in style transfer models (Ulyanov et al., 2016; Huang & Belongie, 2017; Dumoulin et al., 2017). Such an operation is widely known as instance normalization (IN, Ulyanov et al. (2016)). Let $\\boldsymbol { x } ^ { \\prime } \\in \\mathbb { R } ^ { B \\times C \\times \\dot { H } \\times W }$ be a batch of tensors, with $B , C , H$ and $W$ denoting the dimension of batch, channel, height and width, respectively, IN is formulated as ", + "bbox": [ + 174, + 837, + 823, + 892 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 103, + 782, + 118 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/1bcca277c97bac87df3b5ecefb248adf385fad37c24b49b8108b0b8f4ebd2944.jpg", + "text": "$$\n\\mathrm { I N } ( x ) = \\gamma \\frac { x - \\mu ( x ) } { \\sigma ( x ) } + \\beta ,\n$$", + "text_format": "latex", + "bbox": [ + 413, + 121, + 586, + 155 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\gamma , \\beta \\in \\mathbb { R } ^ { C }$ are learnable affine transformation parameters, and $\\mu ( \\boldsymbol { x } ) , \\sigma ( \\boldsymbol { x } ) \\in \\mathbb { R } ^ { B \\times C }$ are mean and standard deviation computed across the spatial dimension within each channel of each instance (tensor), i.e. ", + "bbox": [ + 174, + 159, + 825, + 200 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/fb99b3cee5284125fb7c56d3850f8a40e8f4faade0a3f52d6d8fd4515140a5b9.jpg", + "text": "$$\n\\mu ( x ) _ { b , c } = \\frac { 1 } { H W } \\sum _ { h = 1 } ^ { H } \\sum _ { w = 1 } ^ { W } x _ { b , c , h , w } ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 198, + 611, + 242 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "and ", + "bbox": [ + 173, + 242, + 200, + 257 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/6a50bbe7d108fe07827ea98df07a10608e7e794826c7fb9b310dbc970c77a01f.jpg", + "text": "$$\n\\sigma ( x ) _ { b , c } = \\sqrt { \\frac { 1 } { H W } \\sum _ { h = 1 } ^ { H } \\sum _ { w = 1 } ^ { W } ( x _ { b , c , h , w } - \\mu ( x ) _ { b , c } ) ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 334, + 251, + 663, + 303 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Huang & Belongie (2017) introduced adaptive instance normalization (AdaIN), which simply replaces the scale and shift parameters in Eq. (1) with the feature statistics of style input $y$ to achieve arbitrary style transfer: ", + "bbox": [ + 173, + 309, + 825, + 351 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/3c63f5f72c8d9ea81fce1ab32d869a82feeb503f7c48024c0506a4fc0b6157da.jpg", + "text": "$$\n\\mathrm { A d a I N } ( x ) = \\sigma ( y ) \\frac { x - \\mu ( x ) } { \\sigma ( x ) } + \\mu ( y ) .\n$$", + "text_format": "latex", + "bbox": [ + 375, + 348, + 624, + 383 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 MIXSTYLE", + "text_level": 1, + "bbox": [ + 174, + 397, + 294, + 411 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our method, MixStyle, draws inspiration from AdaIN. However, rather than attaching a decoder for image generation, MixStyle is designed for the purpose of regularizing CNN training by perturbing the style information of source domain training instances. It can be implemented as a plug-andplay module inserted between CNN layers of, e.g., a supervised CNN classifier, without the need to explicitly generate an image of new style. ", + "bbox": [ + 173, + 422, + 825, + 493 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "More specifically, MixStyle mixes the feature statistics of two instances with a random convex weight to simulate new styles. In terms of implementation, MixStyle can be easily integrated into mini-batch training. Given an input batch $x$ , MixStyle first generates a reference batch $\\tilde { x }$ from $x$ . When domain labels are given, $x$ is sampled from two different domains $i$ and $j$ , e.g., $\\bar { \\boldsymbol { x } } = [ x ^ { i } , x ^ { j } ]$ $\\bar { x ^ { i } }$ and $x ^ { j }$ have the same batch size). Then, $\\tilde { x }$ is obtained by swapping the position of $x ^ { i }$ and $x ^ { j }$ , followed by a shuffling operation along the batch dimension applied to each batch, i.e. $\\mathbf { \\widetilde { \\boldsymbol { x } } } = [ \\mathrm { S h u f f e } ( \\boldsymbol { x } ^ { j } ) , \\mathrm { S h u f f e } ( \\boldsymbol { x } ^ { i } ) ]$ . See Fig. 2(a) for an illustration. In cases where domain labels are unknown, $x$ is randomly sampled from the training data, and $\\tilde { x }$ is simply obtained by $\\tilde { x } = \\mathrm { S h u f f e } ( x )$ (see Fig. 2(b)). Fig. 4 shows that sub-domains exist within each domain, so even if two instances of the same domain are sampled, new domain could be synthesized. After shuffling, MixStyle computes the mixed feature statistics by ", + "bbox": [ + 174, + 500, + 581, + 736 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/328a14b30cdf68b6ebf3e522fa5e10ef00672d004ed56b5521dcee7294f42db8.jpg", + "image_caption": [ + "Figure 2: A graphical illustration of how a reference batch is generated. Domain label is denoted by color. " + ], + "image_footnote": [], + "bbox": [ + 599, + 505, + 810, + 643 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/dda5b24328e69e61232fce56170c65ac8f6d17d9ab0153ffcddaf7931cf59390.jpg", + "text": "$$\n\\begin{array} { r } { \\gamma _ { m i x } = \\lambda \\sigma ( x ) + ( 1 - \\lambda ) \\sigma ( \\tilde { x } ) , } \\\\ { \\beta _ { m i x } = \\lambda \\mu ( x ) + ( 1 - \\lambda ) \\mu ( \\tilde { x } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 393, + 738, + 602, + 777 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\lambda \\in \\mathbb { R } ^ { B }$ are instance-wise weights sampled from the Beta distribution, $\\lambda \\sim B e t a ( \\alpha , \\alpha )$ with $\\alpha \\in ( 0 , \\infty )$ being a hyper-parameter. Unless specified otherwise, we set $\\alpha$ to 0.1 throughout this paper. Finally, the mixed feature statistics are applied to the style-normalized $x$ , ", + "bbox": [ + 176, + 779, + 823, + 821 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c55e6d61c724a64df57068c18b04d585859171236745e7c0ae6fceb70291e356.jpg", + "text": "$$\n{ \\mathrm { M i x S t y l e } } ( x ) = \\gamma _ { m i x } { \\frac { x - \\mu ( x ) } { \\sigma ( x ) } } + \\beta _ { m i x } .\n$$", + "text_format": "latex", + "bbox": [ + 366, + 824, + 633, + 858 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In practice, we use a probability of 0.5 to decide if MixStyle is activated or not in the forward pass. At test time, no MixStyle is applied. Note that gradients are blocked in the computational graph of $\\mu ( \\cdot )$ and $\\sigma ( \\cdot )$ . MixStyle can be implemented with only few lines of code. See Algorithm 1 in Appendix A.1 for the PyTorch-like pseudo-code. ", + "bbox": [ + 174, + 867, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/c5ebe274bb1893ac2346fb0d139387bcd5d7bf3e991b91a59bcca22de9df6956.jpg", + "table_caption": [ + "Table 1: Leave-one-domain-out generalization results on PACS. " + ], + "table_footnote": [], + "table_body": "
MethodArtCartoonPhotoSketchAvg
MMD-AAE CCSA JiGen75.2 80.5 79.4 79.8 82.172.7 76.9 75.3 76.896.0 93.6 96.0 96.064.2 66.8 71.6 70.277.0 79.4 80.5 80.7
Epi-FCR Metareg L2A-OT ResNet-1883.7 83.3 77.0±0.677.0 77.2 78.2 75.9±0.693.9 95.5 96.2 96.0±0.173.0 70.3 73.6 69.2±0.681.5 81.7 82.8 79.5
+ Manifold Mixup75.6±0.770.1±0.993.5±0.765.4±0.676.2
+ Cutout74.9±0.474.9±0.695.9±0.367.7±0.978.3
+ CutMix74.6±0.771.8±0.695.6±0.465.3±0.876.8
74.7±1.072.3±0.9
+ Mixup (w/o label interpolation)93.0±0.469.2±0.277.3
+ Mixup76.8±0.774.9±0.795.8±0.366.6±0.778.5
+ DropBlock76.4±0.775.4±0.795.9±0.369.0±0.379.2
+ MixStyle w/ random shuffle82.3±0.279.0±0.396.3±0.373.8±0.982.8
+ MixStyle w/ domain label84.1±0.478.8±0.496.1±0.375.9±0.9
83.7
", + "bbox": [ + 192, + 127, + 807, + 368 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 393, + 326, + 410 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.1 GENERALIZATION IN CATEGORY CLASSIFICATION ", + "text_level": 1, + "bbox": [ + 174, + 424, + 562, + 439 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Dataset and implementation details. We choose the PACS dataset (Li et al., 2017), a commonly used domain generalization (DG) benchmark concerned with domain shift in image classification. PACS consists of four domains, i.e. Art Painting, Cartoon, Photo and Sketch, with totally 9,991 images of 7 classes. As shown in Fig. 1, the domain shift mainly corresponds to image style changes. For evaluation, a model is trained on three domains and tested on the remaining one. Following prior work (Li et al., 2019; Zhou et al., 2020a), we use ResNet-18 (He et al., 2016) as the classifier where MixStyle is inserted after the 1st, 2nd and 3rd residual blocks. Our code is based on Dassl.pytorch (Zhou et al., 2020c).2 ", + "bbox": [ + 173, + 450, + 825, + 563 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Baselines. Our main baselines are general-purpose regularization methods including Mixup (Zhang et al., 2018b), Manifold Mixup (Verma et al., 2019), DropBlock (Ghiasi et al., 2018), CutMix (Yun et al., 2019) and Cutout (DeVries & Taylor, 2017), which are trained using the same training parameters as MixStyle and the optimal hyper-parameter setup as reported in their papers. We also compare with the existing DG methods which reported state-of-the-art performance on PACS. These include domain alignment-based CCSA (Motiian et al., 2017) and MMD-AAE (Li et al., 2018b), Jigsaw puzzle-based JiGen (Carlucci et al., 2019), adversarial gradient-based CrossGrad (Shankar et al., 2018), meta-learning-based Metareg (Balaji et al., 2018) and Epi-FCR (Li et al., 2019), and data augmentation-based L2A-OT (Zhou et al., 2020a). ", + "bbox": [ + 173, + 571, + 825, + 696 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Comparison with general-purpose regularization methods. The results are shown in Table 1. Overall, we observe that the general-purpose regularization methods do not offer any clear advantage over the vanilla ResNet-18 in this DG task, while MixStyle improves upon the vanilla ResNet18 with a significant margin. Compared with Mixup, MixStyle is $5 . 2 \\%$ better on average. Recall that Mixup also interpolates the output space, we further compare with a variant of Mixup in order to demonstrate the advantage of mixing style statistics at the feature level over mixing images at the pixel level for DG—following Sohn et al. (2020), we remove the label interpolation in Mixup and sample the mixing weights from a uniform distribution of $[ 0 , 1 ]$ . Still, MixStyle outperforms this new baseline with a large margin, which justifies our claim. MixStyle and DropBlock share some commonalities in that they are both applied to feature maps at multiple layers, but MixStyle significantly outperforms DropBlock in all test domains. The reason why DropBlock is ineffective here is because dropping out activations mainly encourages a network to mine discriminative patterns, but does not reinforce the ability to cope with unseen styles, which is exactly what MixStyle aims to achieve: by synthesizing “new” styles (domains) MixStyle regularizes the network to become more robust to domain shift. In addition, it is interesting to see that on Cartoon and Photo, MixStyle w/ random shuffle obtains slightly better results. The reason might be because there exist sub-domains in a source domain (see Fig. 4(a-c)), which allow random shuffling to produce more diverse “new” domains that lead to a more domain-generalizable model. ", + "bbox": [ + 174, + 705, + 825, + 900 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Comparison with state-of-the-art DG methods. Overall, MixStyle outperforms most DG methods by a clear margin, despite being a much simpler method. The performance of MixStyle w/ domain label is nearly $1 \\%$ better on average than the recently introduced L2A-OT. From a data augmentation perspective, MixStyle and L2A-OT share a similar goal—to synthesize data from pseudo-novel domains. MixStyle accomplishes this goal through mixing style statistics at the feature level. Whereas L2A-OT works at the pixel level: it trains an image generator by maximizing the domain difference (measured by optimal transport) between the original and the generated images, which introduces much heavier computational overhead than MixStyle in terms of GPU memory and training time. It is worth noting that MixStyle’s domain label-free version is highly competitive: its $8 2 . 8 \\%$ accuracy is on par with L2A-OT’s. ", + "bbox": [ + 174, + 169, + 825, + 308 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 GENERALIZATION IN INSTANCE RETRIEVAL", + "text_level": 1, + "bbox": [ + 174, + 325, + 517, + 339 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Dataset and implementation details. We evaluate MixStyle on the person re-identification (re-ID) problem, which aims to match people across disjoint camera views. As each camera view is itself a distinct domain, person re-ID is essentially a cross-domain image matching problem. Instead of using the standard protocol where training and test data come from the same camera views, we adopt the cross-dataset setting so test camera views are never seen during training. Specifically, we train a model on one dataset and then test its performance on the other dataset. Two commonly used re-ID datasets are adopted: Market1501 (Zheng et al., 2015) and Duke (Ristani et al., 2016; Zheng et al., 2017). Ranking accuracy and mean average precision (mAP) are used as the performance measures (displayed in percentage). We test MixStyle on two CNN architectures: ResNet-50 (He et al., 2016) and OSNet (Zhou et al., 2019). The latter was designed specifically for re-ID. In both architectures, MixStyle is inserted after the 1st and 2nd residual blocks. Our code is based on Torchreid (Zhou & Xiang, 2019).3 ", + "bbox": [ + 173, + 352, + 825, + 518 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Baselines. We compare with three baseline methods: 1) The vanilla model, which serves as a strong baseline; 2) DropBlock, which was the top-performing competitor in Table 1; 3) RandomErase (Zhong et al., 2020), a widely used regularization method in the re-ID literature (similar to Cutout). ", + "bbox": [ + 174, + 529, + 823, + 583 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Results. The results are reported in Table 2. It is clear that only MixStyle consistently outperforms the strong vanilla model under both settings with considerable margins, while DropBlock and RandomErase are unable to show any benefit. Notably, RandomErase, which simulates occlusion by erasing pixels in random rectangular regions with random values, has been used as a default trick when training re-ID CNNs. However, RandomErase shows a detrimental effect in the cross-dataset re-ID setting. Indeed, similar to DropBlock, randomly erasing pixels offers no guarantee to improve the robustness when it comes to domain shift. ", + "bbox": [ + 174, + 594, + 825, + 690 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 GENERALIZATION IN REINFORCEMENT LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 709, + 565, + 723 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Though RL has been greatly advanced by using CNNs for feature learning in raw pixels (Mnih et al., 2013), it has been widely acknowledged that RL agents often overfit training environments while generalize poorly to unseen environments (Cobbe et al., 2019; Igl et al., 2019). ", + "bbox": [ + 174, + 734, + 825, + 777 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Dataset and implementation details. We conduct experiments on Coinrun (Cobbe et al., 2019), a recently introduced RL benchmark for evaluating the generalization performance of RL agents. As shown in Fig. 3(a), the goal in Coinrun is to control a character to collect golden coins while avoiding both stationary and non-stationary obstacles. We follow Igl et al. (2019) to construct and train our RL agent: the CNN architecture used in IMPALA (Espeholt et al., 2018) is adopted as the policy network, and is trained by the Proximal Policy Optimization (PPO) algorithm (Schulman et al., 2017). Please refer to Igl et al. (2019) for further implementation details. MixStyle is inserted after the 1st and 2nd convolutional sequences. Training data are sampled from 500 levels while test data are drawn from new levels of only the highest difficulty. As domain labels are difficult to define, we use the random shuffle version of MixStyle. Our code is built on top of Igl et al. (2019).4 ", + "bbox": [ + 174, + 786, + 825, + 898 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/8e248e2539fca822336240b7fe9d39fd45dacc2101f2bcb3a98c958918e50290.jpg", + "table_caption": [ + "Table 2: Generalization results on the cross-dataset person re-ID task. " + ], + "table_footnote": [], + "table_body": "
ModelMarket1501-→DukeDuke-→Market1501
mAPR1R5R10mAPR1R5R10
ResNet-5019.335.450.356.420.445.263.670.9
+ RandomErase14.327.842.649.116.138.556.864.5
+ DropBlock18.233.249.156.319.745.362.169.1
+ MixStyle w/ random shuffle23.842.258.864.824.151.569.476.2
+ MixStyle w/ domain label23.443.358.964.724.753.070.977.8
OSNet25.944.759.665.424.052.267.574.7
+ RandomErase20.536.252.359.322.449.166.173.0
+ DropBlock23.141.556.562.521.748.265.471.3
+ MixStyle w/ random shuffle27.248.262.768.427.858.174.081.0
+ MixStyle w/ domain label27.347.562.067.129.058.274.980.9
", + "bbox": [ + 192, + 127, + 805, + 300 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/83f0338beadfd5bc519830499778de6828846e5597dc084ac87299494b1148a1.jpg", + "image_caption": [ + "Figure 3: (a) Coinrun benchmark. (b) Test performance in unseen environments. (c) Difference between training and test performance. " + ], + "image_footnote": [], + "bbox": [ + 176, + 333, + 825, + 493 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 578, + 823, + 607 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Baselines. Following Igl et al. (2019), we train strong baseline models and add MixStyle on top of them to see whether MixStyle can bring further improvements. To this end, we train two baseline models: 1) Baseline, which combines weight decay and data augmentation;5 2) IBAC-SNI (the $\\lambda = 0 . 5$ version), the best-performing model in Igl et al. (2019) which is based on selective noise injection. ", + "bbox": [ + 174, + 654, + 825, + 723 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Results. The test performance is shown in Fig. 3(b). Comparing Baseline (blue) with Baseline+MixStyle (orange), we can see that MixStyle brings a significant improvement. Interestingly, the variance is also significantly reduced by using MixStyle, as indicated by the smaller shaded areas (for both orange and red lines). These results strongly demonstrate the effectiveness of MixStyle in enhancing generalization for RL agents. When it comes to the stronger baseline IBAC-SNI (green), MixStyle (red) is able to bring further performance gain, suggesting that MixStyle is complementary to IBAC-SNI. This result also shows the potential of MixStyle as a plug-and-play component to be combined with other advanced RL methods. It is worth noting that Baseline+MixStyle itself is already highly competitive with IBAC-SNI. Fig. 3(c) shows the generalization gap from which it can been seen that the models trained with MixStyle (orange & red) clearly generalize faster and better than those without using MixStyle (blue & green). ", + "bbox": [ + 173, + 770, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/4d2d1d46181970a60036d74149364fc1b7c14abba9cfcdf85a1601e0d521acda.jpg", + "image_caption": [ + "Figure 4: 2-D visualization of flattened feature maps (top) and the corresponding style statistics (bottom). res1-4 denote the four residual blocks in order in a ResNet architecture. We observe that res1 to res3 contain domain-related information while res4 encodes label-related information. " + ], + "image_footnote": [], + "bbox": [ + 176, + 99, + 821, + 313 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/2c49c123023d551ecc0a12856511415bafdfb941fca08cf592f728b54bf5742d.jpg", + "table_caption": [ + "Table 3: Ablation study on where to apply MixStyle in the ResNet architecture. ", + "(a) Category classification on PACS. " + ], + "table_footnote": [], + "table_body": "
ModelAccuracy
ResNet-1879.5
+ MixStyle (res1)80.1
+ MixStyle (res12)81.6
+ MixStyle (res123)82.8
+ MixStyle (res1234)75.6
+ MixStyle (res14)76.3
+ MixStyle (res23)81.7
", + "bbox": [ + 251, + 428, + 506, + 546 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/77ff508fa1e5b4b56a82c9108d2ab847baee3536e4d32e484735f7ac7a2f04a3.jpg", + "table_caption": [ + "(b) Cross-dataset person re-ID. " + ], + "table_footnote": [], + "table_body": "
ModelmAP
ResNet-5019.3
+ MixStyle (res1)22.6
+ MixStyle (res12) + MixStyle (res123)23.8
+MixStyle (res1234)22.0
+ MixStyle (res14)10.2
+MixStyle (res23)11.1 20.6
", + "bbox": [ + 519, + 429, + 746, + 546 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.4 ANALYSIS ", + "text_level": 1, + "bbox": [ + 174, + 590, + 285, + 603 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Where to apply MixStyle? We repeat the experiments on PACS (category classification) and the re-ID datasets (instance retrieval) using the ResNet architecture. Given that a standard ResNet model has four residual blocks denoted by $\\mathtt { r e s l - 4 }$ , we train different models with MixStyle applied to different layers. For notation, res1 means MixStyle is applied after the first residual block; res12 means MixStyle is applied after both the first and second residual blocks; and so forth. The results are shown in Table 3. We have the following observations. 1) Applying MixStyle to multiple lowerlevel layers generally achieves a better performance—for instance, res12 is better than res1 on both tasks. 2) Different tasks favor different combinations—res123 achieves the best performance on PACS, while on the re-ID datasets $\\mathtt { r e s 1 2 }$ is the best. 3) On both tasks, the performance plunges when applying MixStyle to the last residual block. This makes sense because res4 is the closest to the prediction layer and tends to capture semantic content (i.e. label-sensitive) information rather than style. In particular, res4 is followed by an average-pooling layer, which essentially forwards the mean vector to the prediction layer and thus forces the mean vector to capture label-related information. As a consequence, mixing the statistics at res4 breaks the inherent label space. This is clearer in Fig. 4: the features and style statistics in $\\mathtt { r e s l - 3 }$ exhibit clustering patterns based on domains while those in res4 have a high correlation with class labels. ", + "bbox": [ + 174, + 617, + 825, + 840 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/c7e15cbb6312d6d6e57a51169db1f4a01f9832c478ec13254c9df6c7db06a565.jpg", + "image_caption": [ + "Figure 5: Evaluation on the hyper-parameter $\\alpha$ on (a) PACS, (b) person re-ID datasets and (c) Coinrun. In (b), M and D denote Market1501 and Duke respectively. " + ], + "image_footnote": [], + "bbox": [ + 174, + 98, + 825, + 250 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Mixing vs. replacing. Unlike the AdaIN formulation, which completely replaces one style with the other, MixStyle mixes two styles via a convex combination. Table 4 shows that mixing is better than replacing. This is easy to understand: mixing diversifies the styles (imagine an interpolation between two data points). ", + "bbox": [ + 174, + 316, + 612, + 386 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Random vs. fixed shuffle at multiple layers. Applying MixStyle to multiple layers, which has been shown advantageous in Table 3, raises another question of whether to shuffle the mini-batch at different layers or use the same shuffled order for all layers. Table 5 suggests that using random shuffle at different layers gives a better performance, which may be attributed to the increased noise level that gives a better regularization effect. ", + "bbox": [ + 174, + 396, + 614, + 492 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/2c381c18bb58b30c686683356c2292f71c205ce4d70504362b8537940ea26f02.jpg", + "table_caption": [ + "Table 4: Mixing vs. replacing. " + ], + "table_footnote": [], + "table_body": "
Accuracy (%)
Mixing82.8±0.4
Replacing82.1±0.5
", + "bbox": [ + 625, + 325, + 820, + 376 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/76b7d4c0c12c5e282d56837e7a904e7f774b0d08c2aa3a6a179528e5aee67938.jpg", + "table_caption": [ + "Table 5: Random vs. fixed shuffle at multiple layers. " + ], + "table_footnote": [], + "table_body": "
Accuracy (%)
Random82.8±0.4
Fixed82.4±0.5
", + "bbox": [ + 630, + 434, + 815, + 483 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Sensitivity of hyper-parameter. Recall that $\\alpha$ is used to control the shape of Beta distribution, which has a direct effect on how the convex weights $\\lambda$ are sampled. The smaller $\\alpha$ is, the more likely the value in $\\lambda$ is close to the extreme value of 0 or 1. In other words, a smaller $\\alpha$ favors the style statistics in Eqs. (5) & (6) to be dominated by one side. We first evaluate $\\alpha$ on PACS. Fig. 5(a) shows that with $\\alpha$ increasing from 0.1 to 0.4, the accuracy slides from $8 2 . 8 \\%$ to $8 1 . 7 \\%$ . However, further increasing $\\alpha$ does not impact on the accuracy. Therefore, the results suggest that the performance is not too sensitive to $\\alpha$ ; and selecting $\\alpha$ from $\\lbrace 0 . 1 , 0 . 2 , 0 . 3 \\rbrace$ seems to be a good starting point. We further experiment with $\\alpha \\in \\{ 0 . 1 , 0 . 2 , 0 . 3 \\}$ on the re-ID datasets and the Coinrun benchmark. Figs. 5(b) & (c) show that in general the variance for the results of different values is small. Therefore, we suggest practitioners to choose $\\alpha$ from $\\lbrace 0 . 1 , 0 . 2 , 0 . 3 \\rbrace$ , with $\\alpha = 0 . 1$ being a good default setting. ", + "bbox": [ + 174, + 502, + 825, + 655 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For more analyses and discussions, please see Appendix A.2. ", + "bbox": [ + 173, + 662, + 575, + 676 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 696, + 344, + 713 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Domain generalization, or DG, studies out-of-distribution (OOD) generalization given only source data typically composed of multiple related but distinct domains. We refer readers to Zhou et al. (2021) for a comprehensive survey in this topic. Many DG methods are based on the idea of aligning features between different sources, with a hope that the model can be invariant to domain shift given unseen data. For instance, Li et al. (2018b) achieved distribution alignment in the hidden representation of an autoencoder using maximum mean discrepancy; Li et al. (2018c) resorted to adversarial learning with auxiliary domain classifiers to learn features that are domain-agnostic. Some works explored domain-specific parameterization, such as domain-specific weight matrices (Li et al., 2017) and domain-specific BN (Seo et al., 2020). Recently, meta-learning has drawn increasing attention from the DG community (Li et al., 2018a; Balaji et al., 2018; Dou et al., 2019). The main idea is to expose a model to domain shift during training by using pseudo-train and pseudo-test domains, both drawn from source domains. Data augmentation has also been investigated for learning domain-invariant models. Shankar et al. (2018) introduced a cross-gradient training method (CrossGrad) where source data are augmented by adversarial gradients obtained from a domain classifier. ", + "bbox": [ + 174, + 729, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Gong et al. (2019) proposed DLOW (for the DA problem), which models intermediate domains between source and target via a domainness factor and learns an image translation model to generate intermediate-domain images. Very recently, Zhou et al. (2020a) introduced L2A-OT to learn a neural network to map source data to pseudo-novel domains by maximizing an optimal transport-based distance measure. Our MixStyle is related to DLOW and L2A-OT in its efforts to synthesizing novel domains. However, MixStyle differs in the fact that it is done implicitly with a much simpler formulation leveraging the feature-level style statistics and only few lines of extra code on top of a standard supervised classifier while being more effective. Essentially, MixStyle can be seen as feature-level augmentation, which is clearly different from the image-level augmentation-based DLOW and L2A-OT. ", + "bbox": [ + 174, + 103, + 825, + 242 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Generalization in deep RL has been a challenging problem where RL agents often overfit training environments, and as a result, perform poorly in unseen environments with different visual patterns or levels (Zhang et al., 2018a). A natural way to improve generalization, which has been shown effective in (Cobbe et al., 2019; Farebrother et al., 2018), is to use regularization, e.g., weight decay. However, Igl et al. (2019) suggested that stochastic regularization methods like dropout and batch normalization (which uses estimated population statistics) have adverse effect as the training data in RL are essentially model-dependent. As such, they proposed selective noise injection (SNI), which basically combines a stochastic regularization technique with its deterministic counterpart. They further integrated SNI with information bottleneck actor critic (IBAC-SNI) to reduce the variance in gradients. Curriculum learning has been investigated in (Justesen et al., 2018) where the level of training episodes progresses from easy to difficult over the course of training. Gamrian & Goldberg (2019) leveraged the advances in GAN-based image-to-image translation (Liu et al., 2017) to map target data to the source domain which the agent was trained on. Tobin et al. (2017) introduced domain randomization, which diversifies training data by rendering images with different visual effects via a programmable simulator. With a similar goal of data augmentation, Lee et al. (2020) pre-processed input images with a randomly initialized network. Very recent studies (Laskin et al., 2020; Kostrikov et al., 2020) have shown that it is useful to combine a diverse set of label-preserving transformations, such as rotation, shifting and Cutout. Different from the aforementioned methods, our MixStyle works at the feature level and is orthogonal to most existing methods. For instance, we have shown in Sec. 3.3 that MixStyle significantly improves upon IBAC-SNI. ", + "bbox": [ + 173, + 251, + 825, + 527 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 547, + 318, + 563 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We presented a simple yet effective domain generalization method, termed MixStyle. MixStyle mixes the feature statistics of two instances to synthesize novel domains, which is inspired by the observation in style transfer work that the feature statistics encode style/domain-related information. Extensive experiments covering a wide range of tasks were conducted to demonstrate that MixStyle yields new state-of-the-art on three different tasks. ", + "bbox": [ + 174, + 578, + 825, + 647 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ACKNOWLEDGMENTS ", + "text_level": 1, + "bbox": [ + 176, + 664, + 326, + 676 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This work was supported in part by the Shanghai Committee of Science and Technology, China (Grant No. 20DZ1100800), in part by the National Natural Science Foundation of China under Grant (61876176, U1713208), the National Key Research and Development Program of China (No. 2020YFC2004800), and in part by the Guangzhou Research Program (201803010066). ", + "bbox": [ + 176, + 688, + 825, + 743 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 763, + 285, + 779 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Yogesh Balaji, Swami Sankaranarayanan, and Rama Chellappa. Metareg: Towards domain generalization using meta-regularization. In NeurIPS, 2018. 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", + "text_level": 1, + "bbox": [ + 174, + 189, + 527, + 204 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "# x: input features of shape (B, C, H, W) \n# p: probabillity to apply MixStyle (default: 0.5) \n# alpha: hyper-parameter for the Beta distribution (default: 0.1) \n# eps: a small value added before square root for numerical stability (default: 1e-6) ", + "bbox": [ + 171, + 215, + 776, + 252 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "if not in training mode: return x ", + "bbox": [ + 176, + 261, + 344, + 279 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "if random probability $>$ p: return x ", + "bbox": [ + 176, + 287, + 357, + 306 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$\\mathrm { ~ \\textit ~ { ~ B ~ } ~ } = \\mathrm { ~ \\textit ~ { ~ x ~ } ~ }$ .size(0) # batch size ", + "bbox": [ + 174, + 314, + 359, + 325 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "mu $= \\times$ .mean(dim=[2, 3], keepdim $\\cdot ^ { = }$ True) # compute instance mean var $=$ x.var(dim=[2, 3], keepdim $\\cdot ^ { = }$ True) # compute instance variance sig $=$ (var $^ +$ eps).sqrt() # compute instance standard deviation mu, sig $\\qquad = \\quad \\mathtt { m u }$ .detach(), sig.detach() # block gradients x_normed $=$ (x - mu) / sig # normalize input ", + "bbox": [ + 171, + 333, + 633, + 380 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "lmda $=$ Beta(alpha, alpha).sample((B, 1, 1, 1)) # sample instance-wise convex weights ", + "bbox": [ + 166, + 387, + 766, + 398 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "if domain label is given: # in this case, input $\\begin{array} { r } { \\mathrm { ~ ~ \\underline { ~ } { ~ \\sf ~ } { ~ \\sf ~ } { ~ \\sf ~ } { ~ \\sf ~ } = ~ \\mathrm { ~ ~ \\tau ~ } [ \\times \\hat { \\mathrm { ~ ~ \\underline { ~ } { ~ \\bf ~ } { ~ \\underline { ~ } { ~ \\bf ~ } { ~ \\tau ~ } ~ } } } } } \\end{array}$ x perm $=$ torch.arange(B-1, -1, -1) # inverse index perm_j, perm_i $=$ perm.chunk(2) # separate indices perm_j $=$ perm_j[torch.randperm(B // 2)] # shuffling perm_i $=$ perm_i[torch.randperm(B // 2)] # shuffling perm $=$ torch.cat([perm_j, perm_i], 0) # concatenation \nelse: perm $=$ torch.randperm(B) # generate shuffling indices ", + "bbox": [ + 176, + 405, + 571, + 487 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "mu2, sig2 $=$ mu[perm], sig[perm] # shuffling mu_mix $=$ mu $\\star$ lmda $^ +$ mu2 $\\star$ (1 - lmda) # generate mixed mean sig_mix $=$ sig $\\star$ lmda $^ +$ sig2 $\\star$ (1 - lmda) # generate mixed standard deviation return x_normed $\\star$ sig_mix $^ +$ mu_mix # denormalize input using the mixed statistics ", + "bbox": [ + 174, + 496, + 710, + 525 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 531, + 748, + 544 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/c5c91c980ca863ce754049c2ac48657fb6e478a83339d37cdd2bb99ba07e1b07.jpg", + "table_caption": [ + "Table 7: Test results on the source domains on PACS. A: Art. C: Cartoon. P: Photo. S: Sketch. " + ], + "table_footnote": [], + "table_body": "
MethodC,P,SA,P,SA,C,SA,C,PAvg
Vanilla99.49±0.0399.47±0.0499.38±0.0299.65±0.0399.50
MixStyle99.55±0.0299.54±0.0199.47±0.0399.68±0.0399.56
", + "bbox": [ + 233, + 127, + 764, + 175 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/d706ed77580ec62fda2f7d6c0da59fdce21563a4ed5924d61fc6e2035f36f821.jpg", + "table_caption": [ + "Table 8: Leave-one-domain-out generalization results on Digits-DG. " + ], + "table_footnote": [], + "table_body": "
MethodMNIST MNIST-M SVHN SYNAvg
JiGen96.5 61.4 63.7 74.073.9
CCSA95.2 58.2 65.5 79.174.5
MMD-AAECrossGradL2A-OT96.5 58.4 65.0 78.474.675.878.1
96.7 61.1 65.3 80.2
96.7 63.9 68.6 83.2
CNN+ Mixup w/o label interpolation+ Manifold Mixup+ CutMix+ Mixup+ Cutout+ DropBlock+ MixStyle (ours)95.8±0.3 58.8±0.5 61.7±0.5 78.6±0.673.7
93.7±0.6 55.2±1.0 61.6±0.9 74.4±0.871.271.771.8
92.7±0.4 53.1±0.8 64.4±0.2 76.8±0.5
94.9±0.2 50.1±0.5 64.1±0.9 78.1±0.7
94.2±0.5 56.5±0.8 63.3±0.7 76.7±0.672.774.175.376.5
95.8±0.4 58.4±0.6 61.9±0.9 80.6±0.5
96.2±0.1 60.5±0.6 64.1±0.8 80.2±0.6
96.5±0.3 63.5±0.8 64.7±0.7 81.2±0.8
", + "bbox": [ + 194, + 213, + 805, + 415 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.2 FURTHER ANALYSIS ", + "text_level": 1, + "bbox": [ + 176, + 440, + 361, + 454 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "MixStyle between same-domain instances. ", + "text_level": 1, + "bbox": [ + 176, + 467, + 482, + 481 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We are interested in knowing if mixing styles between same-domain instances helps performance. To this end, we sample each mini-batch from a single domain during training when using MixStyle. The results are shown in Table 6 where we observe that mixing styles between same-domain instances is about $1 \\%$ better than the baseline model. This suggests that instancespecific style exists. Nonetheless, the performance is clearly worse than mixing styles between instances of different domains. ", + "bbox": [ + 174, + 481, + 483, + 632 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/e22f37e78ca0e3a60266186cfd9536740f245c97d3f14479137e919c3f8c589e.jpg", + "table_caption": [ + "Table 6: Investigation on the effect of mixing styles between same-domain instances. " + ], + "table_footnote": [], + "table_body": "
Accuracy (%)
ResNet18 79.5
+MixStyle w/ same-domain 80.4
+MixStyle w/random shuffle 82.8
+MixStyle w/ domain label 83.7
", + "bbox": [ + 504, + 522, + 816, + 599 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Performance on source domains. To prove that MixStyle does not sacrifice the performance on seen domains in exchange for gains on unseen domains, we report the test accuracy on the held-out validation set of the source domains on PACS in Table 7. ", + "bbox": [ + 173, + 641, + 825, + 683 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Results on Digits-DG and Office-Home. In addition to the experiments on PACS (in Sec. 3.1), we further evaluate MixStyle’s effectiveness on two DG datasets, namely Digits-DG (Zhou et al., 2020a) and Office-Home (Venkateswara et al., 2017). Digits-DG contains four digit datasets (domains) including MNIST (LeCun et al., 1998), MNIST-M (Ganin & Lempitsky, 2015), SVHN (Netzer et al., 2011) and SYN (Ganin & Lempitsky, 2015). Images from different digit datasets differ drastically in font style, stroke color and background. Office-Home is composed of four domains (Artistic, Clipart, Product and Real World) with around 15,500 images of 65 classes for home and office object recognition. The results are shown in Tables 8 and 9 where no domain labels are used in MixStyle. Similar to the results on PACS, here we observe that MixStyle also brings clear improvements to the baseline CNN model and outperforms all general-purpose regularization methods on both Digits-DG and Office-Home. Compared with more sophisticated DG methods like L2A-OT, MixStyle’s performance is comparable, despite being much simpler to train and consuming much less computing resources. ", + "bbox": [ + 173, + 691, + 825, + 872 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/68a58d31208db3a9f4e6ac1e12e9d0e16e30afd9e38d48a9a578cbeecfbc00e1.jpg", + "table_caption": [ + "Table 9: Leave-one-domain-out generalization results on Office-Home. " + ], + "table_footnote": [], + "table_body": "
MethodArtisticClipartProductReal WorldAvg
JiGen CCSA53.047.571.572.861.2
MMD-AAE59.9 56.549.9 47.374.1 72.175.7 74.864.9 62.7
CrossGrad75.864.4
L2A-OT58.449.473.9
ResNet1860.6 58.9±0.350.1 49.4±0.174.8 74.3±0.177.0 76.2±0.265.6 64.7
+ Manifold Mixup56.2±0.446.3±0.373.6±0.175.2±0.262.8
+ Mixup w/o label interpolation + Cutout57.0±0.2 57.8±0.248.7±0.271.4±0.674.5±0.462.9
48.1±0.373.9±0.275.8±0.363.9
+ CutMix57.9±0.148.3±0.374.5±0.175.6±0.464.1
+ DropBlock58.0±0.148.1±0.174.3±0.375.9±0.464.1
+ Mixup + MixStyle (ours)58.2±0.1 58.7±0.349.3±0.2 53.4±0.274.7±0.1 74.2±0.176.1±0.1 75.9±0.164.6 65.5
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See Algorithm 1 in", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 720, + 304, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 304, + 733 + ], + "score": 1.0, + "content": "Appendix A.1 for the PyTorch-like pseudo-code.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 54.5, + "bbox_fs": [ + 104, + 688, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 118, + 101, + 494, + 292 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 176, + 80, + 433, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 177, + 80, + 434, + 93 + ], + "spans": [ + { + "bbox": [ + 177, + 80, + 434, + 93 + ], + "score": 1.0, + "content": "Table 1: Leave-one-domain-out generalization results on PACS.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 118, + 101, + 494, + 292 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 101, + 494, + 292 + ], + "spans": [ + { + "bbox": [ + 118, + 101, + 494, + 292 + ], + "score": 0.985, + "html": "
MethodArtCartoonPhotoSketchAvg
MMD-AAE CCSA JiGen75.2 80.5 79.4 79.8 82.172.7 76.9 75.3 76.896.0 93.6 96.0 96.064.2 66.8 71.6 70.277.0 79.4 80.5 80.7
Epi-FCR Metareg L2A-OT ResNet-1883.7 83.3 77.0±0.677.0 77.2 78.2 75.9±0.693.9 95.5 96.2 96.0±0.173.0 70.3 73.6 69.2±0.681.5 81.7 82.8 79.5
+ Manifold Mixup75.6±0.770.1±0.993.5±0.765.4±0.676.2
+ Cutout74.9±0.474.9±0.695.9±0.367.7±0.978.3
+ CutMix74.6±0.771.8±0.695.6±0.465.3±0.876.8
74.7±1.072.3±0.9
+ Mixup (w/o label interpolation)93.0±0.469.2±0.277.3
+ Mixup76.8±0.774.9±0.795.8±0.366.6±0.778.5
+ DropBlock76.4±0.775.4±0.795.9±0.369.0±0.379.2
+ MixStyle w/ random shuffle82.3±0.279.0±0.396.3±0.373.8±0.982.8
+ MixStyle w/ domain label84.1±0.478.8±0.496.1±0.375.9±0.9
83.7
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We choose the PACS dataset (Li et al., 2017), a commonly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 369, + 504, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 504, + 380 + ], + "score": 1.0, + "content": "used domain generalization (DG) benchmark concerned with domain shift in image classification.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 379, + 504, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 504, + 392 + ], + "score": 1.0, + "content": "PACS consists of four domains, i.e. Art Painting, Cartoon, Photo and Sketch, with totally 9,991 im-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "ages of 7 classes. As shown in Fig. 1, the domain shift mainly corresponds to image style changes.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 400, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 415 + ], + "score": 1.0, + "content": "For evaluation, a model is trained on three domains and tested on the remaining one. Following", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "prior work (Li et al., 2019; Zhou et al., 2020a), we use ResNet-18 (He et al., 2016) as the classi-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "fier where MixStyle is inserted after the 1st, 2nd and 3rd residual blocks. Our code is based on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 434, + 252, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 252, + 447 + ], + "score": 1.0, + "content": "Dassl.pytorch (Zhou et al., 2020c).2", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 104, + 451, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 505, + 467 + ], + "score": 1.0, + "content": "Baselines. Our main baselines are general-purpose regularization methods including Mixup (Zhang", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "et al., 2018b), Manifold Mixup (Verma et al., 2019), DropBlock (Ghiasi et al., 2018), CutMix (Yun", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "et al., 2019) and Cutout (DeVries & Taylor, 2017), which are trained using the same training parame-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "score": 1.0, + "content": "ters as MixStyle and the optimal hyper-parameter setup as reported in their papers. We also compare", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "with the existing DG methods which reported state-of-the-art performance on PACS. These include", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "domain alignment-based CCSA (Motiian et al., 2017) and MMD-AAE (Li et al., 2018b), Jigsaw", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "puzzle-based JiGen (Carlucci et al., 2019), adversarial gradient-based CrossGrad (Shankar et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 528, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 542 + ], + "score": 1.0, + "content": "2018), meta-learning-based Metareg (Balaji et al., 2018) and Epi-FCR (Li et al., 2019), and data", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 540, + 310, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 310, + 553 + ], + "score": 1.0, + "content": "augmentation-based L2A-OT (Zhou et al., 2020a).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "Comparison with general-purpose regularization methods. The results are shown in Table 1.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "Overall, we observe that the general-purpose regularization methods do not offer any clear advan-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "score": 1.0, + "content": "tage over the vanilla ResNet-18 in this DG task, while MixStyle improves upon the vanilla ResNet-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 376, + 604 + ], + "score": 1.0, + "content": "18 with a significant margin. Compared with Mixup, MixStyle is", + "type": "text" + }, + { + "bbox": [ + 376, + 592, + 398, + 602 + ], + "score": 0.87, + "content": "5 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "better on average. Recall", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 602, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 616 + ], + "score": 1.0, + "content": "that Mixup also interpolates the output space, we further compare with a variant of Mixup in order", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "to demonstrate the advantage of mixing style statistics at the feature level over mixing images at the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 623, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 638 + ], + "score": 1.0, + "content": "pixel level for DG—following Sohn et al. (2020), we remove the label interpolation in Mixup and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 345, + 648 + ], + "score": 1.0, + "content": "sample the mixing weights from a uniform distribution of", + "type": "text" + }, + { + "bbox": [ + 346, + 635, + 366, + 647 + ], + "score": 0.27, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 635, + 506, + 648 + ], + "score": 1.0, + "content": ". Still, MixStyle outperforms this", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "new baseline with a large margin, which justifies our claim. MixStyle and DropBlock share some", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 658, + 504, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 504, + 669 + ], + "score": 1.0, + "content": "commonalities in that they are both applied to feature maps at multiple layers, but MixStyle signifi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "cantly outperforms DropBlock in all test domains. The reason why DropBlock is ineffective here is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "because dropping out activations mainly encourages a network to mine discriminative patterns, but", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "score": 1.0, + "content": "does not reinforce the ability to cope with unseen styles, which is exactly what MixStyle aims to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 702, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 506, + 714 + ], + "score": 1.0, + "content": "achieve: by synthesizing “new” styles (domains) MixStyle regularizes the network to become more", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 29.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 118, + 722, + 363, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 720, + 364, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 720, + 364, + 734 + ], + "score": 1.0, + "content": "2https://github.com/KaiyangZhou/Dassl.pytorch.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 118, + 101, + 494, + 292 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 176, + 80, + 433, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 177, + 80, + 434, + 93 + ], + "spans": [ + { + "bbox": [ + 177, + 80, + 434, + 93 + ], + "score": 1.0, + "content": "Table 1: Leave-one-domain-out generalization results on PACS.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 118, + 101, + 494, + 292 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 101, + 494, + 292 + ], + "spans": [ + { + "bbox": [ + 118, + 101, + 494, + 292 + ], + "score": 0.985, + "html": "
MethodArtCartoonPhotoSketchAvg
MMD-AAE CCSA JiGen75.2 80.5 79.4 79.8 82.172.7 76.9 75.3 76.896.0 93.6 96.0 96.064.2 66.8 71.6 70.277.0 79.4 80.5 80.7
Epi-FCR Metareg L2A-OT ResNet-1883.7 83.3 77.0±0.677.0 77.2 78.2 75.9±0.693.9 95.5 96.2 96.0±0.173.0 70.3 73.6 69.2±0.681.5 81.7 82.8 79.5
+ Manifold Mixup75.6±0.770.1±0.993.5±0.765.4±0.676.2
+ Cutout74.9±0.474.9±0.695.9±0.367.7±0.978.3
+ CutMix74.6±0.771.8±0.695.6±0.465.3±0.876.8
74.7±1.072.3±0.9
+ Mixup (w/o label interpolation)93.0±0.469.2±0.277.3
+ Mixup76.8±0.774.9±0.795.8±0.366.6±0.778.5
+ DropBlock76.4±0.775.4±0.795.9±0.369.0±0.379.2
+ MixStyle w/ random shuffle82.3±0.279.0±0.396.3±0.373.8±0.982.8
+ MixStyle w/ domain label84.1±0.478.8±0.496.1±0.375.9±0.9
83.7
", + "type": "table", + "image_path": "c5ebe274bb1893ac2346fb0d139387bcd5d7bf3e991b91a59bcca22de9df6956.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 118, + 101, + 494, + 164.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 118, + 164.66666666666666, + 494, + 228.33333333333331 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 118, + 228.33333333333331, + 494, + 292.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "title", + "bbox": [ + 107, + 312, + 200, + 325 + ], + "lines": [ + { + "bbox": [ + 104, + 310, + 202, + 326 + ], + "spans": [ + { + "bbox": [ + 104, + 310, + 202, + 326 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 107, + 336, + 344, + 348 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 346, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 346, + 349 + ], + "score": 1.0, + "content": "3.1 GENERALIZATION IN CATEGORY CLASSIFICATION", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 357, + 505, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 370 + ], + "score": 1.0, + "content": "Dataset and implementation details. We choose the PACS dataset (Li et al., 2017), a commonly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 369, + 504, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 504, + 380 + ], + "score": 1.0, + "content": "used domain generalization (DG) benchmark concerned with domain shift in image classification.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 379, + 504, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 504, + 392 + ], + "score": 1.0, + "content": "PACS consists of four domains, i.e. Art Painting, Cartoon, Photo and Sketch, with totally 9,991 im-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "ages of 7 classes. As shown in Fig. 1, the domain shift mainly corresponds to image style changes.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 400, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 415 + ], + "score": 1.0, + "content": "For evaluation, a model is trained on three domains and tested on the remaining one. Following", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 425 + ], + "score": 1.0, + "content": "prior work (Li et al., 2019; Zhou et al., 2020a), we use ResNet-18 (He et al., 2016) as the classi-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "fier where MixStyle is inserted after the 1st, 2nd and 3rd residual blocks. Our code is based on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 434, + 252, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 252, + 447 + ], + "score": 1.0, + "content": "Dassl.pytorch (Zhou et al., 2020c).2", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 105, + 357, + 505, + 447 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 453, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 104, + 451, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 104, + 451, + 505, + 467 + ], + "score": 1.0, + "content": "Baselines. Our main baselines are general-purpose regularization methods including Mixup (Zhang", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 505, + 477 + ], + "score": 1.0, + "content": "et al., 2018b), Manifold Mixup (Verma et al., 2019), DropBlock (Ghiasi et al., 2018), CutMix (Yun", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "et al., 2019) and Cutout (DeVries & Taylor, 2017), which are trained using the same training parame-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 499 + ], + "score": 1.0, + "content": "ters as MixStyle and the optimal hyper-parameter setup as reported in their papers. We also compare", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "with the existing DG methods which reported state-of-the-art performance on PACS. These include", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "domain alignment-based CCSA (Motiian et al., 2017) and MMD-AAE (Li et al., 2018b), Jigsaw", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "puzzle-based JiGen (Carlucci et al., 2019), adversarial gradient-based CrossGrad (Shankar et al.,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 528, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 505, + 542 + ], + "score": 1.0, + "content": "2018), meta-learning-based Metareg (Balaji et al., 2018) and Epi-FCR (Li et al., 2019), and data", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 540, + 310, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 310, + 553 + ], + "score": 1.0, + "content": "augmentation-based L2A-OT (Zhou et al., 2020a).", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 451, + 506, + 553 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 559, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 559, + 505, + 570 + ], + "score": 1.0, + "content": "Comparison with general-purpose regularization methods. The results are shown in Table 1.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "Overall, we observe that the general-purpose regularization methods do not offer any clear advan-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 505, + 594 + ], + "score": 1.0, + "content": "tage over the vanilla ResNet-18 in this DG task, while MixStyle improves upon the vanilla ResNet-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 376, + 604 + ], + "score": 1.0, + "content": "18 with a significant margin. Compared with Mixup, MixStyle is", + "type": "text" + }, + { + "bbox": [ + 376, + 592, + 398, + 602 + ], + "score": 0.87, + "content": "5 . 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 592, + 505, + 604 + ], + "score": 1.0, + "content": "better on average. Recall", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 602, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 616 + ], + "score": 1.0, + "content": "that Mixup also interpolates the output space, we further compare with a variant of Mixup in order", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "to demonstrate the advantage of mixing style statistics at the feature level over mixing images at the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 623, + 506, + 638 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 506, + 638 + ], + "score": 1.0, + "content": "pixel level for DG—following Sohn et al. (2020), we remove the label interpolation in Mixup and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 635, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 345, + 648 + ], + "score": 1.0, + "content": "sample the mixing weights from a uniform distribution of", + "type": "text" + }, + { + "bbox": [ + 346, + 635, + 366, + 647 + ], + "score": 0.27, + "content": "[ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 635, + 506, + 648 + ], + "score": 1.0, + "content": ". Still, MixStyle outperforms this", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 506, + 659 + ], + "score": 1.0, + "content": "new baseline with a large margin, which justifies our claim. MixStyle and DropBlock share some", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 658, + 504, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 504, + 669 + ], + "score": 1.0, + "content": "commonalities in that they are both applied to feature maps at multiple layers, but MixStyle signifi-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 506, + 681 + ], + "score": 1.0, + "content": "cantly outperforms DropBlock in all test domains. The reason why DropBlock is ineffective here is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 692 + ], + "score": 1.0, + "content": "because dropping out activations mainly encourages a network to mine discriminative patterns, but", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 506, + 703 + ], + "score": 1.0, + "content": "does not reinforce the ability to cope with unseen styles, which is exactly what MixStyle aims to", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 702, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 506, + 714 + ], + "score": 1.0, + "content": "achieve: by synthesizing “new” styles (domains) MixStyle regularizes the network to become more", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "robust to domain shift. In addition, it is interesting to see that on Cartoon and Photo, MixStyle w/", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 105 + ], + "score": 1.0, + "content": "random shuffle obtains slightly better results. The reason might be because there exist sub-domains", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "in a source domain (see Fig. 4(a-c)), which allow random shuffling to produce more diverse “new”", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 338, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 338, + 127 + ], + "score": 1.0, + "content": "domains that lead to a more domain-generalizable model.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 559, + 506, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 94 + ], + "score": 1.0, + "content": "robust to domain shift. In addition, it is interesting to see that on Cartoon and Photo, MixStyle w/", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 105 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 506, + 105 + ], + "score": 1.0, + "content": "random shuffle obtains slightly better results. The reason might be because there exist sub-domains", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 506, + 117 + ], + "score": 1.0, + "content": "in a source domain (see Fig. 4(a-c)), which allow random shuffling to produce more diverse “new”", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 338, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 338, + 127 + ], + "score": 1.0, + "content": "domains that lead to a more domain-generalizable model.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 107, + 134, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 134, + 504, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 504, + 147 + ], + "score": 1.0, + "content": "Comparison with state-of-the-art DG methods. Overall, MixStyle outperforms most DG meth-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "ods by a clear margin, despite being a much simpler method. The performance of MixStyle w/", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 202, + 170 + ], + "score": 1.0, + "content": "domain label is nearly", + "type": "text" + }, + { + "bbox": [ + 202, + 156, + 217, + 167 + ], + "score": 0.85, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 156, + 506, + 170 + ], + "score": 1.0, + "content": "better on average than the recently introduced L2A-OT. From a data", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "augmentation perspective, MixStyle and L2A-OT share a similar goal—to synthesize data from", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "pseudo-novel domains. MixStyle accomplishes this goal through mixing style statistics at the fea-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "ture level. Whereas L2A-OT works at the pixel level: it trains an image generator by maximizing the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "score": 1.0, + "content": "domain difference (measured by optimal transport) between the original and the generated images,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "score": 1.0, + "content": "which introduces much heavier computational overhead than MixStyle in terms of GPU memory", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 222, + 504, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 504, + 235 + ], + "score": 1.0, + "content": "and training time. It is worth noting that MixStyle’s domain label-free version is highly competitive:", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 233, + 287, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 118, + 245 + ], + "score": 1.0, + "content": "its", + "type": "text" + }, + { + "bbox": [ + 119, + 233, + 145, + 244 + ], + "score": 0.86, + "content": "8 2 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 234, + 287, + 245 + ], + "score": 1.0, + "content": "accuracy is on par with L2A-OT’s.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5 + }, + { + "type": "title", + "bbox": [ + 107, + 258, + 317, + 269 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 319, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 319, + 271 + ], + "score": 1.0, + "content": "3.2 GENERALIZATION IN INSTANCE RETRIEVAL", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 504, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 504, + 291 + ], + "score": 1.0, + "content": "Dataset and implementation details. We evaluate MixStyle on the person re-identification (re-ID)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "problem, which aims to match people across disjoint camera views. As each camera view is itself", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "a distinct domain, person re-ID is essentially a cross-domain image matching problem. Instead of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "using the standard protocol where training and test data come from the same camera views, we adopt", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "the cross-dataset setting so test camera views are never seen during training. Specifically, we train a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "model on one dataset and then test its performance on the other dataset. Two commonly used re-ID", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 358 + ], + "score": 1.0, + "content": "datasets are adopted: Market1501 (Zheng et al., 2015) and Duke (Ristani et al., 2016; Zheng et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "2017). Ranking accuracy and mean average precision (mAP) are used as the performance measures", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "(displayed in percentage). We test MixStyle on two CNN architectures: ResNet-50 (He et al., 2016)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "and OSNet (Zhou et al., 2019). The latter was designed specifically for re-ID. In both architectures,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "MixStyle is inserted after the 1st and 2nd residual blocks. Our code is based on Torchreid (Zhou &", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 399, + 169, + 413 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 169, + 413 + ], + "score": 1.0, + "content": "Xiang, 2019).3", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "score": 1.0, + "content": "Baselines. We compare with three baseline methods: 1) The vanilla model, which serves as a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "strong baseline; 2) DropBlock, which was the top-performing competitor in Table 1; 3) Ran-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "domErase (Zhong et al., 2020), a widely used regularization method in the re-ID literature (similar", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 452, + 152, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 152, + 464 + ], + "score": 1.0, + "content": "to Cutout).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 504, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 504, + 483 + ], + "score": 1.0, + "content": "Results. The results are reported in Table 2. It is clear that only MixStyle consistently outperforms", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "the strong vanilla model under both settings with considerable margins, while DropBlock and Ran-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "domErase are unable to show any benefit. Notably, RandomErase, which simulates occlusion by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "erasing pixels in random rectangular regions with random values, has been used as a default trick", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "when training re-ID CNNs. However, RandomErase shows a detrimental effect in the cross-dataset", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "re-ID setting. Indeed, similar to DropBlock, randomly erasing pixels offers no guarantee to improve", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 537, + 291, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 291, + 548 + ], + "score": 1.0, + "content": "the robustness when it comes to domain shift.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "title", + "bbox": [ + 107, + 562, + 346, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 347, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 347, + 574 + ], + "score": 1.0, + "content": "3.3 GENERALIZATION IN REINFORCEMENT LEARNING", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "Though RL has been greatly advanced by using CNNs for feature learning in raw pixels (Mnih et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 592, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 607 + ], + "score": 1.0, + "content": "2013), it has been widely acknowledged that RL agents often overfit training environments while", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 605, + 423, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 423, + 618 + ], + "score": 1.0, + "content": "generalize poorly to unseen environments (Cobbe et al., 2019; Igl et al., 2019).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "Dataset and implementation details. We conduct experiments on Coinrun (Cobbe et al., 2019),", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "a recently introduced RL benchmark for evaluating the generalization performance of RL agents.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "As shown in Fig. 3(a), the goal in Coinrun is to control a character to collect golden coins while", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "avoiding both stationary and non-stationary obstacles. We follow Igl et al. (2019) to construct and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "train our RL agent: the CNN architecture used in IMPALA (Espeholt et al., 2018) is adopted as", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "the policy network, and is trained by the Proximal Policy Optimization (PPO) algorithm (Schulman", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "score": 1.0, + "content": "et al., 2017). Please refer to Igl et al. (2019) for further implementation details. MixStyle is inserted", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "after the 1st and 2nd convolutional sequences. Training data are sampled from 500 levels while test", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 45.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 117, + 722, + 379, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 381, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 381, + 734 + ], + "score": 1.0, + "content": "3https://github.com/KaiyangZhou/deep-person-reid.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 127 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 134, + 505, + 244 + ], + "lines": [ + { + "bbox": [ + 106, + 134, + 504, + 147 + ], + "spans": [ + { + "bbox": [ + 106, + 134, + 504, + 147 + ], + "score": 1.0, + "content": "Comparison with state-of-the-art DG methods. Overall, MixStyle outperforms most DG meth-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "spans": [ + { + "bbox": [ + 105, + 145, + 505, + 158 + ], + "score": 1.0, + "content": "ods by a clear margin, despite being a much simpler method. The performance of MixStyle w/", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 156, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 156, + 202, + 170 + ], + "score": 1.0, + "content": "domain label is nearly", + "type": "text" + }, + { + "bbox": [ + 202, + 156, + 217, + 167 + ], + "score": 0.85, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 156, + 506, + 170 + ], + "score": 1.0, + "content": "better on average than the recently introduced L2A-OT. From a data", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 505, + 180 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 180 + ], + "score": 1.0, + "content": "augmentation perspective, MixStyle and L2A-OT share a similar goal—to synthesize data from", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 178, + 505, + 191 + ], + "score": 1.0, + "content": "pseudo-novel domains. MixStyle accomplishes this goal through mixing style statistics at the fea-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 189, + 505, + 202 + ], + "score": 1.0, + "content": "ture level. Whereas L2A-OT works at the pixel level: it trains an image generator by maximizing the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 213 + ], + "score": 1.0, + "content": "domain difference (measured by optimal transport) between the original and the generated images,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 224 + ], + "score": 1.0, + "content": "which introduces much heavier computational overhead than MixStyle in terms of GPU memory", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 222, + 504, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 222, + 504, + 235 + ], + "score": 1.0, + "content": "and training time. It is worth noting that MixStyle’s domain label-free version is highly competitive:", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 233, + 287, + 245 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 118, + 245 + ], + "score": 1.0, + "content": "its", + "type": "text" + }, + { + "bbox": [ + 119, + 233, + 145, + 244 + ], + "score": 0.86, + "content": "8 2 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 234, + 287, + 245 + ], + "score": 1.0, + "content": "accuracy is on par with L2A-OT’s.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 8.5, + "bbox_fs": [ + 105, + 134, + 506, + 245 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 258, + 317, + 269 + ], + "lines": [ + { + "bbox": [ + 106, + 258, + 319, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 319, + 271 + ], + "score": 1.0, + "content": "3.2 GENERALIZATION IN INSTANCE RETRIEVAL", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 279, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 106, + 280, + 504, + 291 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 504, + 291 + ], + "score": 1.0, + "content": "Dataset and implementation details. We evaluate MixStyle on the person re-identification (re-ID)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "problem, which aims to match people across disjoint camera views. As each camera view is itself", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 313 + ], + "score": 1.0, + "content": "a distinct domain, person re-ID is essentially a cross-domain image matching problem. Instead of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "using the standard protocol where training and test data come from the same camera views, we adopt", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 506, + 336 + ], + "score": 1.0, + "content": "the cross-dataset setting so test camera views are never seen during training. Specifically, we train a", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 505, + 346 + ], + "score": 1.0, + "content": "model on one dataset and then test its performance on the other dataset. Two commonly used re-ID", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 344, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 358 + ], + "score": 1.0, + "content": "datasets are adopted: Market1501 (Zheng et al., 2015) and Duke (Ristani et al., 2016; Zheng et al.,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 355, + 506, + 369 + ], + "score": 1.0, + "content": "2017). Ranking accuracy and mean average precision (mAP) are used as the performance measures", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "(displayed in percentage). We test MixStyle on two CNN architectures: ResNet-50 (He et al., 2016)", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 506, + 391 + ], + "score": 1.0, + "content": "and OSNet (Zhou et al., 2019). The latter was designed specifically for re-ID. In both architectures,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "MixStyle is inserted after the 1st and 2nd residual blocks. Our code is based on Torchreid (Zhou &", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 399, + 169, + 413 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 169, + 413 + ], + "score": 1.0, + "content": "Xiang, 2019).3", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 280, + 506, + 413 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 419, + 504, + 462 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "score": 1.0, + "content": "Baselines. We compare with three baseline methods: 1) The vanilla model, which serves as a", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 442 + ], + "score": 1.0, + "content": "strong baseline; 2) DropBlock, which was the top-performing competitor in Table 1; 3) Ran-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 506, + 453 + ], + "score": 1.0, + "content": "domErase (Zhong et al., 2020), a widely used regularization method in the re-ID literature (similar", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 452, + 152, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 152, + 464 + ], + "score": 1.0, + "content": "to Cutout).", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 417, + 506, + 464 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 471, + 505, + 547 + ], + "lines": [ + { + "bbox": [ + 105, + 471, + 504, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 504, + 483 + ], + "score": 1.0, + "content": "Results. The results are reported in Table 2. It is clear that only MixStyle consistently outperforms", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 505, + 494 + ], + "score": 1.0, + "content": "the strong vanilla model under both settings with considerable margins, while DropBlock and Ran-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 505 + ], + "score": 1.0, + "content": "domErase are unable to show any benefit. Notably, RandomErase, which simulates occlusion by", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 504, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 515 + ], + "score": 1.0, + "content": "erasing pixels in random rectangular regions with random values, has been used as a default trick", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "when training re-ID CNNs. However, RandomErase shows a detrimental effect in the cross-dataset", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "re-ID setting. Indeed, similar to DropBlock, randomly erasing pixels offers no guarantee to improve", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 537, + 291, + 548 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 291, + 548 + ], + "score": 1.0, + "content": "the robustness when it comes to domain shift.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 471, + 505, + 548 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 562, + 346, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 561, + 347, + 574 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 347, + 574 + ], + "score": 1.0, + "content": "3.3 GENERALIZATION IN REINFORCEMENT LEARNING", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 582, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "Though RL has been greatly advanced by using CNNs for feature learning in raw pixels (Mnih et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 592, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 607 + ], + "score": 1.0, + "content": "2013), it has been widely acknowledged that RL agents often overfit training environments while", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 605, + 423, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 423, + 618 + ], + "score": 1.0, + "content": "generalize poorly to unseen environments (Cobbe et al., 2019; Igl et al., 2019).", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 582, + 505, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 505, + 712 + ], + "lines": [ + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 637 + ], + "score": 1.0, + "content": "Dataset and implementation details. We conduct experiments on Coinrun (Cobbe et al., 2019),", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "a recently introduced RL benchmark for evaluating the generalization performance of RL agents.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "As shown in Fig. 3(a), the goal in Coinrun is to control a character to collect golden coins while", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "avoiding both stationary and non-stationary obstacles. We follow Igl et al. (2019) to construct and", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 681 + ], + "score": 1.0, + "content": "train our RL agent: the CNN architecture used in IMPALA (Espeholt et al., 2018) is adopted as", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 679, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 691 + ], + "score": 1.0, + "content": "the policy network, and is trained by the Proximal Policy Optimization (PPO) algorithm (Schulman", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 702 + ], + "score": 1.0, + "content": "et al., 2017). Please refer to Igl et al. (2019) for further implementation details. MixStyle is inserted", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "after the 1st and 2nd convolutional sequences. Training data are sampled from 500 levels while test", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "data are drawn from new levels of only the highest difficulty. As domain labels are difficult to define,", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 468, + 476, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 476, + 481 + ], + "score": 1.0, + "content": "we use the random shuffle version of MixStyle. Our code is built on top of Igl et al. (2019).4", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 45.5, + "bbox_fs": [ + 105, + 624, + 506, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 118, + 101, + 493, + 238 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 165, + 80, + 444, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 79, + 445, + 93 + ], + "spans": [ + { + "bbox": [ + 165, + 79, + 445, + 93 + ], + "score": 1.0, + "content": "Table 2: Generalization results on the cross-dataset person re-ID task.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 118, + 101, + 493, + 238 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 101, + 493, + 238 + ], + "spans": [ + { + "bbox": [ + 118, + 101, + 493, + 238 + ], + "score": 0.983, + "html": "
ModelMarket1501-→DukeDuke-→Market1501
mAPR1R5R10mAPR1R5R10
ResNet-5019.335.450.356.420.445.263.670.9
+ RandomErase14.327.842.649.116.138.556.864.5
+ DropBlock18.233.249.156.319.745.362.169.1
+ MixStyle w/ random shuffle23.842.258.864.824.151.569.476.2
+ MixStyle w/ domain label23.443.358.964.724.753.070.977.8
OSNet25.944.759.665.424.052.267.574.7
+ RandomErase20.536.252.359.322.449.166.173.0
+ DropBlock23.141.556.562.521.748.265.471.3
+ MixStyle w/ random shuffle27.248.262.768.427.858.174.081.0
+ MixStyle w/ domain label27.347.562.067.129.058.274.980.9
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(b) Test performance in unseen environments. (c) Difference", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 411, + 264, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 264, + 424 + ], + "score": 1.0, + "content": "between training and test performance.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + } + ], + "index": 6.25 + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 504, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "data are drawn from new levels of only the highest difficulty. As domain labels are difficult to define,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 468, + 476, + 481 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 476, + 481 + ], + "score": 1.0, + "content": "we use the random shuffle version of MixStyle. Our code is built on top of Igl et al. (2019).4", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 518, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "Baselines. Following Igl et al. (2019), we train strong baseline models and add MixStyle on top of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "them to see whether MixStyle can bring further improvements. To this end, we train two baseline", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "models: 1) Baseline, which combines weight decay and data augmentation;5 2) IBAC-SNI (the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 142, + 561 + ], + "score": 0.88, + "content": "\\lambda = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "version), the best-performing model in Igl et al. (2019) which is based on selective noise", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 562, + 147, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 147, + 576 + ], + "score": 1.0, + "content": "injection.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "Results. The test performance is shown in Fig. 3(b). Comparing Baseline (blue) with Base-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "line+MixStyle (orange), we can see that MixStyle brings a significant improvement. Interestingly,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "the variance is also significantly reduced by using MixStyle, as indicated by the smaller shaded areas", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "(for both orange and red lines). These results strongly demonstrate the effectiveness of MixStyle in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "enhancing generalization for RL agents. When it comes to the stronger baseline IBAC-SNI (green),", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 665, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 505, + 680 + ], + "score": 1.0, + "content": "MixStyle (red) is able to bring further performance gain, suggesting that MixStyle is complemen-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "tary to IBAC-SNI. This result also shows the potential of MixStyle as a plug-and-play component", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "to be combined with other advanced RL methods. It is worth noting that Baseline+MixStyle itself", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "is already highly competitive with IBAC-SNI. Fig. 3(c) shows the generalization gap from which", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "it can been seen that the models trained with MixStyle (orange & red) clearly generalize faster and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 720, + 335, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 335, + 733 + ], + "score": 1.0, + "content": "better than those without using MixStyle (blue & green).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "spans": [ + { + "bbox": [ + 105, + 25, + 293, + 39 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 118, + 101, + 493, + 238 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 165, + 80, + 444, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 79, + 445, + 93 + ], + "spans": [ + { + "bbox": [ + 165, + 79, + 445, + 93 + ], + "score": 1.0, + "content": "Table 2: Generalization results on the cross-dataset person re-ID task.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 118, + 101, + 493, + 238 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 101, + 493, + 238 + ], + "spans": [ + { + "bbox": [ + 118, + 101, + 493, + 238 + ], + "score": 0.983, + "html": "
ModelMarket1501-→DukeDuke-→Market1501
mAPR1R5R10mAPR1R5R10
ResNet-5019.335.450.356.420.445.263.670.9
+ RandomErase14.327.842.649.116.138.556.864.5
+ DropBlock18.233.249.156.319.745.362.169.1
+ MixStyle w/ random shuffle23.842.258.864.824.151.569.476.2
+ MixStyle w/ domain label23.443.358.964.724.753.070.977.8
OSNet25.944.759.665.424.052.267.574.7
+ RandomErase20.536.252.359.322.449.166.173.0
+ DropBlock23.141.556.562.521.748.265.471.3
+ MixStyle w/ random shuffle27.248.262.768.427.858.174.081.0
+ MixStyle w/ domain label27.347.562.067.129.058.274.980.9
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(b) Test performance in unseen environments. (c) Difference", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 411, + 264, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 264, + 424 + ], + "score": 1.0, + "content": "between training and test performance.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7.5 + } + ], + "index": 6.25 + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 504, + 481 + ], + "lines": [], + "index": 9.5, + "bbox_fs": [ + 105, + 457, + 505, + 481 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 518, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 530 + ], + "score": 1.0, + "content": "Baselines. Following Igl et al. (2019), we train strong baseline models and add MixStyle on top of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "them to see whether MixStyle can bring further improvements. To this end, we train two baseline", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "models: 1) Baseline, which combines weight decay and data augmentation;5 2) IBAC-SNI (the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 142, + 561 + ], + "score": 0.88, + "content": "\\lambda = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "version), the best-performing model in Igl et al. (2019) which is based on selective noise", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 562, + 147, + 576 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 147, + 576 + ], + "score": 1.0, + "content": "injection.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 518, + 506, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "Results. The test performance is shown in Fig. 3(b). Comparing Baseline (blue) with Base-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 635 + ], + "score": 1.0, + "content": "line+MixStyle (orange), we can see that MixStyle brings a significant improvement. Interestingly,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "the variance is also significantly reduced by using MixStyle, as indicated by the smaller shaded areas", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "(for both orange and red lines). These results strongly demonstrate the effectiveness of MixStyle in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "enhancing generalization for RL agents. When it comes to the stronger baseline IBAC-SNI (green),", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 665, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 104, + 665, + 505, + 680 + ], + "score": 1.0, + "content": "MixStyle (red) is able to bring further performance gain, suggesting that MixStyle is complemen-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "tary to IBAC-SNI. This result also shows the potential of MixStyle as a plug-and-play component", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "to be combined with other advanced RL methods. It is worth noting that Baseline+MixStyle itself", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "is already highly competitive with IBAC-SNI. Fig. 3(c) shows the generalization gap from which", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "it can been seen that the models trained with MixStyle (orange & red) clearly generalize faster and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 720, + 335, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 335, + 733 + ], + "score": 1.0, + "content": "better than those without using MixStyle (blue & green).", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 610, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 108, + 79, + 503, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 108, + 79, + 503, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 79, + 503, + 248 + ], + "spans": [ + { + "bbox": [ + 108, + 79, + 503, + 248 + ], + "score": 0.975, + "type": "image", + "image_path": "4d2d1d46181970a60036d74149364fc1b7c14abba9cfcdf85a1601e0d521acda.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 108, + 79, + 503, + 135.33333333333334 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 108, + 135.33333333333334, + 503, + 191.66666666666669 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 108, + 191.66666666666669, + 503, + 248.00000000000003 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 257, + 506, + 290 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 256, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 256, + 506, + 270 + ], + "score": 1.0, + "content": "Figure 4: 2-D visualization of flattened feature maps (top) and the corresponding style statistics", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 269, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 107, + 269, + 505, + 280 + ], + "score": 1.0, + "content": "(bottom). res1-4 denote the four residual blocks in order in a ResNet architecture. We observe that", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 107, + 280, + 503, + 291 + ], + "spans": [ + { + "bbox": [ + 107, + 280, + 503, + 291 + ], + "score": 1.0, + "content": "res1 to res3 contain domain-related information while res4 encodes label-related information.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "table", + "bbox": [ + 154, + 339, + 310, + 433 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 146, + 305, + 464, + 318 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 145, + 304, + 465, + 319 + ], + "spans": [ + { + "bbox": [ + 145, + 304, + 465, + 319 + ], + "score": 1.0, + "content": "Table 3: Ablation study on where to apply MixStyle in the ResNet architecture.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "table_caption", + "bbox": [ + 165, + 326, + 299, + 337 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 326, + 299, + 338 + ], + "spans": [ + { + "bbox": [ + 165, + 326, + 299, + 338 + ], + "score": 1.0, + "content": "(a) Category classification on PACS.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "table_body", + "bbox": [ + 154, + 339, + 310, + 433 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 154, + 339, + 310, + 433 + ], + "spans": [ + { + "bbox": [ + 154, + 339, + 310, + 433 + ], + "score": 0.952, + "html": "
ModelAccuracy
ResNet-1879.5
+ MixStyle (res1)80.1
+ MixStyle (res12)81.6
+ MixStyle (res123)82.8
+ MixStyle (res1234)75.6
+ MixStyle (res14)76.3
+ MixStyle (res23)81.7
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ModelmAP
ResNet-5019.3
+ MixStyle (res1)22.6
+ MixStyle (res12) + MixStyle (res123)23.8
+MixStyle (res1234)22.0
+ MixStyle (res14)10.2
+MixStyle (res23)11.1 20.6
", + "type": "table", + "image_path": "77ff508fa1e5b4b56a82c9108d2ab847baee3536e4d32e484735f7ac7a2f04a3.jpg" + } + ] + } + ], + "index": 16.5, + "virtual_lines": [ + { + "bbox": [ + 318, + 340, + 457, + 386.5 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 318, + 386.5, + 457, + 433.0 + ], + "spans": [], + "index": 17 + } + ] + } + ], + "index": 12.25 + }, + { + "type": "title", + "bbox": [ + 107, + 468, + 175, + 478 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 177, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 177, + 480 + ], + "score": 1.0, + "content": "3.4 ANALYSIS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 489, + 505, + 666 + ], + "lines": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 505, + 502 + ], + "score": 1.0, + "content": "Where to apply MixStyle? We repeat the experiments on PACS (category classification) and the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 505, + 512 + ], + "score": 1.0, + "content": "re-ID datasets (instance retrieval) using the ResNet architecture. 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ModelAccuracy
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ModelmAP
ResNet-5019.3
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In (b), M and D denote Market1501 and Duke respectively.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 251, + 375, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 376, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 376, + 263 + ], + "score": 1.0, + "content": "Mixing vs. replacing. Unlike the AdaIN formulation, which com-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 262, + 376, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 376, + 275 + ], + "score": 1.0, + "content": "pletely replaces one style with the other, MixStyle mixes two styles", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 273, + 376, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 376, + 285 + ], + "score": 1.0, + "content": "via a convex combination. Table 4 shows that mixing is better than", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 283, + 376, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 376, + 297 + ], + "score": 1.0, + "content": "replacing. This is easy to understand: mixing diversifies the styles", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 295, + 315, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 315, + 308 + ], + "score": 1.0, + "content": "(imagine an interpolation between two data points).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 376, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 376, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 376, + 326 + ], + "score": 1.0, + "content": "Random vs. fixed shuffle at multiple layers. Applying MixStyle", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 325, + 376, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 376, + 336 + ], + "score": 1.0, + "content": "to multiple layers, which has been shown advantageous in Table 3,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 336, + 376, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 376, + 347 + ], + "score": 1.0, + "content": "raises another question of whether to shuffle the mini-batch at dif-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 347, + 376, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 376, + 358 + ], + "score": 1.0, + "content": "ferent layers or use the same shuffled order for all layers. Table 5", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 357, + 377, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 377, + 370 + ], + "score": 1.0, + "content": "suggests that using random shuffle at different layers gives a better", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 369, + 376, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 376, + 380 + ], + "score": 1.0, + "content": "performance, which may be attributed to the increased noise level", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 380, + 264, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 264, + 391 + ], + "score": 1.0, + "content": "that gives a better regularization effect.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13 + }, + { + "type": "table", + "bbox": [ + 383, + 258, + 502, + 298 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 384, + 239, + 503, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 383, + 236, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 383, + 236, + 505, + 253 + ], + "score": 1.0, + "content": "Table 4: Mixing vs. replacing.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "table_body", + "bbox": [ + 383, + 258, + 502, + 298 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 383, + 258, + 502, + 298 + ], + "spans": [ + { + "bbox": [ + 383, + 258, + 502, + 298 + ], + "score": 0.967, + "html": "
Accuracy (%)
Mixing82.8±0.4
Replacing82.1±0.5
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Accuracy (%)
Random82.8±0.4
Fixed82.4±0.5
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Recall that", + "type": "text" + }, + { + "bbox": [ + 293, + 400, + 301, + 408 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "is used to control the shape of Beta distribution,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 325, + 421 + ], + "score": 1.0, + "content": "which has a direct effect on how the convex weights", + "type": "text" + }, + { + "bbox": [ + 326, + 410, + 333, + 419 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 410, + 443, + 421 + ], + "score": 1.0, + "content": "are sampled. The smaller", + "type": "text" + }, + { + "bbox": [ + 444, + 411, + 452, + 419 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "is, the more", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 421, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 184, + 432 + ], + "score": 1.0, + "content": "likely the value in", + "type": "text" + }, + { + "bbox": [ + 184, + 421, + 192, + 430 + ], + "score": 0.73, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 421, + 468, + 432 + ], + "score": 1.0, + "content": "is close to the extreme value of 0 or 1. In other words, a smaller", + "type": "text" + }, + { + "bbox": [ + 468, + 422, + 476, + 430 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 421, + 505, + 432 + ], + "score": 1.0, + "content": "favors", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 430, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 453, + 444 + ], + "score": 1.0, + "content": "the style statistics in Eqs. (5) & (6) to be dominated by one side. We first evaluate", + "type": "text" + }, + { + "bbox": [ + 453, + 433, + 461, + 441 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "on PACS.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 211, + 454 + ], + "score": 1.0, + "content": "Fig. 5(a) shows that with", + "type": "text" + }, + { + "bbox": [ + 212, + 444, + 219, + 452 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 442, + 434, + 454 + ], + "score": 1.0, + "content": "increasing from 0.1 to 0.4, the accuracy slides from", + "type": "text" + }, + { + "bbox": [ + 434, + 442, + 462, + 453 + ], + "score": 0.86, + "content": "8 2 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 442, + 474, + 454 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 474, + 442, + 501, + 453 + ], + "score": 0.85, + "content": "8 1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 442, + 505, + 454 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 221, + 466 + ], + "score": 1.0, + "content": "However, further increasing", + "type": "text" + }, + { + "bbox": [ + 221, + 455, + 229, + 463 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "does not impact on the accuracy. Therefore, the results suggest that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 264, + 477 + ], + "score": 1.0, + "content": "the performance is not too sensitive to", + "type": "text" + }, + { + "bbox": [ + 264, + 466, + 272, + 474 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 464, + 333, + 477 + ], + "score": 1.0, + "content": "; and selecting", + "type": "text" + }, + { + "bbox": [ + 333, + 466, + 340, + 474 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 464, + 365, + 477 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 365, + 464, + 423, + 476 + ], + "score": 0.92, + "content": "\\lbrace 0 . 1 , 0 . 2 , 0 . 3 \\rbrace", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "seems to be a good", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 276, + 487 + ], + "score": 1.0, + "content": "starting point. We further experiment with", + "type": "text" + }, + { + "bbox": [ + 276, + 475, + 353, + 487 + ], + "score": 0.92, + "content": "\\alpha \\in \\{ 0 . 1 , 0 . 2 , 0 . 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "on the re-ID datasets and the Coinrun", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "benchmark. Figs. 5(b) & (c) show that in general the variance for the results of different values is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 104, + 496, + 321, + 510 + ], + "score": 1.0, + "content": "small. Therefore, we suggest practitioners to choose", + "type": "text" + }, + { + "bbox": [ + 321, + 499, + 329, + 507 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 496, + 353, + 510 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 353, + 497, + 411, + 509 + ], + "score": 0.84, + "content": "\\lbrace 0 . 1 , 0 . 2 , 0 . 3 \\rbrace", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 496, + 435, + 510 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 436, + 497, + 471, + 507 + ], + "score": 0.88, + "content": "\\alpha = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "being a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 508, + 190, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 190, + 521 + ], + "score": 1.0, + "content": "good default setting.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 352, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 353, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 353, + 538 + ], + "score": 1.0, + "content": "For more analyses and discussions, please see Appendix A.2.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 108, + 552, + 211, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 213, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 213, + 567 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Domain generalization, or DG, studies out-of-distribution (OOD) generalization given only source", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "data typically composed of multiple related but distinct domains. We refer readers to Zhou et al.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 598, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 614 + ], + "score": 1.0, + "content": "(2021) for a comprehensive survey in this topic. Many DG methods are based on the idea of aligning", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "features between different sources, with a hope that the model can be invariant to domain shift given", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 504, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 504, + 633 + ], + "score": 1.0, + "content": "unseen data. For instance, Li et al. (2018b) achieved distribution alignment in the hidden represen-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "tation of an autoencoder using maximum mean discrepancy; Li et al. (2018c) resorted to adversarial", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "learning with auxiliary domain classifiers to learn features that are domain-agnostic. Some works", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "explored domain-specific parameterization, such as domain-specific weight matrices (Li et al., 2017)", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "and domain-specific BN (Seo et al., 2020). Recently, meta-learning has drawn increasing attention", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "from the DG community (Li et al., 2018a; Balaji et al., 2018; Dou et al., 2019). The main idea", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "is to expose a model to domain shift during training by using pseudo-train and pseudo-test do-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "score": 1.0, + "content": "mains, both drawn from source domains. Data augmentation has also been investigated for learning", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "domain-invariant models. Shankar et al. 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In (b), M and D denote Market1501 and Duke respectively.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 107, + 251, + 375, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 376, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 376, + 263 + ], + "score": 1.0, + "content": "Mixing vs. replacing. Unlike the AdaIN formulation, which com-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 262, + 376, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 376, + 275 + ], + "score": 1.0, + "content": "pletely replaces one style with the other, MixStyle mixes two styles", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 273, + 376, + 285 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 376, + 285 + ], + "score": 1.0, + "content": "via a convex combination. Table 4 shows that mixing is better than", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 283, + 376, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 283, + 376, + 297 + ], + "score": 1.0, + "content": "replacing. This is easy to understand: mixing diversifies the styles", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 295, + 315, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 295, + 315, + 308 + ], + "score": 1.0, + "content": "(imagine an interpolation between two data points).", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 250, + 376, + 308 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 376, + 390 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 376, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 376, + 326 + ], + "score": 1.0, + "content": "Random vs. fixed shuffle at multiple layers. 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Table 5", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 357, + 377, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 377, + 370 + ], + "score": 1.0, + "content": "suggests that using random shuffle at different layers gives a better", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 369, + 376, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 376, + 380 + ], + "score": 1.0, + "content": "performance, which may be attributed to the increased noise level", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 380, + 264, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 264, + 391 + ], + "score": 1.0, + "content": "that gives a better regularization effect.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 313, + 377, + 391 + ] + }, + { + "type": "table", + "bbox": [ + 383, + 258, + 502, + 298 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 384, + 239, + 503, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 383, + 236, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 383, + 236, + 505, + 253 + ], + "score": 1.0, + "content": "Table 4: Mixing vs. replacing.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "table_body", + "bbox": [ + 383, + 258, + 502, + 298 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 383, + 258, + 502, + 298 + ], + "spans": [ + { + "bbox": [ + 383, + 258, + 502, + 298 + ], + "score": 0.967, + "html": "
Accuracy (%)
Mixing82.8±0.4
Replacing82.1±0.5
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Accuracy (%)
Random82.8±0.4
Fixed82.4±0.5
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Recall that", + "type": "text" + }, + { + "bbox": [ + 293, + 400, + 301, + 408 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 397, + 506, + 411 + ], + "score": 1.0, + "content": "is used to control the shape of Beta distribution,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 410, + 505, + 421 + ], + "spans": [ + { + "bbox": [ + 106, + 410, + 325, + 421 + ], + "score": 1.0, + "content": "which has a direct effect on how the convex weights", + "type": "text" + }, + { + "bbox": [ + 326, + 410, + 333, + 419 + ], + "score": 0.8, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 410, + 443, + 421 + ], + "score": 1.0, + "content": "are sampled. The smaller", + "type": "text" + }, + { + "bbox": [ + 444, + 411, + 452, + 419 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 410, + 505, + 421 + ], + "score": 1.0, + "content": "is, the more", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 421, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 421, + 184, + 432 + ], + "score": 1.0, + "content": "likely the value in", + "type": "text" + }, + { + "bbox": [ + 184, + 421, + 192, + 430 + ], + "score": 0.73, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 421, + 468, + 432 + ], + "score": 1.0, + "content": "is close to the extreme value of 0 or 1. 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We first evaluate", + "type": "text" + }, + { + "bbox": [ + 453, + 433, + 461, + 441 + ], + "score": 0.77, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 430, + 506, + 444 + ], + "score": 1.0, + "content": "on PACS.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 442, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 211, + 454 + ], + "score": 1.0, + "content": "Fig. 5(a) shows that with", + "type": "text" + }, + { + "bbox": [ + 212, + 444, + 219, + 452 + ], + "score": 0.8, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 442, + 434, + 454 + ], + "score": 1.0, + "content": "increasing from 0.1 to 0.4, the accuracy slides from", + "type": "text" + }, + { + "bbox": [ + 434, + 442, + 462, + 453 + ], + "score": 0.86, + "content": "8 2 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 442, + 474, + 454 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 474, + 442, + 501, + 453 + ], + "score": 0.85, + "content": "8 1 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 442, + 505, + 454 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 452, + 506, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 221, + 466 + ], + "score": 1.0, + "content": "However, further increasing", + "type": "text" + }, + { + "bbox": [ + 221, + 455, + 229, + 463 + ], + "score": 0.75, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 452, + 506, + 466 + ], + "score": 1.0, + "content": "does not impact on the accuracy. Therefore, the results suggest that", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 464, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 264, + 477 + ], + "score": 1.0, + "content": "the performance is not too sensitive to", + "type": "text" + }, + { + "bbox": [ + 264, + 466, + 272, + 474 + ], + "score": 0.74, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 464, + 333, + 477 + ], + "score": 1.0, + "content": "; and selecting", + "type": "text" + }, + { + "bbox": [ + 333, + 466, + 340, + 474 + ], + "score": 0.79, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 464, + 365, + 477 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 365, + 464, + 423, + 476 + ], + "score": 0.92, + "content": "\\lbrace 0 . 1 , 0 . 2 , 0 . 3 \\rbrace", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 464, + 505, + 477 + ], + "score": 1.0, + "content": "seems to be a good", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 475, + 276, + 487 + ], + "score": 1.0, + "content": "starting point. We further experiment with", + "type": "text" + }, + { + "bbox": [ + 276, + 475, + 353, + 487 + ], + "score": 0.92, + "content": "\\alpha \\in \\{ 0 . 1 , 0 . 2 , 0 . 3 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "on the re-ID datasets and the Coinrun", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 486, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 505, + 498 + ], + "score": 1.0, + "content": "benchmark. Figs. 5(b) & (c) show that in general the variance for the results of different values is", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 104, + 496, + 321, + 510 + ], + "score": 1.0, + "content": "small. Therefore, we suggest practitioners to choose", + "type": "text" + }, + { + "bbox": [ + 321, + 499, + 329, + 507 + ], + "score": 0.78, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 496, + 353, + 510 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 353, + 497, + 411, + 509 + ], + "score": 0.84, + "content": "\\lbrace 0 . 1 , 0 . 2 , 0 . 3 \\rbrace", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 496, + 435, + 510 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 436, + 497, + 471, + 507 + ], + "score": 0.88, + "content": "\\alpha = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "being a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 508, + 190, + 521 + ], + "spans": [ + { + "bbox": [ + 106, + 508, + 190, + 521 + ], + "score": 1.0, + "content": "good default setting.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 397, + 506, + 521 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 525, + 352, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 353, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 353, + 538 + ], + "score": 1.0, + "content": "For more analyses and discussions, please see Appendix A.2.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 524, + 353, + 538 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 552, + 211, + 565 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 213, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 213, + 567 + ], + "score": 1.0, + "content": "4 RELATED WORK", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "Domain generalization, or DG, studies out-of-distribution (OOD) generalization given only source", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 601 + ], + "score": 1.0, + "content": "data typically composed of multiple related but distinct domains. We refer readers to Zhou et al.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 598, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 614 + ], + "score": 1.0, + "content": "(2021) for a comprehensive survey in this topic. Many DG methods are based on the idea of aligning", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 624 + ], + "score": 1.0, + "content": "features between different sources, with a hope that the model can be invariant to domain shift given", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 622, + 504, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 504, + 633 + ], + "score": 1.0, + "content": "unseen data. For instance, Li et al. (2018b) achieved distribution alignment in the hidden represen-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 505, + 645 + ], + "score": 1.0, + "content": "tation of an autoencoder using maximum mean discrepancy; Li et al. (2018c) resorted to adversarial", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "learning with auxiliary domain classifiers to learn features that are domain-agnostic. Some works", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "explored domain-specific parameterization, such as domain-specific weight matrices (Li et al., 2017)", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "and domain-specific BN (Seo et al., 2020). Recently, meta-learning has drawn increasing attention", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "from the DG community (Li et al., 2018a; Balaji et al., 2018; Dou et al., 2019). The main idea", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "is to expose a model to domain shift during training by using pseudo-train and pseudo-test do-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 506, + 713 + ], + "score": 1.0, + "content": "mains, both drawn from source domains. Data augmentation has also been investigated for learning", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "domain-invariant models. Shankar et al. (2018) introduced a cross-gradient training method (Cross-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "Grad) where source data are augmented by adversarial gradients obtained from a domain classifier.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 43.5, + "bbox_fs": [ + 104, + 577, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Gong et al. 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A natural way to improve generalization, which has been shown", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 245 + ], + "score": 1.0, + "content": "effective in (Cobbe et al., 2019; Farebrother et al., 2018), is to use regularization, e.g., weight decay.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 242, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 242, + 505, + 255 + ], + "score": 1.0, + "content": "However, Igl et al. 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Metareg: Towards domain gener-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 116, + 634, + 335, + 646 + ], + "spans": [ + { + "bbox": [ + 116, + 634, + 335, + 646 + ], + "score": 1.0, + "content": "alization using meta-regularization. In NeurIPS, 2018.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 108, + 651, + 504, + 674 + ], + "lines": [ + { + "bbox": [ + 106, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 106, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "Gilles Blanchard, Gyemin Lee, and Clayton Scott. Generalizing from several related classification", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 115, + 663, + 325, + 674 + ], + "spans": [ + { + "bbox": [ + 115, + 663, + 325, + 674 + ], + "score": 1.0, + "content": "tasks to a new unlabeled sample. 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They", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "further integrated SNI with information bottleneck actor critic (IBAC-SNI) to reduce the variance", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 506, + 309 + ], + "score": 1.0, + "content": "in gradients. Curriculum learning has been investigated in (Justesen et al., 2018) where the level of", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "training episodes progresses from easy to difficult over the course of training. Gamrian & Goldberg", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 333 + ], + "score": 1.0, + "content": "(2019) leveraged the advances in GAN-based image-to-image translation (Liu et al., 2017) to map", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 342 + ], + "score": 1.0, + "content": "target data to the source domain which the agent was trained on. Tobin et al. (2017) introduced", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 341, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 506, + 354 + ], + "score": 1.0, + "content": "domain randomization, which diversifies training data by rendering images with different visual", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "effects via a programmable simulator. With a similar goal of data augmentation, Lee et al. (2020)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "pre-processed input images with a randomly initialized network. Very recent studies (Laskin et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 372, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 506, + 388 + ], + "score": 1.0, + "content": "2020; Kostrikov et al., 2020) have shown that it is useful to combine a diverse set of label-preserving", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "transformations, such as rotation, shifting and Cutout. Different from the aforementioned methods,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 506, + 409 + ], + "score": 1.0, + "content": "our MixStyle works at the feature level and is orthogonal to most existing methods. For instance,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 407, + 433, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 433, + 419 + ], + "score": 1.0, + "content": "we have shown in Sec. 3.3 that MixStyle significantly improves upon IBAC-SNI.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 197, + 506, + 419 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 434, + 195, + 446 + ], + "lines": [ + { + "bbox": [ + 105, + 432, + 197, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 197, + 450 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 458, + 505, + 513 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "We presented a simple yet effective domain generalization method, termed MixStyle. MixStyle", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 505, + 482 + ], + "score": 1.0, + "content": "mixes the feature statistics of two instances to synthesize novel domains, which is inspired by the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 505, + 492 + ], + "score": 1.0, + "content": "observation in style transfer work that the feature statistics encode style/domain-related information.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 505, + 504 + ], + "score": 1.0, + "content": "Extensive experiments covering a wide range of tasks were conducted to demonstrate that MixStyle", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 503, + 309, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 309, + 514 + ], + "score": 1.0, + "content": "yields new state-of-the-art on three different tasks.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 458, + 505, + 514 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 526, + 200, + 536 + ], + "lines": [ + { + "bbox": [ + 107, + 527, + 200, + 537 + ], + "spans": [ + { + "bbox": [ + 107, + 527, + 200, + 537 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 108, + 545, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "This work was supported in part by the Shanghai Committee of Science and Technology, China", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "spans": [ + { + "bbox": [ + 106, + 555, + 506, + 568 + ], + "score": 1.0, + "content": "(Grant No. 20DZ1100800), in part by the National Natural Science Foundation of China under", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 565, + 506, + 581 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 506, + 581 + ], + "score": 1.0, + "content": "Grant (61876176, U1713208), the National Key Research and Development Program of China", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 578, + 478, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 478, + 591 + ], + "score": 1.0, + "content": "(No. 2020YFC2004800), and in part by the Guangzhou Research Program (201803010066).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38.5, + "bbox_fs": [ + 104, + 545, + 506, + 591 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 605, + 175, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 176, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 176, + 618 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 41 + }, + { + "type": "text", + "bbox": [ + 107, + 623, + 504, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 622, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 505, + 636 + ], + "score": 1.0, + "content": "Yogesh Balaji, Swami Sankaranarayanan, and Rama Chellappa. 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Omni-scale feature learning for", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 356, + 279, + 367 + ], + "spans": [ + { + "bbox": [ + 115, + 356, + 279, + 367 + ], + "score": 1.0, + "content": "person re-identification. In ICCV, 2019.", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "Kaiyang Zhou, Yongxin Yang, Timothy Hospedales, and Tao Xiang. Learning to generate novel", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 116, + 385, + 331, + 397 + ], + "spans": [ + { + "bbox": [ + 116, + 385, + 331, + 397 + ], + "score": 1.0, + "content": "domains for domain generalization. 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A: Art. C: Cartoon. P: Photo. S: Sketch.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 143, + 101, + 468, + 139 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 143, + 101, + 468, + 139 + ], + "spans": [ + { + "bbox": [ + 143, + 101, + 468, + 139 + ], + "score": 0.937, + "html": "
MethodC,P,SA,P,SA,C,SA,C,PAvg
Vanilla99.49±0.0399.47±0.0499.38±0.0299.65±0.0399.50
MixStyle99.55±0.0299.54±0.0199.47±0.0399.68±0.0399.56
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MethodMNIST MNIST-M SVHN SYNAvg
JiGen96.5 61.4 63.7 74.073.9
CCSA95.2 58.2 65.5 79.174.5
MMD-AAECrossGradL2A-OT96.5 58.4 65.0 78.474.675.878.1
96.7 61.1 65.3 80.2
96.7 63.9 68.6 83.2
CNN+ Mixup w/o label interpolation+ Manifold Mixup+ CutMix+ Mixup+ Cutout+ DropBlock+ MixStyle (ours)95.8±0.3 58.8±0.5 61.7±0.5 78.6±0.673.7
93.7±0.6 55.2±1.0 61.6±0.9 74.4±0.871.271.771.8
92.7±0.4 53.1±0.8 64.4±0.2 76.8±0.5
94.9±0.2 50.1±0.5 64.1±0.9 78.1±0.7
94.2±0.5 56.5±0.8 63.3±0.7 76.7±0.672.774.175.376.5
95.8±0.4 58.4±0.6 61.9±0.9 80.6±0.5
96.2±0.1 60.5±0.6 64.1±0.8 80.2±0.6
96.5±0.3 63.5±0.8 64.7±0.7 81.2±0.8
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Accuracy (%)
ResNet18 79.5
+MixStyle w/ same-domain 80.4
+MixStyle w/random shuffle 82.8
+MixStyle w/ domain label 83.7
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To prove that MixStyle does not sacrifice the performance on", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "seen domains in exchange for gains on unseen domains, we report the test accuracy on the held-out", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 529, + 336, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 336, + 542 + ], + "score": 1.0, + "content": "validation set of the source domains on PACS in Table 7.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 106, + 548, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "Results on Digits-DG and Office-Home. In addition to the experiments on PACS (in Sec. 3.1),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 560, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 571 + ], + "score": 1.0, + "content": "we further evaluate MixStyle’s effectiveness on two DG datasets, namely Digits-DG (Zhou et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 570, + 504, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 504, + 582 + ], + "score": 1.0, + "content": "2020a) and Office-Home (Venkateswara et al., 2017). Digits-DG contains four digit datasets (do-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "mains) including MNIST (LeCun et al., 1998), MNIST-M (Ganin & Lempitsky, 2015), SVHN (Net-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "zer et al., 2011) and SYN (Ganin & Lempitsky, 2015). Images from different digit datasets differ", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "drastically in font style, stroke color and background. Office-Home is composed of four domains", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "(Artistic, Clipart, Product and Real World) with around 15,500 images of 65 classes for home and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 626, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 637 + ], + "score": 1.0, + "content": "office object recognition. The results are shown in Tables 8 and 9 where no domain labels are used", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "in MixStyle. Similar to the results on PACS, here we observe that MixStyle also brings clear im-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 646, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 104, + 646, + 506, + 660 + ], + "score": 1.0, + "content": "provements to the baseline CNN model and outperforms all general-purpose regularization methods", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "on both Digits-DG and Office-Home. Compared with more sophisticated DG methods like L2A-OT,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "MixStyle’s performance is comparable, despite being much simpler to train and consuming much", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 679, + 212, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 212, + 694 + ], + "score": 1.0, + "content": "less computing resources.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 292, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2021", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "14", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 143, + 101, + 468, + 139 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 112, + 80, + 494, + 92 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 116, + 79, + 495, + 92 + ], + "spans": [ + { + "bbox": [ + 116, + 79, + 495, + 92 + ], + "score": 1.0, + "content": "Table 7: Test results on the source domains on PACS. A: Art. C: Cartoon. P: Photo. S: Sketch.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 143, + 101, + 468, + 139 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 143, + 101, + 468, + 139 + ], + "spans": [ + { + "bbox": [ + 143, + 101, + 468, + 139 + ], + "score": 0.937, + "html": "
MethodC,P,SA,P,SA,C,SA,C,PAvg
Vanilla99.49±0.0399.47±0.0499.38±0.0299.65±0.0399.50
MixStyle99.55±0.0299.54±0.0199.47±0.0399.68±0.0399.56
", + "type": "table", + "image_path": "c5c91c980ca863ce754049c2ac48657fb6e478a83339d37cdd2bb99ba07e1b07.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 143, + 101, + 468, + 113.66666666666667 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 143, + 113.66666666666667, + 468, + 126.33333333333334 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 143, + 126.33333333333334, + 468, + 139.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "table", + "bbox": [ + 119, + 169, + 493, + 329 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 167, + 149, + 443, + 161 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 167, + 147, + 443, + 163 + ], + "spans": [ + { + "bbox": [ + 167, + 147, + 443, + 163 + ], + "score": 1.0, + "content": "Table 8: Leave-one-domain-out generalization results on Digits-DG.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "table_body", + "bbox": [ + 119, + 169, + 493, + 329 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 119, + 169, + 493, + 329 + ], + "spans": [ + { + "bbox": [ + 119, + 169, + 493, + 329 + ], + "score": 0.984, + "html": "
MethodMNIST MNIST-M SVHN SYNAvg
JiGen96.5 61.4 63.7 74.073.9
CCSA95.2 58.2 65.5 79.174.5
MMD-AAECrossGradL2A-OT96.5 58.4 65.0 78.474.675.878.1
96.7 61.1 65.3 80.2
96.7 63.9 68.6 83.2
CNN+ Mixup w/o label interpolation+ Manifold Mixup+ CutMix+ Mixup+ Cutout+ DropBlock+ MixStyle (ours)95.8±0.3 58.8±0.5 61.7±0.5 78.6±0.673.7
93.7±0.6 55.2±1.0 61.6±0.9 74.4±0.871.271.771.8
92.7±0.4 53.1±0.8 64.4±0.2 76.8±0.5
94.9±0.2 50.1±0.5 64.1±0.9 78.1±0.7
94.2±0.5 56.5±0.8 63.3±0.7 76.7±0.672.774.175.376.5
95.8±0.4 58.4±0.6 61.9±0.9 80.6±0.5
96.2±0.1 60.5±0.6 64.1±0.8 80.2±0.6
96.5±0.3 63.5±0.8 64.7±0.7 81.2±0.8
", + "type": "table", + "image_path": "d706ed77580ec62fda2f7d6c0da59fdce21563a4ed5924d61fc6e2035f36f821.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 119, + 169, + 493, + 222.33333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 119, + 222.33333333333334, + 493, + 275.6666666666667 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 119, + 275.6666666666667, + 493, + 329.0 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 5.0 + }, + { + "type": "title", + "bbox": [ + 108, + 349, + 221, + 360 + ], + "lines": [ + { + "bbox": [ + 106, + 349, + 221, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 221, + 361 + ], + "score": 1.0, + "content": "A.2 FURTHER ANALYSIS", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 108, + 370, + 295, + 381 + ], + "lines": [ + { + "bbox": [ + 106, + 369, + 297, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 369, + 297, + 382 + ], + "score": 1.0, + "content": "MixStyle between same-domain instances.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 381, + 296, + 501 + ], + "lines": [ + { + "bbox": [ + 106, + 380, + 297, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 297, + 393 + ], + "score": 1.0, + "content": "We are interested in knowing if mixing styles", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 392, + 296, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 296, + 403 + ], + "score": 1.0, + "content": "between same-domain instances helps perfor-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 403, + 297, + 414 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 297, + 414 + ], + "score": 1.0, + "content": "mance. To this end, we sample each mini-batch", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 414, + 297, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 297, + 425 + ], + "score": 1.0, + "content": "from a single domain during training when us-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 425, + 298, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 298, + 435 + ], + "score": 1.0, + "content": "ing MixStyle. The results are shown in Table 6", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 436, + 297, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 297, + 447 + ], + "score": 1.0, + "content": "where we observe that mixing styles between", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 446, + 297, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 236, + 458 + ], + "score": 1.0, + "content": "same-domain instances is about", + "type": "text" + }, + { + "bbox": [ + 236, + 447, + 251, + 457 + ], + "score": 0.85, + "content": "1 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 446, + 297, + 458 + ], + "score": 1.0, + "content": "better than", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 457, + 297, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 297, + 469 + ], + "score": 1.0, + "content": "the baseline model. This suggests that instance-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 469, + 297, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 297, + 480 + ], + "score": 1.0, + "content": "specific style exists. Nonetheless, the perfor-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 480, + 297, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 480, + 297, + 492 + ], + "score": 1.0, + "content": "mance is clearly worse than mixing styles be-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 491, + 258, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 258, + 502 + ], + "score": 1.0, + "content": "tween instances of different domains.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15, + "bbox_fs": [ + 105, + 380, + 298, + 502 + ] + }, + { + "type": "table", + "bbox": [ + 309, + 414, + 500, + 475 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 304, + 383, + 504, + 406 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 303, + 380, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 303, + 380, + 506, + 396 + ], + "score": 1.0, + "content": "Table 6: Investigation on the effect of mixing", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 304, + 395, + 463, + 406 + ], + "spans": [ + { + "bbox": [ + 304, + 395, + 463, + 406 + ], + "score": 1.0, + "content": "styles between same-domain instances.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "table_body", + "bbox": [ + 309, + 414, + 500, + 475 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 309, + 414, + 500, + 475 + ], + "spans": [ + { + "bbox": [ + 309, + 414, + 500, + 475 + ], + "score": 0.977, + "html": "
Accuracy (%)
ResNet18 79.5
+MixStyle w/ same-domain 80.4
+MixStyle w/random shuffle 82.8
+MixStyle w/ domain label 83.7
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To prove that MixStyle does not sacrifice the performance on", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 106, + 520, + 505, + 532 + ], + "score": 1.0, + "content": "seen domains in exchange for gains on unseen domains, we report the test accuracy on the held-out", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 529, + 336, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 336, + 542 + ], + "score": 1.0, + "content": "validation set of the source domains on PACS in Table 7.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 507, + 505, + 542 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 548, + 505, + 691 + ], + "lines": [ + { + "bbox": [ + 106, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "Results on Digits-DG and Office-Home. In addition to the experiments on PACS (in Sec. 3.1),", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 560, + 505, + 571 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 505, + 571 + ], + "score": 1.0, + "content": "we further evaluate MixStyle’s effectiveness on two DG datasets, namely Digits-DG (Zhou et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 570, + 504, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 570, + 504, + 582 + ], + "score": 1.0, + "content": "2020a) and Office-Home (Venkateswara et al., 2017). Digits-DG contains four digit datasets (do-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 505, + 594 + ], + "score": 1.0, + "content": "mains) including MNIST (LeCun et al., 1998), MNIST-M (Ganin & Lempitsky, 2015), SVHN (Net-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 592, + 505, + 605 + ], + "score": 1.0, + "content": "zer et al., 2011) and SYN (Ganin & Lempitsky, 2015). Images from different digit datasets differ", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 604, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 604, + 505, + 615 + ], + "score": 1.0, + "content": "drastically in font style, stroke color and background. Office-Home is composed of four domains", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 626 + ], + "score": 1.0, + "content": "(Artistic, Clipart, Product and Real World) with around 15,500 images of 65 classes for home and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 626, + 505, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 637 + ], + "score": 1.0, + "content": "office object recognition. The results are shown in Tables 8 and 9 where no domain labels are used", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 636, + 505, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 505, + 648 + ], + "score": 1.0, + "content": "in MixStyle. Similar to the results on PACS, here we observe that MixStyle also brings clear im-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 646, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 104, + 646, + 506, + 660 + ], + "score": 1.0, + "content": "provements to the baseline CNN model and outperforms all general-purpose regularization methods", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 670 + ], + "score": 1.0, + "content": "on both Digits-DG and Office-Home. Compared with more sophisticated DG methods like L2A-OT,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "spans": [ + { + "bbox": [ + 105, + 668, + 505, + 681 + ], + "score": 1.0, + "content": "MixStyle’s performance is comparable, despite being much simpler to train and consuming much", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 679, + 212, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 212, + 694 + ], + "score": 1.0, + "content": "less computing resources.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 37, + "bbox_fs": [ + 104, + 548, + 506, + 694 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 118, + 336, + 494, + 495 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 163, + 315, + 448, + 327 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 162, + 315, + 449, + 328 + ], + "spans": [ + { + "bbox": [ + 162, + 315, + 449, + 328 + ], + "score": 1.0, + "content": "Table 9: Leave-one-domain-out generalization results on Office-Home.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 118, + 336, + 494, + 495 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 336, + 494, + 495 + ], + "spans": [ + { + "bbox": [ + 118, + 336, + 494, + 495 + ], + "score": 0.984, + "html": "
MethodArtisticClipartProductReal WorldAvg
JiGen CCSA53.047.571.572.861.2
MMD-AAE59.9 56.549.9 47.374.1 72.175.7 74.864.9 62.7
CrossGrad75.864.4
L2A-OT58.449.473.9
ResNet1860.6 58.9±0.350.1 49.4±0.174.8 74.3±0.177.0 76.2±0.265.6 64.7
+ Manifold Mixup56.2±0.446.3±0.373.6±0.175.2±0.262.8
+ Mixup w/o label interpolation + Cutout57.0±0.2 57.8±0.248.7±0.271.4±0.674.5±0.462.9
48.1±0.373.9±0.275.8±0.363.9
+ CutMix57.9±0.148.3±0.374.5±0.175.6±0.464.1
+ DropBlock58.0±0.148.1±0.174.3±0.375.9±0.464.1
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MethodArtCartoonPhotoSketchAvg
MMD-AAE CCSA JiGen75.2 80.5 79.4 79.8 82.172.7 76.9 75.3 76.896.0 93.6 96.0 96.064.2 66.8 71.6 70.277.0 79.4 80.5 80.7
Epi-FCR Metareg L2A-OT ResNet-1883.7 83.3 77.0±0.677.0 77.2 78.2 75.9±0.693.9 95.5 96.2 96.0±0.173.0 70.3 73.6 69.2±0.681.5 81.7 82.8 79.5
+ Manifold Mixup75.6±0.770.1±0.993.5±0.765.4±0.676.2
+ Cutout74.9±0.474.9±0.695.9±0.367.7±0.978.3
+ CutMix74.6±0.771.8±0.695.6±0.465.3±0.876.8
74.7±1.072.3±0.9
+ Mixup (w/o label interpolation)93.0±0.469.2±0.277.3
+ Mixup76.8±0.774.9±0.795.8±0.366.6±0.778.5
+ DropBlock76.4±0.775.4±0.795.9±0.369.0±0.379.2
+ MixStyle w/ random shuffle82.3±0.279.0±0.396.3±0.373.8±0.982.8
+ MixStyle w/ domain label84.1±0.478.8±0.496.1±0.375.9±0.9
83.7
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ModelMarket1501-→DukeDuke-→Market1501
mAPR1R5R10mAPR1R5R10
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Fixed82.4±0.5
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Replacing82.1±0.5
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MethodMNIST MNIST-M SVHN SYNAvg
JiGen96.5 61.4 63.7 74.073.9
CCSA95.2 58.2 65.5 79.174.5
MMD-AAECrossGradL2A-OT96.5 58.4 65.0 78.474.675.878.1
96.7 61.1 65.3 80.2
96.7 63.9 68.6 83.2
CNN+ Mixup w/o label interpolation+ Manifold Mixup+ CutMix+ Mixup+ Cutout+ DropBlock+ MixStyle (ours)95.8±0.3 58.8±0.5 61.7±0.5 78.6±0.673.7
93.7±0.6 55.2±1.0 61.6±0.9 74.4±0.871.271.771.8
92.7±0.4 53.1±0.8 64.4±0.2 76.8±0.5
94.9±0.2 50.1±0.5 64.1±0.9 78.1±0.7
94.2±0.5 56.5±0.8 63.3±0.7 76.7±0.672.774.175.376.5
95.8±0.4 58.4±0.6 61.9±0.9 80.6±0.5
96.2±0.1 60.5±0.6 64.1±0.8 80.2±0.6
96.5±0.3 63.5±0.8 64.7±0.7 81.2±0.8
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Accuracy (%)
ResNet18 79.5
+MixStyle w/ same-domain 80.4
+MixStyle w/random shuffle 82.8
+MixStyle w/ domain label 83.7
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MethodC,P,SA,P,SA,C,SA,C,PAvg
Vanilla99.49±0.0399.47±0.0499.38±0.0299.65±0.0399.50
MixStyle99.55±0.0299.54±0.0199.47±0.0399.68±0.0399.56
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MethodArtisticClipartProductReal WorldAvg
JiGen CCSA53.047.571.572.861.2
MMD-AAE59.9 56.549.9 47.374.1 72.175.7 74.864.9 62.7
CrossGrad75.864.4
L2A-OT58.449.473.9
ResNet1860.6 58.9±0.350.1 49.4±0.174.8 74.3±0.177.0 76.2±0.265.6 64.7
+ Manifold Mixup56.2±0.446.3±0.373.6±0.175.2±0.262.8
+ Mixup w/o label interpolation + Cutout57.0±0.2 57.8±0.248.7±0.271.4±0.674.5±0.462.9
48.1±0.373.9±0.275.8±0.363.9
+ CutMix57.9±0.148.3±0.374.5±0.175.6±0.464.1
+ DropBlock58.0±0.148.1±0.174.3±0.375.9±0.464.1
+ Mixup + MixStyle (ours)58.2±0.1 58.7±0.349.3±0.2 53.4±0.274.7±0.1 74.2±0.176.1±0.1 75.9±0.164.6 65.5
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The original formulation, however, requires labeled data to identify the superfluous information. In this work, we extend this ability to the multi-view unsupervised setting, where two views of the same underlying entity are provided but the label is unknown. This enables us to identify superfluous information as that not shared by both views. A theoretical analysis leads to the definition of a new multi-view model that produces state-of-the-art results on the Sketchy dataset and label-limited versions of the MIR-Flickr dataset. We also extend our theory to the single-view setting by taking advantage of standard data augmentation techniques, empirically showing better generalization capabilities when compared to common unsupervised approaches for representation learning. + +# 1 INTRODUCTION + +The goal of deep representation learning (LeCun et al., 2015) is to transform a raw observational input, x, into a, typically lower-dimensional, representation, z, that contains the information relevant for a given task or set of tasks. Significant progress has been made in deep learning via supervised representation learning, where the labels, y, for the downstream task are known while $p ( \mathbf { y } \vert \mathbf { x } )$ is learned directly (Sutskever et al., 2012; Hinton et al., 2012). Due to the cost of acquiring large labeled datasets, a recently renewed focus on unsupervised representation learning seeks to generate representations, $\mathbf { z }$ , that are useful for a wide variety of different tasks where little to no labeled data is available (Devlin et al., 2018; Radford et al., 2019). + +Our work is based on the information bottleneck principle (Tishby et al., 2000) where a representation becomes less affected by nuisances by discarding all information from the input that is not useful for a given task, resulting in increased robustness. In the supervised setting, one can directly apply the information bottleneck principle by minimizing the mutual information between the data $\mathbf { x }$ and its representation $\mathbf { z }$ , $I ( \mathbf { x } ; \mathbf { z } )$ , while simultaneously maximizing the mutual information between $\mathbf { z }$ and the label y (Alemi et al., 2017). In the unsupervised setting, discarding only superfluous information is more challenging, as without labels the model cannot directly identify which information is relevant. Recent literature (Devon Hjelm et al., 2019; van den Oord et al., 2018) has focused on the InfoMax objective maximizing $I ( \mathbf { x } , \mathbf { z } )$ instead of minimizing it, to guarantee that all the predictive information is retained by the representation, but doing nothing to discard the irrelevant information. + +In this paper, we extend the information bottleneck method to the unsupervised multi-view setting. To do this, we rely on a basic assumption of the multi-view literature – that each view provides the same task-relevant information (Zhao et al., 2017). Hence, one can improve generalization by discarding all the information not shared by both views from the representation. We do this by maximizing the mutual information between the representations of the two views (Multi-View + +InfoMax objective) while at the same time eliminating the information not shared between them, since it is guaranteed to be superfluous. The resulting representations are more robust for the given task as they have eliminated view specific nuisances. + +Our contributions are three-fold: (1) We extend the information bottleneck principle to the unsupervised multi-view setting and provide a rigorous theoretical analysis of its application. (2) We define a new model 1 that empirically leads to state-of-the-art results in the low-label setting on two standard multi-view datasets, Sketchy and MIR-Flickr. (3) By exploiting data augmentation techniques, we empirically show that the representations learned by our model in single-view settings are more robust than existing unsupervised representation learning methods, connecting our theory to the choice of augmentation strategy. + +# 2 PRELIMINARIES AND FRAMEWORK + +The challenge of representation learning can be formulated as finding a distribution $p ( \mathbf { z } | \mathbf { x } )$ that maps data observations $\mathbf { x } \in \mathbb { X }$ into a representation $\mathbf { z } \in \mathbb { Z }$ , capturing some desired characteristics. Whenever the end goal involves predicting a label y, we consider only $\mathbf { z }$ that are discriminative enough to identify y. This requirement can be quantified by considering the amount of label information that remains accessible after encoding the data, and is known as sufficiency of $\mathbf { z }$ for $\mathbf { y }$ (Achille & Soatto, 2018): + +Definition 1. Sufficiency: A representation $\mathbf { z }$ of $\mathbf { x }$ is sufficient for $\mathbf { y }$ if and only if $I ( \mathbf { x } ; \mathbf { y } | \mathbf { z } ) = 0$ + +Any model that has access to a sufficient representation $\mathbf { z }$ must be able to predict $\mathbf { y }$ at least as accurately as if it has access to the original data $\mathbf { x }$ instead. In fact, $\mathbf { z }$ is sufficient for $\mathbf { y }$ if and only if the amount of information regarding the task is unchanged by the encoding procedure (see Proposition B.1 in the Appendix): + +$$ +I ( \mathbf { x } ; \mathbf { y } | \mathbf { z } ) = 0 \iff I ( \mathbf { x } ; \mathbf { y } ) = I ( \mathbf { y } ; \mathbf { z } ) . +$$ + +Among sufficient representations, the ones that result in better generalization for unlabeled data instances are particularly appealing. When $\mathbf { x }$ has higher information content than $\mathbf { y }$ , some of the information in $\mathbf { x }$ must be irrelevant for the prediction task. This can be better understood by subdividing $I ( \mathbf { x } ; \mathbf { z } )$ into two components by using the chain rule of mutual information (see Appendix A): + +$$ +\begin{array} { r } { I ( \mathbf { x } ; \mathbf { z } ) = \underbrace { I ( \mathbf { x } ; \mathbf { z } | \mathbf { y } ) } _ { \mathrm { s u p e r f l u o u s ~ i n f o r m a t i o n ~ \mu ~ } } + \underbrace { I ( \mathbf { y } ; \mathbf { z } ) } _ { \mathrm { p r e d i c t i v e ~ i n f o r m a t i o n ~ } } . } \end{array} +$$ + +Conditional mutual information $I ( \mathbf { x } ; \mathbf { z } | \mathbf { y } )$ represents the information in $\mathbf { z }$ that is not predictive of $\mathbf { y }$ , i.e. superfluous information. While $I ( \mathbf { y } ; \mathbf { z } )$ determines how much label information is accessible from the representation. Note that this last term is independent of the representation as long as $\mathbf { z }$ is sufficient for $\mathbf { y }$ (see Equation 1). As a consequence, a sufficient representation contains minimal data information whenever $I ( \mathbf { x } ; \mathbf { z } | \mathbf { y } )$ is minimized. + +Minimizing the amount of superfluous information can be done directly only in supervised settings. In fact, reducing $I ( \mathbf { x } ; \mathbf { z } )$ without violating the sufficiency constraint necessarily requires making some additional assumptions on the predictive task (see Theorem B.1 in the Appendix). In the next section we describe the basis of our technique, a strategy to safely reduce the information content of a representation even when the label $\mathbf { y }$ is not observed, by exploiting redundant information in the form of an additional view on the data. + +# 3 MULTI-VIEW INFORMATION BOTTLENECK + +As a motivating example, consider $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ to be two images of the same object from different view-points and let $\mathbf { y }$ be its label. Assuming that the object is clearly distinguishable from both $\mathbf { v } _ { 1 }$ and let $\mathbf { v } _ { 2 }$ , any representation $\mathbf { z }$ containing all information accessible from both views would also contain the necessary label information. Furthermore, if $\mathbf { z }$ captures only the details that are visible from both pictures, it would eliminate the view-specific details and reduce the sensitivity of the representation to view-changes. The theory to support this intuition is described in the following where $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ are jointly observed and referred to as data-views. + +# 3.1 SUFFICIENCY AND ROBUSTNESS IN THE MULTI-VIEW SETTING + +In this section we extend our analysis of sufficiency and minimality to the multi-view setting. + +Intuitively, we can guarantee that $\mathbf { z }$ is sufficient for predicting $\mathbf { y }$ even without knowing y by ensuring that $\mathbf { z }$ maintains all information which is shared by $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ . This intuition relies on a basic assumption of the multi-view environment – that the two views provide the same predictive information. To formalize this we define redundancy. + +Definition 2. Redundancy: $\mathbf { v } _ { 1 }$ is redundant with respect to $\mathbf { v } _ { 2 }$ for $\mathbf { y }$ if and only if $I ( \mathbf { y } ; \mathbf { v } _ { 1 } | \mathbf { v } _ { 2 } ) = 0$ + +Intuitively, a view $\mathbf { v } _ { 1 }$ is redundant for a task whenever it is irrelevant for the prediction of $\mathbf { y }$ if $\mathbf { v } _ { 2 }$ is already observed. Whenever $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ are mutually redundant $\mathbf { \bar { v } } _ { 1 }$ is redundant with respect to $\mathbf { v } _ { 2 }$ for $\mathbf { y }$ , and vice-versa), we can show the following: + +Corollary 1. Let $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ be two mutually redundant views for a target y and let $\mathbf { z } _ { 1 }$ be a representation of $\mathbf { v } _ { 1 }$ . If $\mathbf { \dot { z } } _ { 1 }$ is sufficient for $v _ { 2 } \left( I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) = 0 \right)$ then $\mathbf { z } _ { 1 }$ is as predictive for y as the joint observation of the two views $( I ( \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ; \mathbf { y } ) = I ( \mathbf { y } ; \mathbf { z } _ { 1 } ) )$ . + +In other words, whenever it is possible to assume mutual redundancy, any representation which contains all the information shared by both views (the redundant information) is as predictive as their joint observation. + +By factorizing the mutual information between $\mathbf { v } _ { 1 }$ and $\mathbf { z } _ { 1 }$ analogously to Equation 2, we can identify two components: + +$$ +\begin{array} { r } { I ( { \bf v } _ { 1 } ; { \bf z } _ { 1 } ) = \underbrace { I ( { \bf v } _ { 1 } ; { \bf z } _ { 1 } | { \bf v } _ { 2 } ) } _ { \mathrm { s u p e r f l u o u s ~ i n f o r m a t i o n } } + \underbrace { I ( { \bf v } _ { 2 } ; { \bf z } _ { 1 } ) } _ { \mathrm { p r e d i c t i v e ~ i n f o r m a t i o n ~ f o r ~ } { \bf v } _ { 2 } } . } \end{array} +$$ + +Since $I ( \mathbf { v } _ { 2 } ; \mathbf { z } _ { 1 } )$ has to be maximal if we want the representation to be sufficient for the label, we conclude that $I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } )$ can be reduced by minimizing $I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } )$ . This term intuitively represents the information $\mathbf { z } _ { 1 }$ contains which is unique to $\mathbf { v } _ { 1 }$ and is not predictable by observing $\mathbf { v } _ { 2 }$ . Since we assumed mutual redundancy between the two views, this information must be irrelevant for the predictive task and, therefore, it can be safely discarded. The proofs and formal assertions for the above statements and Corollary 1 can be found in Appendix B. + +The less the two views have in common, the more $I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } )$ can be reduced without violating sufficiency for the label, and consequently, the more robust the resulting representation. At the extreme, $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ share only label information, in which case we can show that $\mathbf { z } _ { 1 }$ is minimal for y and our method is identical to the supervised information bottleneck method without needing to access the labels. Conversely, if $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ are identical, then our method degenerates to the InfoMax principle since no information can be safely discarded (see Appendix E). + +# 3.2 THE MULTI-VIEW INFORMATION BOTTLENECK LOSS FUNCTION + +Given $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ that satisfy the mutual redundancy condition for a label $\mathbf { y }$ , we would like to define an objective function for the representation $\mathbf { z } _ { 1 }$ of $\mathbf { v } _ { 1 }$ that discards as much information as possible without losing any label information. In Section 3.1 we showed that we can obtain sufficiency for $\mathbf { y }$ by ensuring that the representation $\mathbf { z } _ { 1 }$ of $\mathbf { v } _ { 1 }$ is sufficient for $\mathbf { v } _ { 2 }$ , and that decreasing $I ( \mathbf { z } _ { 1 } ; \mathbf { v } _ { 1 } | \mathbf { v } _ { 2 } )$ will increase the robustness of the representation by discarding irrelevant information. So we can combine these two requirements using a relaxed Lagrangian objective to obtain the minimal sufficient representation $\mathbf { z } _ { 1 }$ for $\mathbf { v } _ { 2 }$ : + +$$ +\begin{array} { r } { \mathcal L _ { 1 } ( \boldsymbol \theta ; \lambda _ { 1 } ) = I _ { \boldsymbol \theta } ( \mathbf z _ { 1 } ; \mathbf v _ { 1 } | \mathbf v _ { 2 } ) - \lambda _ { 1 } I _ { \boldsymbol \theta } ( \mathbf v _ { 2 } ; \mathbf z _ { 1 } ) , } \end{array} +$$ + +where $\theta$ denotes the dependency on the parameters of the encoder $p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } )$ , and $\lambda _ { 1 }$ represents the Lagrangian multiplier introduced by the constrained optimization. Symmetrically, we define a loss $\mathcal { L } _ { 2 }$ to optimize the parameters $\psi$ of a conditional distribution $p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } )$ that defines a minimal sufficient representation $\mathbf { z } _ { 2 }$ of the second view $\mathbf { v } _ { 2 }$ for $\mathbf { v } _ { 1 }$ : + +$$ +\begin{array} { r } { \mathcal { L } _ { 2 } ( \psi ; \lambda _ { 2 } ) = I _ { \psi } ( \mathbf { z } _ { 2 } ; \mathbf { v } _ { 2 } | \mathbf { v } _ { 1 } ) - \lambda _ { 2 } I _ { \psi } ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 2 } ) , } \end{array} +$$ + +By defining $\mathbf { z } _ { 1 }$ and $\mathbf { z } _ { 2 }$ on the same domain $\mathbb { Z }$ and re-parametrizing the Lagrangian multipliers, the average of the two loss functions $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ can be upper bounded as follows: + +![](images/1477c2774fb08cee976c3a6575401d2e64f7ffe2704825b5af1736a3d74f0f68.jpg) +Figure 1: Visualization our Multi-View Information Bottleneck model for both multi-view and single-view settings, where $\hat { I } _ { \xi } ( z _ { 1 } ; z _ { 2 } )$ refers to the sample-based parametric mutual information estimation. Whenever $p ( \mathbf { v } _ { 1 } )$ and $p ( \mathbf { v } _ { 2 } )$ have the same distribution, the two encoders can share their parameters. + +$$ +\displaystyle \mathcal { L } _ { M I B } ( \theta , \psi ; \beta ) = - I _ { \theta \psi } ( { \bf z } _ { 1 } ; { \bf z } _ { 2 } ) + \beta D _ { S K L } ( p _ { \theta } ( { \bf z } _ { 1 } | { \bf v } _ { 1 } ) | | p _ { \psi } ( { \bf z } _ { 2 } | { \bf v } _ { 2 } ) ) , +$$ + +where $D _ { S K L }$ represents the symmetrized KL divergence obtained by averaging the expected value of $D _ { \mathrm { K L } } ( p _ { \boldsymbol \theta } ( \mathbf { z } _ { 1 } | \mathbf { \bar { v } } _ { 1 } ) | | p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) )$ and $D _ { \mathrm { K L } } ( p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) | | p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) )$ for joint observations of the two views, while the coefficient $\beta$ defines the trade-off between sufficiency and robustness of the representation, which is a hyper-parameter in this work. The resulting Multi-View Infomation Bottleneck (MIB) model (Equation 5) is visualized in Figure 1, while the batch-based computation of the loss function is summarized in Algorithm 1. + +The symmetrized KL divergence $D _ { S K L } ( p _ { \boldsymbol { \theta } } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) | | p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) )$ can be computed directly whenever $\dot { p _ { \theta } } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } )$ and $p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } )$ have a known density, while the mutual information between the two representations $I _ { \theta \psi } ( \mathbf { z } _ { 1 } ; \mathbf { z } _ { 2 } )$ can be maximized by using any sample-based differentiable mutual information lower bound. We tried the Jensen-Shannon $I _ { \mathrm { J S } }$ (Devon Hjelm et al., 2019; Poole et al., 2019) and the InfoNCE $I _ { \mathrm { N C E } }$ (van den Oord et al., 2018) estimators. These both require introducing an auxiliary parameteric model $C _ { \xi } ( z _ { 1 } , z _ { 2 } )$ which is jointly optimized during the training procedure using re-parametrized samples from $p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } )$ and $p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } )$ . The full derivation for the MIB loss function can be found in Appendix F. + +# 3.3 SELF-SUPERVISION AND INVARIANCE + +Our method can also be applied when multiple views are not available by taking advantage of standard data augmentation techniques. This allows learning invariances directly from the augmented data, rather than requiring them to be built into the model architecture. + +By picking a class $\mathbb { T }$ of data augmentation functions $t : \mathbb { X } \to \mathbb { W }$ that do not affect label information, it is possible to artificially build views that satisfy mutual redundancy for $\mathbf { y }$ . Let $\mathbf { t } _ { 1 }$ and $\mathbf { t } _ { 2 }$ be two random variables over $\mathbb { T }$ , then $\mathbf { v } _ { 1 } : = \mathbf { t } _ { 1 } ( \mathbf { x } )$ and $\mathbf { v } _ { 2 } : = \mathbf { t } _ { 2 } ( \mathbf { x } )$ must be mutually redundant for $\mathbf { y }$ . Since data augmentation functions in $\mathbb { T }$ do not affect label information $( I ( { \bf v } _ { 1 } ; { \bf \bar { y } } ) = I ( { \bf v } _ { 2 } ; { \bf y } ) =$ $I ( \mathbf { x } ; \mathbf { y } ) )$ , a representation $\mathbf { z } _ { 1 }$ of $\mathbf { v } _ { 1 }$ that is sufficient for $\mathbf { v } _ { 2 }$ must contain same amount of predictive information as $\mathbf { x }$ . Formal proofs for this statement can be found in Appendix B.4. + +Whenever the two transformations for the same observation are independent $( I ( \mathbf { t } _ { 1 } ; \mathbf { t } _ { 2 } | \mathbf { x } ) = 0 )$ , they introduce uncorrelated variations in the two views, which will be discarded when creating a representation using our training objective. As an example, if $\mathbb { T }$ represents a set of small translations, the two resulting views will differ by a small shift. Since this information is not shared, any $\mathbf { z } _ { 1 }$ which is optimal according to the MIB objective must discard fine-grained details regarding the position. + +To enable parameter sharing between the encoders, we generate the two views $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ by independently sampling two functions from the same function class $\mathbb { T }$ with uniform probability. As a result, $\mathbf { t } _ { 1 }$ and $\mathbf { t } _ { 2 }$ will have the same distribution, and so the two generated views will also have the same marginals $( p ( \mathbf { v } _ { 1 } ) = p ( \mathbf { v } _ { 2 } ) )$ . For this reason, the two conditional distributions $p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } )$ and $p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } )$ can share their parameters and only one encoder is necessary. Full (or partial) parameter sharing can be also applied in the multi-view settings whenever the two views have the same (or similar) marginal distributions. + +# 4 RELATED WORK + +The relationship between our method and past work on representation learning is best described using the Information Plane (Tishby et al., 2000). In this setting, each representation $\mathbf { z }$ of $\mathbf { x }$ for a predictive task y can be characterised by the amount of information regarding the raw observation $I ( \mathbf { x } ; \mathbf { z } )$ and the corresponding measure of accessible predictive information $I ( \mathbf { y } ; \mathbf { z } )$ $x$ and $y$ axis respectively on Figure 2). Ideally, a good representation would be maximally informative about the label while retaining a minimal amount of information from the observations (top left corner of the parallelogram). Further details on the Information Plane and the bounds visualized in Figure 2 are described in Appendix C. + +Thanks to recent progress in mutual information estimation (Nguyen et al., 2008; Ishmael Belghazi et al., 2018; Poole et al., 2019), the InfoMax principle (Linsker, 1988) has gained attention for unsupervised representation learning (Devon Hjelm et al., 2019; van den Oord et al., 2018). Since the InfoMax objective involves maximizing $I ( \mathbf { x } ; \mathbf { z } )$ , the resulting representation aims to preserve all the information regarding the raw observations (top right corner in Figure 2). + +Concurrent work has applied the InfoMax principle in the Multi-View setting (Ji et al., 2019; Henaff et al., 2019; Tian et al., 2019; Bachman ´ et al., 2019), aiming to maximize mutual infor + +![](images/8293dec173cf38f592d351c0bf24a9a1218f0f2d6584288a4a2daa3512081767.jpg) +Figure 2: Information Plane determined by $I ( \mathbf { x } ; \mathbf { z } )$ $\mathbf { \dot { x } }$ -axis) and $I ( \mathbf { y } ; \mathbf { z } )$ (y-axis). Different objectives are compared based on their target. + +mation between the representation $\mathbf { z }$ of a first data-view $\mathbf { x }$ and a second one $\mathbf { v } _ { 2 }$ . The target representation for the Multi-View InfoMax (MV-InfoMax) models should contain at least the amount of information in $\mathbf { x }$ that is predictive for $\mathbf { v } _ { 2 }$ , targeting the region $I ( \mathbf { z } ; \mathbf { x } ) \ge I ( \mathbf { x } ; \mathbf { v } _ { 2 } )$ on the Information Plane (purple dotted line in Figure 2). Since the MV-InfoMax has no incentive to discard any information regarding $\mathbf { x }$ from z, a representation that is optimal according to the InfoMax principle is also optimal for any MV-InfoMax model. Our model with $\beta = 0$ (Equation 5) belongs to this family of objectives since the incentive to remove superfluous information is removed. Despite their success, Tschannen et al. (2019) has shown that the effectiveness of the InfoMax models is due to inductive biases introduced by the architecture and estimators rather than the training objective itself, since the InfoMax and MV-InfoMax objectives can be trivially maximized by using invertible encoders. + +On the other hand, Variational Autoencoders (VAEs) (Kingma & Welling, 2014) define a training objective that balances compression and reconstruction error (Alemi et al., 2018) through an hyper-parameter $\beta$ . Whenever $\beta$ is close to 0, the VAE objective aims for a lossless representation, approaching the same region of the Information Plane as the one targeted by InfoMax (Barber & Agakov, 2003). When $\beta$ approaches large values, the representation becomes more compressed, showing increased generalization and disentanglement (Higgins et al., 2017; Burgess et al., 2018), and, as $\beta$ approaches infinity, $I ( \mathbf { z } ; \mathbf { x } )$ goes to zero. During this transition from low to high $\beta$ , however, there are no guarantees that VAEs will retain label information (Theorem B.1 in the Appendix). The path between the two regimes depends on how well the label information aligns with the inductive bias introduced by encoder (Jimenez Rezende & Mohamed, 2015; Kingma et al., 2016), prior (Tomczak & Welling, 2018) and decoder architectures (Gulrajani et al., 2017; Chen et al., 2017). + +The idea of discarding irrelevant information was introduced in Tishby et al. (2000) and identified as one of the possible reasons behind the generalization capabilities of deep neural networks by Tishby & Zaslavsky (2015) and Achille & Soatto (2018). Representations based on the information bottleneck principle explicitly minimize the amount of superfluous information in the representation while retaining all the label information from the data (top-left corner of the Information Plane in Figure 2). This direction of research has been explored for both single-view (Alemi et al., 2018) and multi-view settings (Wang et al., 2019), even if explicit label supervision is required to train the representation $\mathbf { z }$ . + +In contrast to all of the above, our work is the first to explicitly identify and discard superfluous information from the representation in the unsupervised multi-view setting. This is because unsupervised models based on the $\beta$ -VAE objective remove information indiscriminately without identifying which part is relevant for teh predictive task, and the InfoMax and Multi-View InfoMax methods do not explicitly try to remove superfluous information at all. The MIB objective, on the other hand, results in the representation with the least superfluous information, i.e. the most robust among the representations that are optimal according to Multi-View InfoMax, without requiring any additional label supervision. + +# 5 EXPERIMENTS + +In this section we demonstrate the effectiveness of our model against state-of-the-art baselines in both the multi-view and single-view setting. In the single-view setting, we also estimate the coordinates on the Information Plane for each of the baseline methods as well as our method to validate the theory in Section 3. + +The results reported in the following sections are obtained using the Jensen-Shannon $I _ { J S }$ (Devon Hjelm et al., 2019; Poole et al., 2019) estimator, which resulted in better performance for MIB and the other InfoMax-based models (Table 2 in the supplementary material). In order to facilitate the comparison between the effect of the different loss functions, the same estimator is used across the different models. + +# 5.1 MULTI-VIEW TASKS + +We compare MIB on the sketch-based image retrieval (Sangkloy et al., 2016) and Flickr multiclass image classification (Huiskes & Lew, 2008) tasks with domain specific and prior multi-view learning methods. + +# 5.1.1 SKETCH-BASED IMAGE RETRIEVAL + +Dataset. The Sketchy dataset (Sangkloy et al., 2016) consists of 12,500 images and 75,471 handdrawn sketches of objects from 125 classes. As in Liu et al. (2017), we also include another 60,502 images from the ImageNet (Deng et al., 2009) from the same classes, which results in total 73,002 natural object images. As per the experimental protocol of Zhang et al. (2018), a total of 6,250 sketches (50 sketches per category) are randomly selected and removed from the training set for testing purpose, which leaves 69,221 sketches for training the model. + +Experimental Setup. The sketch-based image retrieval task is a ranking of 73,002 natural images according to the unseen test (query) sketch. Retrieval is done for our model by generating representations for the query sketch as well as all natural images, and ranking the image by the euclidean distance of their representation from the sketch representation. The baselines use various domain specific ranking methodologies. Model performance is computed based on the class of the ranked pictures corresponding to the query sketch. The training set consists of pairs of image $\mathbf { v } _ { 1 }$ and sketch $\mathbf { v } _ { 2 }$ randomly selected from the same class, to ensure that both views contain the equivalent label information (mutual redundancy). + +Following recent work (Zhang et al., 2018; Dutta & Akata, 2019), we use features extracted from images and sketches by a VGG (Simonyan & Zisserman, 2014) architecture trained for classification on the TU-Berlin dataset (Eitz et al., 2012). The resulting flattened 4096-dimensional feature vectors are fed to our image and sketch encoders to produce a 64-dimensional representation. Both encoders consist of neural networks with hidden layers of 2048 and 1024 units respectively. Size of the representation and regularization strength $\beta$ are tuned on a validation sub-split. We evaluate MIB on five different train/test splits and report mean and standard deviation in Table 5.1.1. Further details on our training procedure and architecture are in Appendix G. + +![](images/726f2e918ff1392fa39ef9bdb9741b3bd6f5d6c84acce49da00d6f18d2bfa125.jpg) +Table 1: Examples of the two views and class label from the Sketchy dataset (on the left) and comparison between MIB and other popular models in literature on the sketch-based image retrieval task (on the right). ∗ denotes models that use a 64-bits binary representation. The results for MIB corresponds to $\beta = 1$ . + +Results. Table 5.1.1 shows that the our model achieves strong performance for both mean average precision $( \mathrm { m A P } @ \mathrm { a l l } )$ and precision at 200 (Prec $@ 2 0 0 )$ , suggesting that the representation is able to capture the common class information between the paired pictures and sketches. The effectiveness of MIB on the retrieval task can be mostly attributed to the regularization introduced with the symmetrized KL divergence between the two encoded views. In addition to discarding view-private information, this term actively aligns the representations of $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ , making the MIB model especially suitable for retrieval tasks + +# 5.1.2 MIR-FLICKR + +Dataset. The MIR-Flickr dataset (Huiskes & Lew, 2008) consists of 1M images annotated with 800K distinct user tags. Each image is represented by a vector of 3,857 hand-crafted image features $( \mathbf { v } _ { 1 } )$ , while the 2,000 most frequent tags are used to produce a 2000-dimensional multi-hot encoding $\left( \mathbf { v } _ { 2 } \right)$ for each picture. The dataset is divided into labeled and unlabeled sets that respectively contain 975K and 25K images, where the labeled set also contains 38 distinct topic classes together with the user tags. Training images with less than two tags are removed, which reduces the total number of training samples to 749,647 pairs (Sohn et al., 2014; Wang et al., 2016). The labeled set contains 5 different splits of train, validation and test sets of size 10K/5K/10K respectively. + +Experimental Setup. Following standard procedure in the literature (Srivastava & Salakhutdinov, 2014; Wang et al., 2016), we train our model on the unlabeled pairs of images and tags. Then a multi-label logistic classifier is trained from the representation of 10K labeled train images to the corresponding macro-categories. The quality of the representation is assessed based on the performance of the trained logistic classifier on the labeled test set. Each encoder consists of a multi-layer perceptron of 4 hidden layers with ReLU activations learning two 1024-dimensional representations $\mathbf { z } _ { 1 }$ and $\mathbf { z } _ { 2 }$ for images $\mathbf { v } _ { 1 }$ and tags $\mathbf { v } _ { 2 }$ respectively. Examples of the two views, labels, and further details on the training procedure are in Appendix G. + +Results. Our MIB model is compared with other popular multi-view learning models in Figure 3 for $\beta = 0$ (Multi-View InfoMax), $\beta = 1$ and $\beta \stackrel { \cdot } { = } 1 0 ^ { - 3 }$ (best on validation set). Although the tuned MIB performs similarly to Multi-View InfoMax with a large number of labels, it outperforms it when fewer labels are available. Furthermore, by choosing a larger $\beta$ the accuracy of our model drastically increases in scarce label regimes, while slightly reducing the accuracy when all the labels are observed (see right side of Figure 3). This effect is likely due to a violation of the mutual redundancy constraint (see Figure 6 in the supplementary material) which can be compensated with smaller values of $\beta$ for less aggressive compression. + +![](images/93e7041326b04365403c56c3f7d87401c68d1dcba01f704729dbfa33da116434.jpg) +Figure 3: Left: mean average precision (mAP) of the classifier trained on different multi-view representations for the MIR-Flickr task. Right: comparing the performance for different values of $\beta$ and percentages of given labeled examples (from $1 \%$ up to $100 \%$ ). Each model uses encoders of comparable size, producing a 1024 dimensional representation. † results from Wang et al. (2016). + +![](images/546e59c737c14c10bb507eb44771651342c9a72b8a20bd6bb7385c5c68344ff4.jpg) + +A possible reason for the effectiveness of MIB against some of the other baselines may be its ability to use mutual information estimators that do not require reconstruction. Both Multi-View VAE (MVAE) and Deep Variational CCA (VCCA) rely on a reconstruction term to capture cross-modal information, which can introduce bias that decreases performance. + +# 5.2 SELF-SUPERVISED SINGLE-VIEW TASK + +In this section, we compare the performance of different unsupervised learning models by measuring their data efficiency and empirically estimating the coordinates of their representation on the Information Plane. Since accurate estimation of mutual information is extremely expensive (McAllester & Stratos, 2018), we focus on relatively small experiments that aim to uncover the difference between popular approaches for representation learning. + +Dataset. The dataset is generated from MNIST by creating the two views, $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ , via the application of data augmentation consisting of small affine transformations and independent pixel corruption to each image. These are kept small enough to ensure that label information is not effected. Each pair of views is generated from the same underlying image, so no label information is used in this process (details in Appendix G). + +Experimental Setup. To evaluate, we train the encoders using the unlabeled multi-view dataset just described, and then fix the representation model. A logistic regression model is trained using the resulting representations along with a subset of labels for the training set, and we report the accuracy of this model on a disjoint test set as is standard for the unsupervised representation learning literature (Tschannen et al., 2019; Tian et al., 2019; van den Oord et al., 2018). We estimate $I ( \mathbf { x } ; \mathbf { z } )$ and $I ( \mathbf { y } ; \mathbf { z } )$ using mutual information estimation networks trained from scratch on the final representations using batches of joint samples $\{ ( { \pmb x } ^ { ( i ) } , { \pmb y } ^ { ( i ) } , z ^ { ( i ) } ) \} _ { i = 1 } ^ { B } \sim p ( { \pmb x } , { \pmb y } ) p _ { \theta } ( { \pmb z } | { \pmb x } )$ . All models are trained using the same encoder architecture consisting of 2 layers of 1024 hidden units with ReLU activations, resulting in 64-dimensional representations. The same data augmentation procedure was also applied for single-view architectures and models were trained for 1 million iterations with batch size $B = 6 4$ . + +Results. Figure 4 summarizes the results. The empirical measurements of mutual information reported on the Information Plane are consistent with the theoretical analysis reported in Section 4: models that retain less information about the data while maintaining the maximal amount of predictive information, result in better classification performance at low-label regimes, confirming the hypothesis that discarding irrelevant information yields robustness and more data-efficient representations. Notably, the MIB model with $\beta = 1$ retains almost exclusively label information, hardly decreasing the classification performance when only one label is used for each data point. + +![](images/041c04c65ea645bac713211faa7fa73f64ba7d3e4c249ab0746f6c9ab1eefe4f.jpg) +Figure 4: Comparing the representations obtained with different objectives on MNIST dataset. The empirical estimation of the coordinates on the Information Plane (in nats on the left) is followed by the respective classification accuracy for different number of randomly sampled labels (from 1 example per label up to 6000 examples per label). Representations that discard more observational information tend to perform better in scarce label regimes. The measurements used to produce the two graphs are reported in Appedix G.4.1. + +# 6 CONCLUSIONS AND FUTURE WORK + +In this work, we introduce Multi-View Information Bottleneck, a novel method for taking advantage of multiple data-views to produce robust representations for downstream tasks. In our experiments, we compared MIB empirically against other approaches in the literature on three such tasks: sketchbased image retrieval, multi-view and unsupervised representation learning. The strong performance obtained in the different areas show that Multi-View Information Bottleneck can be practically applied to various tasks for which the paired observations are either readily available or artificially produced. Furthermore, the positive results on the MIR-Flickr dataset show that our model can work well in practice even when mutual redundancy holds only approximately. + +There are multiple extensions that we would like to explore in future work. One interesting direction would be considering more than two views. In Appendix D we discuss why the mutual redundancy condition cannot be trivially extended to more than two views, but we still believe such an extension is possible. Secondly, we believe that exploring the role played by different choices of data augmentation could bridge the gap between the Information Bottleneck principle and the literature on invariant neural networks (Bloem-Reddy & Whye Teh, 2019), which are able to exploit known symmetries and structure of the data to remove superfluous information. + +# ACKNOWLEDGMENTS + +We thank Andy Keller, Karen Ullrich, Maximillian Ilse and the anonymous reviewers for their feedback and insightful comments. This work has received funding from the ERC under the Horizon 2020 program (grant agreement No. 853489). The Titan Xp and Titan V used for this research were donated by the NVIDIA Corporation. + +# REFERENCES + +Alessandro Achille and Stefano Soatto. Emergence of Invariance and Disentanglement in Deep Representations. JMLR, 2018. + +Alexander A. Alemi, Ian Fischer, Joshua V. Dillon, and Kevin Murphy. Deep Variational Information Bottleneck. In ICLR, 2017. + +Alexander A. 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In ECCV, 2018. + +Jing Zhao, Xijiong Xie, Xin Xu, and Shiliang Sun. Multi-view learning overview: Recent progress and new challenges. IF, 2017. + +# A PROPERTIES OF MUTUAL INFORMATION AND ENTROPY + +In this section we enumerate some of the properties of mutual information that are used to prove the theorems reported in this work. For any random variables $\mathbf { w } , \mathbf { x } , \mathbf { y }$ and $\mathbf { z }$ : + +$( P _ { 1 } )$ ) Positivity: + +$$ +I ( \mathbf { x } ; \mathbf { y } ) \geq 0 , I ( \mathbf { x } ; \mathbf { y } | \mathbf { z } ) \geq 0 +$$ + +$( P _ { 2 } )$ Chain rule: + +$$ +I ( \mathbf { x y } ; \mathbf { z } ) = I ( \mathbf { y } ; \mathbf { z } ) + I ( \mathbf { x } ; \mathbf { z } | \mathbf { y } ) +$$ + +$( P _ { 3 } )$ Chain rule (Multivariate Mutual Information): + +$$ +I ( \mathbf { x } ; \mathbf { y } ; \mathbf { z } ) = I ( \mathbf { y } ; \mathbf { z } ) - I ( \mathbf { y } ; \mathbf { z } | \mathbf { x } ) +$$ + +$( P _ { 4 } )$ Positivity of discrete entropy: For discrete $\mathbf { x }$ + +$$ +H ( \mathbf { x } ) \geq 0 , H ( \mathbf { x } | \mathbf { y } ) \geq 0 +$$ + +$( P _ { 5 } )$ Entropy and Mutual Information + +$$ +H ( \mathbf { x } ) = H ( \mathbf { x } | \mathbf { y } ) + I ( \mathbf { x } ; \mathbf { y } ) +$$ + +# B THEOREMS AND PROOFS + +In the following section we prove the statements reported in the main text of the paper. Whenever a random variable $\mathbf { z }$ is defined to be a representation of another random variable $\mathbf { x }$ , we state that $\mathbf { z }$ is conditionally independent from any other variable in the system once $\mathbf { x }$ is observed. This does not imply that $\mathbf { z }$ must be a deterministic function of $\mathbf { x }$ , but that the source of stochasticity for $\mathbf { z }$ is independent of the other random variables. As a result whenever $\mathbf { z }$ is a representation of $\mathbf { x }$ : + +$$ +I ( \mathbf { z } ; \mathbf { a } | \mathbf { x b } ) = 0 , +$$ + +for any variable (or groups of variables) a and $\mathbf { b }$ in the system. + +# B.1 ON SUFFICIENCY + +Proposition B.1. Let x and y be random variables with joint distribution $p ( \mathbf { x } , \mathbf { y } )$ . Let z be a representation of x, then z is sufficient for y if and only if $I ( \mathbf { x ; y } ) = I ( \mathbf { y ; z } )$ + +Hypothesis: + +$( H _ { 1 }$ ) z is a representation of x: $I ( \mathbf { y } ; \mathbf { z } | \mathbf { x } ) = 0$ + +Thesis: + +$$ +( T _ { 1 } ) \ I ( \mathbf { x } ; \mathbf { y } | \mathbf { z } ) = 0 \iff I ( \mathbf { x } ; \mathbf { y } ) = I ( \mathbf { y } ; \mathbf { z } ) +$$ + +Proof. + +$$ +\begin{array} { r l } & { I ( \mathbf { x } ; \mathbf { y } | \mathbf { z } ) \overset { ( P _ { 3 } ) } { = } I ( \mathbf { x } ; \mathbf { y } ) - I ( \mathbf { x } ; \mathbf { y } ; \mathbf { z } ) \overset { ( P _ { 3 } ) } { = } I ( \mathbf { x } ; \mathbf { y } ) - I ( \mathbf { y } ; \mathbf { z } ) - I ( \mathbf { y } ; \mathbf { z } | \mathbf { x } ) } \\ & { \overset { ( H _ { 1 } ) } { = } I ( \mathbf { x } ; \mathbf { y } ) - I ( \mathbf { y } ; \mathbf { z } ) } \end{array} +$$ + +Since both $I ( \mathbf { x } ; \mathbf { y } )$ and $I ( \mathbf { y } ; \mathbf { z } )$ are non-negative $P _ { 1 } ) , I ( \mathbf { x } ; \mathbf { y } | \mathbf { z } ) = 0 \iff I ( \mathbf { y } ; \mathbf { z } ) = I ( \mathbf { x } ; \mathbf { y } )$ + +# B.2 NO FREE GENERALIZATION + +Theorem B.1. Let x, z and y be random variables with joint distribution $p ( \mathbf { x } , \mathbf { y } , \mathbf { z } )$ . Let $\mathbf { z } ^ { \prime }$ be a representation of x that satisfies $I ( \mathbf { x } ; \mathbf { z } ) > I ( \mathbf { x } ; \mathbf { z } ^ { \prime } )$ , then it is always possible to find a label y for which $\mathbf { z } ^ { \prime }$ is not predictive for y while z is. + +# Hypothesis: + +$\left( H _ { 1 } \right) \textbf { z } ^ { \prime }$ is a representation of x: $I ( \mathbf { y } ; \mathbf { z } ^ { \prime } | \mathbf { x } ) = 0$ $( H _ { 2 } ) I ( { \bf x ; z } ) > I ( { \bf x ; z ^ { \prime } } )$ + +Thesis: + +$$ +( T _ { 1 } ) \ I ( \mathbf { x } ; \mathbf { z } ^ { \prime } ) < I ( \mathbf { x } ; \mathbf { z } ) \implies \exists \mathbf { y } . I ( \mathbf { y } ; \mathbf { z } ) > I ( \mathbf { y } ; \mathbf { z } ^ { \prime } ) = 0 +$$ + +Proof. By construction. + +1. We first factorize $\mathbf { x }$ as a function of two independent random variables (Proposition 2.1 Achille & Soatto (2018)) by picking $\mathbf { y }$ such that: + +$$ +\begin{array} { r } { \left( C _ { 1 } \right) I ( \mathbf { y } ; \mathbf { z } ^ { \prime } ) = 0 } \\ { \left( C _ { 2 } \right) \mathbf { x } = f ( \mathbf { z } ^ { \prime } , \mathbf { y } ) } \end{array} +$$ + +for some deterministic function $f$ . Note that such $\mathbf { y }$ always exists. + +2. Since $\mathbf { x }$ is a function of $\mathbf { y }$ and $\mathbf { z } ^ { \prime }$ : + +$$ +\begin{array} { r l } { \left( C _ { 4 } \right) } & { { } I ( \mathbf { x } ; \mathbf { z } | \mathbf { y } \mathbf { z } ^ { \prime } ) = 0 } \end{array} +$$ + +Considering $I ( \mathbf { y } ; \mathbf { z } )$ : + +$$ +\begin{array} { r l } & { \mathcal { U } _ { 2 , \Psi } ^ { \{ S \} , \{ S \} } ( \frac { \mathcal { H } _ { 2 , \Psi } ^ { \{ S \} } } { 2 } \langle J ( x , x ) | \phi \rangle + I ( x , x ) \langle \phi \rangle } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) + I ( x , y ; x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) , } \\ & { \quad \stackrel { \mathrm { O D } } { \underset { \mathrm { i } \leq 1 } { \sum } } I ( x , y ; x ) , } \\ & \quad \stackrel { \mathrm { O D } } \underset \mathrm { i } \leq \end{array} +$$ + +Since $I ( \mathbf { y } ; \mathbf { z } ^ { \prime } ) = 0$ by construction and $I ( \mathbf { y } ; \mathbf { z } ) > 0$ , the y built in 1. satisfies the conditions reported in the thesis. □ + +Corollary B.1.1. Let $\mathbf { z } ^ { \prime }$ be a representation of x that discards observational information. There is always a label y for which a $\mathbf { z } ^ { \prime }$ is not predictive, while the original observations are. Hypothesis: + +$( H _ { 1 }$ ) x is discrete $\left( H _ { 2 } \right) \textbf { z } ^ { \prime }$ discards information regarding x: $I ( \mathbf { z } ^ { \prime } ; \mathbf { x } ) < H ( \mathbf { x } )$ + +Thesis: + +$$ +( T _ { 1 } ) \ \exists \mathbf { y } . I ( \mathbf { y } ; \mathbf { x } ) > I ( \mathbf { y } ; \mathbf { z } ^ { \prime } ) = 0 +$$ + +Proof. By construction using Theorem B.1. + +1. Set $\mathbf { z } = \mathbf { x }$ : + +$$ +( C _ { 1 } ) \ I ( \mathbf { x } ; \mathbf { z } ) \ { \stackrel { ( P _ { 5 } ) } { = } } \ H ( \mathbf { x } ) - H ( \mathbf { x } | \mathbf { z } ) \ { \stackrel { ( H _ { 1 } ) } { = } } \ H ( \mathbf { x } ) +$$ + +Since the hypothesis are met, we conclude that there exist $\mathbf { y }$ such that $I ( \mathbf { y } ; \mathbf { x } ) > I ( \mathbf { y } ; \mathbf { z } ^ { \prime } ) = 0$ + +# B.3 MULTI-VIEW + +B.3.1 MULTI-VIEW REDUNDANCY AND SUFFICIENCY + +Proposition B.2. Let $\mathbf { v } _ { 1 }$ , $\mathbf { v } _ { 2 }$ , y be random variables with joint distribution $p ( \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { y } )$ . Let $\mathbf { z } _ { 1 }$ be a representation of $\mathbf { v } _ { 1 }$ , then: + +$$ +I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) \leq I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) +$$ + +Hypothesis: + +$( H _ { 1 } ) \ \mathbf { z } _ { 1 }$ is a representation of $\mathbf { v } _ { 1 } \colon I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } \mathbf { v } _ { 1 } ) = 0$ + +Thesis: + +$$ +( T _ { 1 } ) \ I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) \leq I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) +$$ + +Proof. Since $\mathbf { z } _ { 1 }$ is a representation of $\mathbf { v } _ { 1 }$ + +$$ +( C _ { 1 } ) \ I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } \mathbf { v } _ { 1 } ) = 0 +$$ + +Therefore: + +$$ +\begin{array} { r l } { I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) } & { \overset { ( P _ { 3 } ) } { = } I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { z } _ { 1 } \mathbf { v } _ { 2 } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) } \\ & { \overset { ( E ) } { = } I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) - I ( \mathbf { v } _ { 1 } ; \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) } \\ & { \overset { ( E ) } { = } I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) - I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } ) + I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } \mathbf { y } _ { 1 } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) } \\ & { \overset { ( P _ { 1 } ) } { \leq } I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) + I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } ) + I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) } \\ & { \overset { ( P _ { 1 } ) } { \leq } I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) + I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } \mathbf { y } _ { 1 } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) } \\ & { \overset { ( R ) } { = } I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) } \\ & \overset { ( E ) } { \leq } I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf \end{array} +$$ + +Proposition B.3. Let $\mathbf { v } _ { 1 }$ be a redundant view with respect to $\mathbf { v } _ { 2 }$ for y. Any representation $\mathbf { z } _ { 1 }$ of $\mathbf { v } _ { 1 }$ that is sufficient for $\mathbf { v } _ { 2 }$ is also sufficient for y. + +Hypothesis: + +$' H _ { 1 }$ ) $\mathbf { z } _ { 1 }$ is a representation of $\mathbf { v } _ { 1 }$ $\mathbf { \sigma } _ { 1 } \mathbf { : } I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } \mathbf { v } _ { 1 } ) = 0$ $\left( H _ { 2 } \right) \ \mathbf { v } _ { 1 }$ is redundant with respect to $\mathbf { v } _ { 2 }$ for y: $I ( \mathbf { y } ; \mathbf { v } _ { 1 } | \mathbf { v } _ { 2 } ) = 0$ + +Thesis: + +$$ +( T _ { 1 } ) \ I ( { \bf v } _ { 1 } ; { \bf v } _ { 2 } | { \bf z } _ { 1 } ) = 0 \ \Longrightarrow \ I ( { \bf v } _ { 1 } ; { \bf y } | { \bf z } _ { 1 } ) = 0 +$$ + +Proof. Using the results from Theorem B.2: + +$$ +I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) \overset { ( T h _ { B . 2 } ) } { \leq } I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) \overset { ( H _ { 2 } ) } { = } I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) +$$ + +Therefore $I ( { \bf v } _ { 1 } ; { \bf v } _ { 2 } | { \bf z } _ { 1 } ) = 0 \implies I ( { \bf v } _ { 1 } ; { \bf y } | { \bf z } _ { 1 } ) = 0$ + +Theorem B.2. Let $\mathbf { v } _ { 1 }$ , $\mathbf { v } _ { 2 }$ and y be random variables with distribution $p ( \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { y } )$ . Let z be $a$ representation of $\mathbf { v } _ { 1 }$ , then + +$$ +I ( \mathbf { y } ; \mathbf { z } _ { 1 } ) \geq I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ) - I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) - I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) - I ( \mathbf { v } _ { 2 } ; \mathbf { y } | \mathbf { v } _ { 1 } ) +$$ + +Hypothesis: + +$( H _ { 1 } ) \ \mathbf { z } _ { 1 }$ is a representation of $\mathbf { v } _ { 1 }$ $\mathbf { \sigma } _ { 1 } \mathbf { : } I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ) = 0$ + +Thesis: + +$$ +( T _ { 1 } ) \ I ( \mathbf { y } ; \mathbf { z } _ { 1 } ) \geq I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ) - I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) - I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) - I ( \mathbf { v } _ { 2 } ; \mathbf { y } | \mathbf { v } _ { 1 } ) +$$ + +Proof. + +$$ + \begin{array} { r l } { I ( \mathbf { y } ; \mathbf { z } _ { 1 } ) } & { ( { \frac { e ^ { \mathbf { y } } } { 2 } } ) \ I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } \mathbf { y } _ { 2 } ) + I ( \mathbf { y } ; \mathbf { y } _ { 1 } \mathbf { v } _ { 2 } ; \mathbf { z } _ { 1 } ) } \\ & { ( { \frac { e ^ { \mathbf { y } } } { 2 } } ) \ I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { y } _ { 2 } ; \mathbf { z } _ { 1 } ) } \\ & { ( { \frac { e ^ { \mathbf { y } } } { 2 } } ) \ I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ; \mathbf { z } _ { 1 } ) } \\ & { ( \mathbf { z } ) \ I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ) - I ( \mathbf { y } ; \mathbf { y } _ { 1 } \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) } \\ & { ( { \frac { e ^ { \mathbf { y } } } { 2 } } ) \ I ( \mathbf { y } ; \mathbf { x } _ { 1 } \mathbf { v } _ { 2 } ) - I ( \mathbf { y } ; \mathbf { y } _ { 1 } \mathbf { z } _ { 1 } ) - I ( \mathbf { y } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } \mathbf { v } _ { 1 } ) } \\ & { ( { \frac { e ^ { \mathbf { y } } } { 2 } } ) \ I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ) - I ( \mathbf { y } ; \mathbf { x } _ { 1 } | \mathbf { z } _ { 1 } ) - I ( \mathbf { y } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) + I ( \mathbf { y } ; \mathbf { y } _ { 2 } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) } \\ & { ( { \frac { e ^ { \mathbf { y } } } { 2 } } ) \ I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ) - I ( \mathbf { y } ; \mathbf { y } _ { 1 } | \mathbf { z } _ { 1 } ) - I ( \mathbf { y } ; \mathbf { v } _ { 2 } | \mathbf { v } _ { 1 } ) + I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) } \\ & ( { \frac { e ^ { \mathbf { y } } } { 2 } } ) \end{array} +$$ + +Corollary B.2.1. Let $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ be mutually redundant views for y. Let $\mathbf { z } _ { 1 }$ be a representation of $\mathbf { v } _ { 1 }$ that is sufficient for $\mathbf { v } _ { 2 }$ . Then: + +$$ +I ( \mathbf { y } ; \mathbf { z } _ { 1 } ) = I ( \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ; \mathbf { y } ) +$$ + +Hypothesis: + +$( H _ { 1 }$ ) $\mathbf { z } _ { 1 }$ is a representation of $\mathbf { v } _ { 1 } \colon I ( \mathbf { y } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ) = 0$ +$( H _ { 2 }$ ) $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ are mutually redundant for y: $I ( \mathbf { y } ; \mathbf { v } _ { 1 } | \mathbf { v } _ { 2 } ) + I ( \mathbf { y } ; \mathbf { v } _ { 2 } | \mathbf { v } _ { 1 } ) = 0$ +$( H _ { 3 } ) \ \mathbf { z } _ { 1 }$ is sufficient for $\mathbf { v } _ { 2 }$ : $I ( \mathbf { v } _ { 2 } ; \mathbf { v } _ { 1 } | \mathbf { z } ) = 0$ + +Thesis: + +$$ +( T _ { 1 } ) \ I ( { \bf y } ; { \bf z } _ { 1 } ) = I ( { \bf v } _ { 1 } { \bf v } _ { 2 } ; { \bf y } ) +$$ + +Proof. Using Theorem B.2 + +$$ +\begin{array} { r l } & { I ( \mathbf { y } ; \mathbf { z } _ { 1 } ) \overset { ( T h _ { B , 2 } ) } { \geq } I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ) - I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) - I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) - I ( \mathbf { y } ; \mathbf { v } _ { 2 } | \mathbf { v } _ { 1 } ) } \\ & { \overset { ( H _ { 2 } ) } { = } I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ) - I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) } \\ & { \overset { ( H _ { 3 } ) } { = } I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } ) } \end{array} +$$ + +Since $I ( \mathbf { y } ; \mathbf { z } _ { 1 } ) \leq I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } )$ is a consequence of the data processing inequality, we conclude that $I ( \mathbf { y } ; \mathbf { z } _ { 1 } ) = I ( \mathbf { y } ; \mathbf { v } _ { 1 } \mathbf { v } _ { 2 } )$ □ + +# B.4 SUFFICIENCY AND AUGMENTATION + +Let $\mathbf { x }$ and $\mathbf { y }$ be random variables with domain $\mathbb { X }$ and $\mathbb { Y }$ respectively. Let $\mathbb { T }$ be a class of functions $t : \mathbb { X } \to \mathbb { W }$ and let $\mathbf { t } _ { 1 }$ and $\mathbf { t } _ { 2 }$ be a random variables over $\mathbb { T }$ that depends only on x. For the theorems and corollaries discussed in this section, we are going to consider the independence assumption that can be derived from the graphical model $\mathcal { G }$ reported in Figure 5. + +![](images/909aaf259ac4731b46cf28b94c79fc401805146a5d9c4d1f118b0bde376f35cf.jpg) +Figure 5: Visualization of the graphical model $\mathcal { G }$ that relates the observations $\mathbf { x }$ , label $\mathbf { y }$ , functions used for augmentation $\mathbf { t } _ { 1 }$ , $\mathbf { t } _ { 2 }$ and the representation $\mathbf { z } _ { 1 }$ . + +Proposition B.4. Whenever $I ( \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { y } ) = I ( \mathbf { t } _ { 2 } ( \mathbf { x } ) ; \mathbf { y } ) = I ( \mathbf { x } ; \mathbf { y } )$ the two views $\mathbf { t } _ { 1 } ( \mathbf { x } )$ and $\mathbf { t } _ { 2 } ( \mathbf { x } )$ must be mutually redundant for y. + +Hypothesis: + +$\left( H _ { 1 } \right)$ Independence relations determined by $\mathcal { G }$ + +Thesis: + +$$ +( T _ { 1 } ) \ I ( \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { y } ) = I ( \mathbf { t } _ { 2 } ( \mathbf { x } ) ; \mathbf { y } ) = I ( \mathbf { x } ; \mathbf { y } ) \implies I ( \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { y } | \mathbf { t } _ { 2 } ( \mathbf { x } ) ) + I ( \mathbf { t } _ { 2 } ( \mathbf { x } ) ; \mathbf { y } | \mathbf { t } _ { 1 } ( \mathbf { x } ) ) = 0 +$$ + +Proof. + +1. Considering $\mathcal { G }$ we have: + +$$ +\begin{array} { l } { { ( C _ { 1 } ) \ I ( \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { y } | \mathbf { x } \mathbf { t } _ { 2 } ( \mathbf { x } ) ) = 0 } } \\ { { ( C _ { 2 } ) \ I ( \mathbf { y } ; \mathbf { t } _ { 2 } ( \mathbf { x } ) | \mathbf { x } ) = 0 } } \end{array} +$$ + +2. Since ${ \bf t } _ { 2 } ( { \bf x } )$ is uniquely determined by $\mathbf { x }$ and $\mathbf { t } _ { 2 }$ : + +$$ +( C _ { 3 } ) I ( \bf t _ { 2 } ( \bf x ) ; \bf y | \bf x t _ { 2 } ) = 0 +$$ + +3. Consider $I ( \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { y } | \mathbf { t } _ { 2 } ( \mathbf { x } ) )$ + +$$ +\begin{array} { r l } { T [ \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { y } ] \mathbf { t } _ { 2 } ( \mathbf { x } ) } & { \stackrel { \mathrm { f o r ~ } } { \Longleftrightarrow } } & { T [ \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { y } ] \mathbf { t } _ { 2 } ( \mathbf { x } ) } \\ & { \stackrel { \mathrm { f o r ~ } } { \Longleftrightarrow } } & { \frac { \omega } { 2 } \delta ( t _ { 1 } ( \mathbf { x } ) ; \mathbf { y } , \mathbf { x } ; \mathbf { z } ) [ \mathbf { t } _ { 2 } ( \mathbf { x } ) ] } \\ & { \stackrel { \mathrm { f o r ~ } } { \Longleftrightarrow } } & { \frac { \omega } { 2 } \delta ( t _ { 2 } ( \mathbf { x } ) ; \mathbf { z } , \mathbf { x } | \mathbf { z } ) } \\ & { \stackrel { \mathrm { f o r ~ } } { \Longleftrightarrow } } & { \frac { \omega } { 2 } \delta ( t _ { 3 } ; \mathbf { x } | \mathbf { t } _ { 4 } ( \mathbf { x } ) ) - f [ \mathbf { y } ; \mathbf { x } | \mathbf { t } _ { 4 } ( \mathbf { x } ) ; \mathbf { t } _ { 5 } ( \mathbf { x } ) ) } \\ & { \stackrel { \mathrm { f o r ~ } } { \Longleftrightarrow } } & { \frac { \omega } { 2 } \delta ( t _ { 3 } ; \mathbf { x } | \mathbf { t } _ { 5 } ( \mathbf { x } ) ) } \\ & { \stackrel { \mathrm { f o r ~ } } { \Longleftrightarrow } } & { \frac { \omega } { 1 } \delta ( t _ { 5 } ; \mathbf { x } ) - T [ \mathbf { y } ; \mathbf { x } , \mathbf { t } _ { 6 } ( \mathbf { x } ) ) } \\ & { \stackrel { \mathrm { f o r ~ } } { \Longleftrightarrow } } & { \frac { \omega } { 1 } \delta ( t _ { 5 } ; \mathbf { x } ) - T [ \mathbf { y } ; \mathbf { t } _ { 6 } ( \mathbf { x } ) ; \mathbf { t } _ { 7 } ( \mathbf { x } ) ] } \\ & { \stackrel { \mathrm { f o r ~ } } { \Longleftrightarrow } } & { \frac { \omega } { 2 } \delta ( t _ { 5 } ; \mathbf { x } ) - f [ \mathbf { y } ; \mathbf { t } _ { 6 } ( \mathbf { x } ) ] + T [ \mathbf { y } ; \mathbf { t } _ { 2 } ( \mathbf { x } ) ] \mathbf { t } _ { 2 } \delta ( t _ { 5 } ; \mathbf { t } _ { 6 } ( \mathbf { x } ) ; \mathbf { t } _ { 7 } ( \mathbf { x } ) ] } \\ & \stackrel \ \end{array} +$$ + +Therefore $I ( \mathbf { y } ; \mathbf { x } ) = I ( \mathbf { y } ; \mathbf { t } _ { 2 } ( \mathbf { x } ) ) \implies I ( \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { y } | \mathbf { t } _ { 2 } ( \mathbf { x } ) ) = 0$ + +The proof for ${ \cal I } ( { \bf y } ; { \bf x } ) = { \cal I } ( { \bf y } ; { \bf t } _ { 1 } ( { \bf x } ) ) \Longrightarrow { \cal I } ( { \bf t } _ { 2 } ( { \bf x } ) ; { \bf y } | { \bf t } _ { 1 } ( { \bf x } ) ) = 0$ is symmetric, therefore we conclude I ${ \bf \bar { \Psi } ( t _ { 1 } ( x ) ; y ) } = I ( { \bf t } _ { 2 } ( { \bf x } ) ; { \bf y } ) = I ( { \bf x } ; { \bf y } ) \Longrightarrow I ( { \bf t } _ { 1 } ( { \bf x } ) ; { \bf y } | { \bf t } _ { 2 } ( { \bf x } ) ) + I ( { \bf t } _ { 2 } ( { \bf x } ) ; { \bf y } | { \bf t } _ { 1 } ( { \bf x } ) ) = I ( { \bf t } _ { 1 } ( { \bf x } ) ; { \bf y } ) ,$ 0 + +Theorem B.3. Let $I ( \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { y } ) = I ( \mathbf { t } _ { 2 } ( \mathbf { x } ) ; \mathbf { y } ) = I ( \mathbf { x } ; \mathbf { y } )$ . Let $\mathbf { z } _ { 1 }$ be a representation of $\mathbf { t } _ { 1 } ( \mathbf { x } )$ . I f $\mathbf { z } _ { 1 }$ is sufficient for ${ \bf t } _ { 2 } ( { \bf x } )$ then $I ( \mathbf { x } ; \mathbf { y } ) = I ( \mathbf { y } ; \mathbf { z } _ { 1 } )$ . + +Hypothesis: + +$\left( H _ { 1 } \right)$ Independence relations determined by $\mathcal { G }$ + +$$ +( H _ { 2 } ) \ I ( \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { y } ) = I ( \mathbf { t } _ { 2 } ( \mathbf { x } ) ; \mathbf { y } ) = I ( \mathbf { x } ; \mathbf { y } ) +$$ + +Thesis: + +$$ +( T _ { 1 } ) \ I ( \mathbf { t } _ { 1 } ( \mathbf { x } ) ; \mathbf { t } _ { 2 } ( \mathbf { x } ) | \mathbf { z } _ { 1 } ) = 0 \ \Longrightarrow \ I ( \mathbf { x } ; \mathbf { y } ) = I ( \mathbf { y } ; \mathbf { z } _ { 1 } ) +$$ + +Proof. Since $\mathbf { t } _ { 1 } ( \mathbf { x } )$ is redundant for ${ \bf t } _ { 2 } ( { \bf x } )$ (Proposition B.4) any representation $\mathbf { z } _ { 1 }$ of $\mathbf { t } _ { 1 } ( \mathbf { x } )$ that is sufficient for ${ \bf t } _ { 2 } ( { \bf x } )$ must also be sufficient for $\mathbf { y }$ (Theorem B.2). Using Proposition B.1 we have ${ \cal I } ( { \bf y } ; { \bf z } _ { 1 } ) = { \cal I } ( { \bf y } ; { \bf t } _ { 1 } ( { \bf x } ) )$ . Since $I ( \mathbf { y } ; \mathbf { t } _ { 1 } ( \mathbf { x } ) ) ~ = ~ I ( \mathbf { y } ; \mathbf { x } )$ by hypothesis, we conclude $I ( \mathbf { x } ; \mathbf { y } ) \ =$ $I ( \mathbf { y } ; \mathbf { z } _ { 1 } )$ + +# C INFORMATION PLANE + +Every representation $\mathbf { z }$ of $\mathbf { x }$ must satisfy the following constraints: + +• $0 \leq I ( \mathbf { y } ; \mathbf { z } ) \leq I ( \mathbf { x } ; \mathbf { y } )$ : The amount of label information ranges from 0 to the total predictive information accessible from the raw observations $I ( \mathbf { x } ; \mathbf { y } )$ . + +• $I ( \mathbf { y } ; \mathbf { z } ) \leq I ( \mathbf { x } ; \mathbf { z } ) \leq I ( \mathbf { y } ; \mathbf { z } ) + H ( \mathbf { x } | \mathbf { y } )$ : The representation must contain more information about the observations than about the label. When $\mathbf { x }$ is discrete, the amount of discarded label information $I ( \mathbf { x } ; \mathbf { y } ) - I ( \mathbf { y } ; \mathbf { z } )$ must be smaller than the amount of discarded observational information $H ( \mathbf { x } ) - I ( \mathbf { x } ; \mathbf { z } )$ , which implies $I ( \mathbf { x } ; \mathbf { z } ) \leq I ( \mathbf { y } ; \mathbf { z } ) + H ( \mathbf { x } | \mathbf { y } )$ . + +Proof. Since $\mathbf { z }$ is a representation of $\mathbf { x }$ + +$$ +( C _ { 1 } ) \ I ( { \bf y } ; { \bf z } | { \bf x } ) = 0 +$$ + +Considering the four bounds separately: + +1. $I ( \mathbf { y } ; \mathbf { z } ) \geq 0$ : Follows from $P _ { 1 }$ + +2. $I ( \mathbf { x ; z } ) \geq I ( \mathbf { y ; z } )$ : Follows from: + +$$ +\begin{array} { l } { I ( \mathbf { x } ; \mathbf { z } ) \overset { ( P _ { 3 } ) } { = } I ( \mathbf { x } ; \mathbf { z } | \mathbf { y } ) + I ( \mathbf { x } ; \mathbf { y } ; \mathbf { z } ) } \\ { \begin{array} { l } { ( P _ { 1 } ) } \\ { \geq } \\ { \overset { ( P _ { 3 } ) } { = } I ( \mathbf { y } ; \mathbf { z } ) - I ( \mathbf { y } ; \mathbf { z } | \mathbf { x } ) } \end{array} } \\ { \begin{array} { l } { ( \underline { { C _ { 1 } } } ) } \\ { = } \end{array} } \end{array} +$$ + +3. $I ( \mathbf { y } ; \mathbf { z } ) \leq I ( \mathbf { y } ; \mathbf { x } )$ : Data processing inequality + +$$ +\begin{array} { r l } & { I ( \mathbf { y } ; \mathbf { z } ) \overset { ( P _ { 3 } ) } { = } I ( \mathbf { y } ; \mathbf { z } | \mathbf { x } ) + I ( \mathbf { y } ; \mathbf { z } ; \mathbf { x } ) } \\ & { \overset { ( C _ { 1 } ) } { = } I ( \mathbf { y } ; \mathbf { z } ; \mathbf { x } ) } \\ & { \overset { ( P _ { 3 } ) } { = } I ( \mathbf { x } ; \mathbf { y } ) - I ( \mathbf { x } ; \mathbf { y } | \mathbf { z } ) } \\ & { \overset { ( P _ { 1 } ) } { \leq } I ( \mathbf { x } ; \mathbf { y } ) } \end{array} +$$ + +4. $I ( \mathbf { x } ; \mathbf { z } ) \leq I ( \mathbf { y } ; \mathbf { z } ) + H ( \mathbf { x } | \mathbf { y } )$ : For discrete $\mathbf { x }$ : + +$$ +\begin{array} { r l } { I ( \mathbf { x } ; \mathbf { z } ) \overset { ( P _ { 3 } ) } { = } I ( \mathbf { x } ; \mathbf { z } | \mathbf { y } ) + I ( \mathbf { x } ; \mathbf { y } ; \mathbf { z } ) } & { } \\ & { \overset { ( P _ { 3 } ) } { = } I ( \mathbf { x } ; \mathbf { z } | \mathbf { y } ) + I ( \mathbf { y } ; \mathbf { z } ) - I ( \mathbf { y } ; \mathbf { z } | \mathbf { x } ) } \\ & { \overset { ( C _ { 1 } ) } { = } I ( \mathbf { x } ; \mathbf { z } | \mathbf { y } ) + I ( \mathbf { y } ; \mathbf { z } ) } \\ & { \overset { ( P _ { 4 } ) } { \leq } I ( \mathbf { x } ; \mathbf { z } | \mathbf { y } ) + H ( \mathbf { x } | \mathbf { y } \mathbf { z } ) + I ( \mathbf { y } ; \mathbf { z } ) } \\ & { \overset { ( P _ { 5 } ) } { = } I ( \mathbf { x } | \mathbf { y } ) + I ( \mathbf { y } ; \mathbf { z } ) } \end{array} +$$ + +Note that the discreetness of $\mathbf { x }$ is required only to prove bound 4. For continuous $\mathbf { x }$ bounds $1 , 2$ and 3 still hold. + +# D NON-TRANSITIVITY OF MUTUAL REDUNDANCY + +The mutual redundancy condition between two views $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ for a label y can not be trivially extended to an arbitrary number of views, as the relation is not transitive because of some higher order interaction between the different views and the label. This can be shown with a simple example. + +Given three views $\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }$ and $\mathbf { v } _ { 3 }$ and a task y such that: + +• $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ are mutually redundant for y • $\mathbf { v } _ { 2 }$ and $\mathbf { v } _ { 3 }$ are mutually redundant for y + +Then, we show that $\mathbf { v } _ { 1 }$ is not necessarily mutually redundant with respect to $\mathbf { v } _ { 3 }$ for y. + +Let $\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }$ and $\mathbf { v } _ { 3 }$ be fair and independent binary random variables. Defining y as the exclusive or operator applied to $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 3 }$ ( $\mathbf { y } : = \mathbf { v } _ { 1 } \mathrm { X O R } \mathbf { v } _ { 3 } )$ ), we have that $I ( \mathbf { v } _ { 1 } ; \mathbf { y } ) = I ( \mathbf { v } _ { 3 } ; \mathbf { y } ) = 0$ . In this settings, $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ are mutually redundant for y: + +$$ +\begin{array} { r } { I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) = H ( \mathbf { v } _ { 1 } | \mathbf { v } _ { 2 } ) - H ( \mathbf { v } _ { 1 } | \mathbf { v } _ { 2 } \mathbf { y } ) = H ( \mathbf { v } _ { 1 } ) - H ( \mathbf { v } _ { 1 } ) = 0 } \\ { I ( \mathbf { v } _ { 2 } ; \mathbf { y } | \mathbf { v } _ { 1 } ) = H ( \mathbf { v } _ { 2 } | \mathbf { v } _ { 1 } ) - H ( \mathbf { v } _ { 2 } | \mathbf { v } _ { 1 } \mathbf { y } ) = H ( \mathbf { v } _ { 2 } ) - H ( \mathbf { v } _ { 2 } ) = 0 } \end{array} +$$ + +Analogously, $\mathbf { v } _ { 2 }$ and $\mathbf { v } _ { 3 }$ are also mutually redundant for $\mathbf { y }$ as the three random variables are not predictive for each other. Nevertheless, $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 3 }$ are not mutually redundant for $\mathbf { y }$ : + +$$ +\begin{array} { r l } & { I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 3 } ) = H ( \mathbf { v } _ { 1 } | \mathbf { v } _ { 3 } ) - \underbrace { H ( \mathbf { v } _ { 1 } | \mathbf { v } _ { 3 } \mathbf { y } ) } _ { 0 } = H ( \mathbf { v } _ { 1 } ) = 1 } \\ & { I ( \mathbf { v } _ { 3 } ; \mathbf { y } | \mathbf { v } _ { 1 } ) = H ( \mathbf { v } _ { 3 } | \mathbf { v } _ { 1 } ) - \underbrace { H ( \mathbf { v } _ { 3 } | \mathbf { v } _ { 1 } \mathbf { y } ) } _ { 0 } = H ( \mathbf { v } _ { 3 } ) = 1 } \end{array} +$$ + +Where $H ( \mathbf { v } _ { 1 } | \mathbf { v } _ { 3 } \mathbf { y } ) = H ( \mathbf { v } _ { 3 } | \mathbf { v } _ { 1 } \mathbf { y } ) = 0$ follows from $\mathbf { v } _ { 1 } = \mathbf { v } _ { 3 }$ XOR y and $\mathbf { v } _ { 3 } = \mathbf { v } _ { 1 }$ XOR y, while $H ( \mathbf { v } _ { 1 } ) = H ( \mathbf { v } _ { 3 } ) = 1$ holds by construction. + +This counter-intuitive higher order interaction between multiple views makes our theory non-trivial to generalize to more than two views. + +# E EQUIVALENCES OF DIFFERENT OBJECTIVES + +Different objectives in literature can be seen as a special case of the Multi-View Information Bottleneck principle. In this section we show that the supervised version of Information Bottleneck is equivalent to the corresponding Multi-View version whenever the two redundant views have only label information in common. A second subsection show equivalence between InfoMax and MultiView Information Bottleneck whenever the two views are identical. + +# E.1 MULTI-VIEW INFORMATION BOTTLENECK AND SUPERVISED INFORMATION BOTTLENECK + +Whenever the two mutually redundant views $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ have only label information in common (or when one of the two views is the label itself) the Multi-View Information Bottleneck objective is equivalent to the respective supervised version. This can be shown by proving that $I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } ) =$ $I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { y } )$ , i.e. a representation $\mathbf { z } _ { 1 }$ of $\mathbf { v } _ { 1 }$ that is sufficient and minimal for $\mathbf { v } _ { 2 }$ is also sufficient and minimal for y. + +Proposition E.1. Let $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ be mutually redundant views for a label y that share only label information. Then a sufficient representation $\mathbf { z } _ { 1 }$ of $\mathbf { v } _ { 1 } f o r \mathbf { v } _ { 2 }$ that is minimal for $\mathbf { v } _ { 2 }$ is also a minimal representation for y. + +# Hypothesis: + +$( H _ { 1 }$ ) $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ are mutually redundant for y: $I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { v } _ { 2 } ) + I ( \mathbf { v } _ { 2 } ; \mathbf { y } | \mathbf { v } _ { 1 } ) = 0$ + +$\left( H _ { 2 } \right) \ \mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ share only label information: $I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } ) = I ( \mathbf { v } _ { 1 } ; \mathbf { y } )$ + +$( H _ { 3 } ) \ \mathbf { z } _ { 1 }$ is sufficient for $\mathbf { v } _ { 2 }$ : $I ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) = 0$ + +Thesis: + +$$ +( T _ { 1 } ) \ I ( { \bf v } _ { 1 } ; { \bf z } _ { 1 } | { \bf v } _ { 2 } ) = I ( { \bf v } _ { 1 } ; { \bf z } _ { 1 } | { \bf y } ) +$$ + +Proof. + +1. Consider $I ( \mathbf { v } _ { 1 } ; \mathbf { z } )$ : + +$$ +\begin{array} { l } { { { I } ( { \bf { v } } _ { 1 } ; { \bf { z } } _ { 1 } ) \stackrel { { ( P _ { 3 } ) } } { = } I ( { \bf { v } } _ { 1 } ; { \bf { z } } _ { 1 } | { \bf { v } } _ { 2 } ) + I ( { \bf { v } } _ { 1 } ; { \bf { v } } _ { 2 } ; { \bf { z } } _ { 1 } ) } \ ~ } \\ { { \displaystyle ~ \stackrel { { ( P _ { 3 } ) } } { = } I ( { \bf { v } } _ { 1 } ; { \bf { z } } _ { 1 } | { \bf { v } } _ { 2 } ) + I ( { \bf { v } } _ { 1 } ; { \bf { v } } _ { 2 } ) - I ( { \bf { v } } _ { 1 } ; { \bf { v } } _ { 2 } | { \bf { z } } _ { 1 } ) } \ ~ } \\ { { \displaystyle ~ \stackrel { { ( H _ { 3 } ) } } { = } I ( { \bf { v } } _ { 1 } ; { \bf { z } } _ { 1 } | { \bf { v } } _ { 2 } ) + I ( { \bf { v } } _ { 1 } ; { \bf { v } } _ { 2 } ) } \ ~ } \\ { { \displaystyle ~ \stackrel { { ( H _ { 1 } ) } } { = } I ( { \bf { v } } _ { 1 } ; { \bf { z } } _ { 1 } | { \bf { v } } _ { 2 } ) + I ( { \bf { v } } _ { 1 } ; { \bf { y } } ) } \ ~ } \end{array} +$$ + +2. Using Corollary 1, from $\left( H _ { 2 } \right)$ and $\left( H _ { 3 } \right)$ follows $I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) = 0$ + +3. $I ( \mathbf { v } _ { 1 } ; \mathbf { z } )$ can be alternatively expressed as: + +$$ +\begin{array} { r l } & { I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } ) \overset { ( P _ { 3 } ) } { = } I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { y } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { y } ; \mathbf { z } _ { 1 } ) } \\ & { \overset { ( P _ { 3 } ) } { = } I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { y } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { y } ) - I ( \mathbf { v } _ { 1 } ; \mathbf { y } | \mathbf { z } _ { 1 } ) } \\ & { \overset { ( C o r _ { 1 } ) } { = } I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { y } ) + I ( \mathbf { v } _ { 1 } ; \mathbf { y } ) } \end{array} +$$ + +Equating 1 and 3, we conclude $I ( { \bf v } _ { 1 } ; { \bf z } _ { 1 } | { \bf v } _ { 2 } ) = I ( { \bf v } _ { 1 } ; { \bf z } _ { 1 } | { \bf y } )$ , therefore $\mathbf { z } _ { 1 }$ which minimizes $I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } )$ is also minimizing $I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { y } )$ . When $I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { y } )$ is minimal, $I ( \mathbf { y } ; \mathbf { z } _ { 1 } )$ is also minimal (see equation 2). □ + +# E.2 MULTI-VIEW INFORMATION BOTTLENECK AND INFOMAX + +Whenever $\mathbf { v } _ { 1 } = \mathbf { v } _ { 2 }$ , a representation $\mathbf { z } _ { 1 }$ of $\mathbf { v } _ { 1 }$ that is sufficient for $\mathbf { v } _ { 2 }$ must contain all the original information regarding $\mathbf { v } _ { 1 }$ . Furthermore since $I ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } ) = 0$ for every representation, no superfluous information can be identified and removed. As a consequence, a minimal sufficient representation $\mathbf { z } _ { 1 }$ of $\mathbf { v } _ { 1 }$ for $\mathbf { v } _ { 2 }$ is any representation for which mutual information is maximal, hence InfoMax. + +# F LOSS COMPUTATION + +Starting from Equation 3, we consider the average of the losses $\mathcal { L } _ { 1 } ( \theta ; \lambda _ { 1 } )$ and $\mathcal { L } _ { 2 } ( \psi ; \lambda _ { 2 } )$ that aim to create the minimal sufficient representations $\mathbf { z } _ { 1 }$ and $\mathbf { z } _ { 2 }$ respectively: + +$$ +\mathscr { L } _ { \frac { 1 + 2 } { 2 } } ( \theta , \psi ; \lambda _ { 1 } , \lambda _ { 2 } ) = \frac { I _ { \theta } ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } ) + I _ { \psi } ( \mathbf { v } _ { 2 } ; \mathbf { z } _ { 2 } | \mathbf { v } _ { 1 } ) } { 2 } + \frac { \lambda _ { 1 } I _ { \theta } ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) + \lambda _ { 2 } I _ { \psi } ( \mathbf { v } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 1 } ) } { 2 } +$$ + +Considering $\mathbf { z } _ { 1 }$ and $\mathbf { z } _ { 2 }$ on the same domain $\mathbb { Z }$ , $I _ { \theta } ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } )$ can be expressed as: + +$$ +\begin{array} { r l } & { I _ { \theta } ( \mathbf { v } _ { 1 } ; \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } ) = \mathbb { E } _ { v _ { 1 } , v _ { 2 } \sim p ( \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } ) } \mathbb { E } _ { z \sim p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) } \left[ \log \frac { p _ { \theta } ( \mathbf { z } _ { 1 } = z | \mathbf { v } _ { 1 } = v _ { 1 } ) } { p _ { \theta } ( \mathbf { z } _ { 1 } = z | \mathbf { v } _ { 2 } = v _ { 2 } ) } \right] } \\ & { \qquad = \mathbb { E } _ { v _ { 1 } , v _ { 2 } \sim p ( \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } ) } \mathbb { E } _ { z \sim p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) } \left[ \log \frac { p _ { \theta } ( \mathbf { z } _ { 1 } = z | \mathbf { v } _ { 1 } = v _ { 1 } ) } { p _ { \psi } ( \mathbf { z } _ { 2 } = z | \mathbf { v } _ { 2 } = v _ { 2 } ) } \frac { p _ { \psi } ( \mathbf { z } _ { 2 } = z | \mathbf { v } _ { 2 } = v _ { 2 } ) } { p _ { \theta } ( \mathbf { z } _ { 1 } = z | \mathbf { v } _ { 2 } = v _ { 2 } ) } \right] } \\ & { \qquad = D _ { \mathrm { K L } } ( p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) | | p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) ) - D _ { \mathrm { K L } } ( p _ { \theta } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 1 } ) | | p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) ) } \\ & { \qquad \leq D _ { \mathrm { K L } } ( p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) | | p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) ) } \end{array} +$$ + +Note that the bound is tight whenever $p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } )$ coincides with $p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 2 } )$ . This happens whenever $\mathbf { z } _ { 1 }$ and $\mathbf { z } _ { 2 }$ produce a consistent encoding. Analogously $I _ { \psi } ( \mathbf { v } _ { 2 } ; \mathbf { z } _ { 2 } | \mathbf { v } _ { 1 } )$ is upper bounded by $D _ { \mathrm { K L } } ( p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) | | p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) )$ . + +$I _ { \theta } ( \mathbf { v } _ { 2 } ; \mathbf { z } _ { 1 } )$ can be rephrased as: + +$$ +\begin{array} { r l } & { I _ { \boldsymbol { \theta } } ( \mathbf { z } _ { 1 } ; \mathbf { v } _ { 2 } ) \overset { ( P _ { 2 } ) } { = } I _ { \boldsymbol { \theta } \boldsymbol { \psi } } ( \mathbf { z } _ { 1 } ; \mathbf { z } _ { 2 } \mathbf { v } _ { 2 } ) - I _ { \boldsymbol { \theta } \boldsymbol { \psi } } ( \mathbf { z } _ { 1 } ; \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) } \\ & { = ^ { \ast } I _ { \boldsymbol { \theta } \boldsymbol { \psi } } ( \mathbf { z } _ { 1 } ; \mathbf { z } _ { 2 } \mathbf { v } _ { 2 } ) } \\ & { = I _ { \boldsymbol { \theta } \boldsymbol { \psi } } ( \mathbf { z } _ { 1 } ; \mathbf { z } _ { 2 } ) + I _ { \boldsymbol { \theta } \boldsymbol { \psi } } ( \mathbf { z } _ { 1 } ; \mathbf { v } _ { 2 } | \mathbf { z } _ { 2 } ) } \\ & { \geq I _ { \boldsymbol { \theta } \boldsymbol { \psi } } ( \mathbf { z } _ { 1 } ; \mathbf { z } _ { 2 } ) } \end{array} +$$ + +Where ∗ follows from $\mathbf { z } _ { 2 }$ representation of $\mathbf { v } _ { 2 }$ . The bound reported in this equation is tight whenever $\mathbf { z } _ { 2 }$ is sufficient for ${ \bf z } _ { 1 } ( { \cal I } _ { \theta \psi } ( { \bf z } _ { 1 } ; { \bf v } _ { 2 } | { \bf z } _ { 2 } ) = 0 )$ . This happens whenever $\mathbf { z } _ { 2 }$ contains all the information regarding $\mathbf { z } _ { 1 }$ (and therefore $\mathbf { v } _ { 1 }$ ). Once again, the same bound can symmetrically be used to show $I _ { \theta } ( \mathbf { z } _ { 2 } ; \mathbf { v } _ { 1 } ) \ge I _ { \theta \psi } ( \mathbf { z } _ { 1 } ; \mathbf { z } _ { 2 } )$ . Therefore, the loss function in Equation 6 can be upper-bounded with: + +$$ +\mathcal { L } _ { \frac { 1 + 2 } { 2 } } ( \theta , \psi ; \lambda _ { 1 } , \lambda _ { 2 } ) \leq D _ { S K L } ( p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) | | p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) ) - \frac { \lambda _ { 1 } + \lambda _ { 2 } } { 2 } I _ { \theta \psi } ( \mathbf { z } _ { 1 } ; \mathbf { z } _ { 2 } ) +$$ + +Where: + +$$ +\partial _ { S K L } ( p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) | | p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) ) : = \frac { 1 } { 2 } D _ { \mathrm { K L } } ( p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) | | p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) ) + \frac { 1 } { 2 } D _ { \mathrm { K L } } ( p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) | | p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) ) +$$ + +Lastly, multiplying both terms with $\begin{array} { r } { \beta : = \frac { 2 } { \lambda _ { 1 } + \lambda _ { 2 } } } \end{array}$ and re-parametrizing the objective, we obtain: + +$$ +\mathcal { L } _ { M I B } ( \theta , \psi ; \beta ) = - I _ { \theta \psi } ( \mathbf { z } _ { 1 } ; \mathbf { z } _ { 2 } ) + \beta D _ { S K L } ( p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) | | p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) ) +$$ + +# G EXPERIMENTAL PROCEDURE AND DETAILS + +# G.1 MODELING + +The two stochastic encoders $p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } )$ and $p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } )$ are modeled by Normal distributions parametrized with neural networks $( \mu _ { \theta } , \sigma _ { \theta } ^ { 2 } )$ and $( \mu _ { \psi } , \sigma _ { \psi } ^ { 2 } )$ respectively: + +$$ +\begin{array} { r } { p _ { \theta } ( \mathbf { z } _ { 1 } | \mathbf { v } _ { 1 } ) : = \mathcal { N } \left( \mathbf { z } _ { 1 } | \pmb { \mu } _ { \theta } ( \mathbf { v } _ { 1 } ) , \pmb { \sigma } _ { \theta } ^ { 2 } ( \mathbf { v } _ { 1 } ) \right) } \\ { p _ { \psi } ( \mathbf { z } _ { 2 } | \mathbf { v } _ { 2 } ) : = \mathcal { N } \left( \mathbf { z } _ { 2 } | \pmb { \mu } _ { \psi } ( \mathbf { v } _ { 2 } ) , \pmb { \sigma } _ { \psi } ^ { 2 } ( \mathbf { v } _ { 2 } ) \right) } \end{array} +$$ + +Since the density of the two encoders can be evaluated, the symmetrized KL-divergence in equation 4 can be directly computed. On the other hand, $I _ { \theta \psi } ( \mathbf { z } _ { 1 } ; \mathbf { z } _ { 2 } )$ requires the use of a mutual information estimator. + +To facilitate the optimization, the hyper-parameter $\beta$ is slowly increased during training, starting from a small value $\approx 1 0 ^ { - 4 }$ to its final value with an exponential schedule. This is because the mutual information estimator is trained together with the other architectures and, since it starts from a random initialization, it requires an initial warm-up. Starting with bigger $\beta$ results in the encoder collapsing into a fixed representation. The update policy for the hyper-parameter during training has not shown strong influence on the representation, as long as the mutual information estimator network has reached full capacity. + +All the experiments have been performed using the Adam optimizer with a learning rate of $1 0 ^ { - 4 }$ for both encoders and the estimation network. Higher learning rate can result in instabilities in the training procedure. The results reported in the main text relied on the Jensen-Shannon mutual information estimator (Devon Hjelm et al., 2019) since the InfoNCE counterpart (van den Oord et al., 2018) generally resulted in worse performance that could be explained by the effect of the factorization of the critic network (Poole et al., 2019). + +# G.2 SKETCHY EXPERIMENTS + +• Input: The two views for the sketch-based classification task consist of 4096 dimensional sketch and image features extracted from two distinct VGG-16 network models which were pre-trained on images and sketches from the TU-Berlin dataset Eitz et al. (2012) for endto-end classification. The feature extractors are frozen during the training procedure of for the two representations. Each training iteration used batches of size $B = 1 2 8$ . Encoder and Critic architectures: Both sketch and image encoders consist of multi-layer perceptrons of 2 hidden ReLU units of size 2,048 and 1,024 respectively with an output of size $2 \mathrm { x } 6 4$ that parametrizes mean and variance for the two Gaussian posteriors. The critic architecture also consists of a multi layer perceptron of 2 hidden ReLU units of size 512. $\beta$ update policy: The initial value of $\beta$ is set to $1 0 ^ { - 4 }$ . Starting from the $1 0 { , } 0 0 0 ^ { \mathrm { t h } }$ training iteration, the value of $\beta$ is exponentially increased up to 1.0 during the following 250,000 training iterations. The value of $\beta$ is then kept fixed to one until the end of the training procedure (500,000 iterations). + +• Evaluation: All natural images are used as both training sets and retrieval galleries. The 64 dimensional real outputs of sketch and image representation are compared using Euclidean distance. For having a fair comparison other methods that rely on binary hashing (Liu et al., 2017; Zhang et al., 2018), we used Hamming distance on a binarized representation (obtained by applying iterative quantization Gong et al. (2013) on our real valued representation). We report the mean average precision $( \mathrm { m A P } @ \mathrm { a l l } )$ and precision at toprank 200 (Prec $@ 2 0 0$ ) Su et al. (2015) on both the real and binary representation to evaluate our method and compare it with prior works. + +# G.3 MIR-FLICKR EXPERIMENTS + +Figure 6: Examples of pictures $\mathbf { v } _ { 1 }$ , tags $\mathbf { v } _ { 2 }$ and category labels y for the MIR-Flickr dataset (Srivastava & Salakhutdinov, 2014). As visualized is the second row, the tags are not always predictive of the label. For this reason, the mutual redundancy assumption holds only approximately. + +![](images/0d8f12fdddf514eb4f72b6b35413f89d2358eeac502f25d9701f440c26b770c1.jpg) + +• Input: Whitening is applied to the handcrafted image features. Batches of size $B = 1 2 8$ are used for each update step. + +• Encoders and Critic architectures: The two encoders consists of a multi layer perceptron of 4 hidden ReLU units of size 1,024, which exactly resemble the architecture used in Wang et al. (2016). Both representations $\mathbf { z } _ { 1 }$ and $\mathbf { z } _ { 2 }$ have a size of 1,024, therefore the two architecture output a total of $^ { 2 \mathrm { x } 1 , 0 2 4 }$ parameters that define mean and variance of the respective factorized Gaussian posterior. Similarly to the Sketchy experiments, the critic is consists of a multi-layer perceptron of 2 hidden ReLU units of size 512. + +• $\beta$ update policy: The initial value of $\beta$ is set to $1 0 ^ { - 8 }$ . Starting from $1 5 0 0 0 0 ^ { \mathrm { t h } }$ iteration, $\beta$ is set to exponentially increase up to 1.0 (and $1 0 ^ { - 3 }$ ) during the following 150,000 iterations. + +• Evaluation: Once the models are trained on the unlabeled set, the representation of the 25,000 labeled images is computed. The resulting vectors are used for training and evaluating a multi-label logistic regression classifier on the respective splits. The optimal parameters (such as $\beta$ ) for our model are chosen based on the performance on the validation set. In Table 3, we report the aggregated mean of the 5 test splits as the final value mean average precision value. + +# G.4 MNIST EXPERIMENTS + +• Input: The two views $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ for the MNIST dataset are generated by applying small translation $( [ 0 - 1 0 ] \% )$ , rotation ([-15,15] degrees), scale $( [ 9 0 , 1 1 0 ] \% )$ , shear ([-15,15] degrees) and pixel corruption $( 2 0 \% )$ . Batches of size $B = 6 4$ samples are used during training. + +• Encoders, Decoders and Critic architectures: All the encoders used for the MNIST experiments consist of neural networks with two hidden layers of 1,024 units and ReLU activations, producing a $2 \mathrm { x } 6 4$ -dimensional parameter vector that is used to parameterize mean and variance for the Gaussian posteriors. The decoders used for the VAE experiments also consist of the networks of the same size. Similarly, the critic architecture used for mutual information estimation consists of two hidden layers of 1,204 units each and ReLU activations. + +• $\beta$ update policy: The initial value of $\beta$ is set to $1 0 ^ { - 3 }$ , which is increased with an exponential schedule starting from the $5 0 { , } 0 0 0 ^ { \mathrm { t h } }$ until 1the $5 0 { , } 0 0 0 ^ { \mathrm { t h } }$ iteration. The value of $\beta$ is then kept constant until the $1 , 0 0 0 { , } 0 0 0 ^ { \mathrm { t h } }$ iteration. The same annealing policy is used to trained the different $\beta$ -VAEs reported in this work. + +• Evaluation: The trained representation are evaluated following the well-known protocol described in Tschannen et al. (2019); Tian et al. (2019); Bachman et al. (2019); van den Oord et al. (2018). Each logistic regression is trained 5 different balanced splits of the training set for different percentages of training examples, ranging from 1 example per label to the whole training set. The accuracy reported in this work has been computed on the disjoint test set. Mean and standard deviation are computed according to the 5 different subsets used for training the logistic regression. Mean and variance for the mutual information estimation reported on the Information Plane (Figure 4) are computed by training two estimation networks from scratch on the final representation of the non-augmented train set. The two estimation architectures consist of 2 hidden layers of 2048 and 1024 units each, and have been trained with batches of size $B = 2 5 6$ for a total of approximately 25,000 iterations. The Jensen-Shannon mutual information lower bound is maximized during training, while the numerical estimation are computed using an energy-based bound (Poole et al., 2019; Devon Hjelm et al., 2019). The final values for $I ( \mathbf { x } ; \mathbf { z } )$ and $I ( \mathbf { y } ; \mathbf { z } )$ are computed by averaging the mutual information estimation on the whole dataset. In order to reduce the variance of the estimator, the lowest and highest $5 \%$ are removed before averaging. This practical detail makes the estimation more consistent and less susceptible to numerical instabilities. + +# G.4.1 RESULTS AND VISUALIZATION + +In this section we include additional quantitative results and visualizations which refer to the singleview MNIST experiments reported in section 5.2. + +Table 2 reports the quantitative results used for to produce the visualizations reported in Figure 4, including the comparison between the performance resulting from different mutual information estimators. As the Jensen-Shannon estimator generally resulted in better performance for the InfoMax, MV-InfoMax and MIB models, all the experiments reported on the main text make use of this estimator. Note that the InfoMax model with the $I _ { \mathrm { J S } }$ estimator is equivalent to the global model reported in Devon Hjelm et al. (2019), while MV-InfoMax with the $I _ { \mathrm { N C E } }$ estimator results in a similar architecture to the one introduced in Tian et al. (2019). + +Table 2: Comparison of the amount of input information $I ( \mathbf { x } ; \mathbf { z } )$ , label information $I ( \mathbf { z } ; \mathbf { y } )$ , and accuracy of a linear classifier trained with different amount of labeled Examples $\left( \operatorname { E x } \right)$ for the models reported in Figure 4. Both the results obtained using the Jensen-Shannon $I _ { \mathrm { J S D } }$ (Devon Hjelm et al., 2019; Poole et al., 2019) and the InfoNCE $I _ { \mathrm { N C E } }$ (van den Oord et al., 2018) estimators are reported. + +
ModelI(x; z) [nats]I(z;y)[nats]Test Accuracy [%]
10 Ex50Ex3750Ex60000 Ex
VAE (beta=0)12.5 ± 0.72.3± 0.243.8±1.665.6±3.389.0± 0.491.3 ±0.1
VAE (beta=4)7.5 ±1.02.0±0.255.9 ± 2.681.4 ± 4.094.2 ± 0.396.0 ± 0.2
VAE (beta=8)3.0 ± 0.51.0 ± 0.143.8 ±2.861.1 ± 4.881.9 ± 1.187.2 ± 0.6
InfoMax (INCE)12.8 ± 0.52.3 ±0.225.4 ± 1.939.6 ± 3.369.2 ± 0.774.6 ± 0.6
InfoMax(IJs)13.7 ± 0.72.2 ±0.235.0± 2.852.5 ± 2.874.4 ± 1.178.2 ±1.2
MV-InfoMax(INCE)12.2 ± 0.72.3 ±0.250.2 ± 3.675.8±3.894.6 ± 0.496.5 ± 0.1
MV-InfoMax (IJs)11.1 ± 1.02.3 ± 0.254.0 ± 6.178.3 ± 4.494.1 ± 0.395.90 ± 0.08
MIB (β=1,INCE)4.6 ± 0.72.1 ± 0.281.8 ± 5.092.7 ± 0.997.19 ± 0.0897.75 ± 0.05
MIB(β=1,Is)2.4 ± 0.22.1 ± 0.297.1 ± 0.297.2 ± 0.297.70 ± 0.0697.82 ± 0.01
+ +Figure 7 reports the linear projection of the embedding obtained using the MIB model. The latent space appears to roughly consists of ten clusters which corresponds to the different digits. This observation is consistent with the empirical measurement of input and label information $\bar { I } ( { \bf { x } } ; { \bf { z } } ) \approx$ $I ( \mathbf { z } ; \mathbf { y } ) \approx \log 1 0$ , and the performance of the linear classifier in scarce label regimes. As the cluster are distinct and concentrated around the respective centroids, 10 labeled examples are sufficient to align the centroid coordinates with the digit labels. + +![](images/f76dfca9ce31325a34acae4c0ad7ac8778fd112016ade2e04378631dbe9ba589.jpg) +Figure 7: Linear projection of the embedding obtained by applying the MIB encoder to the MNIST test set. The 64 dimensional representation is projected onto the two principal components. Different colors are used to represent the 10 digit classes. + +# H ABLATION STUDIES + +# H.1 DIFFERENT RANGES OF DATA AUGMENTATION + +Figure 8 visualizes the effect of different ranges of corruption probabily as data augmentation strategy to produce the two views $\mathbf { v } _ { 1 }$ and $\mathbf { v } _ { 2 }$ . The MV-InfoMax Model does not seem to get any advantage from the use increasing amount of corruption, and it representation remains approximately in the same region of the information plane. On the other hand, the models trained with the MIB objective are able to take advantage of the augmentation to remove irrelevant data information and the representation transitions from the top right corner of the Information Plane (no-augmentation) to the top-left. When the amount of corruption approaches $100 \%$ , the mutual redundancy assumption is clearly violated, and the performances of MIB deteriorate. In the initial part of the transitions between the two regimes (which corresponds to extremely low probability of corruption) the MIB models drops some label information that is quickly re-gained when pixel corruption becomes more frequent. We hypothesize that this behavior is due to a problem with the optimization procedure, since the corruption are extremely unlikely, the Monte-Carlo estimation for the symmetrized Kullback-Leibler divergence is more biased. Using more examples of views produced from the same data-point within the same batch could mitigate this issue. + +![](images/33faa873a6686b66ec62de391891ff78f26f3fa6c9429a7ee8cc067cef8b2108.jpg) +Figure 8: Visualization of the coordinates on the Information Plane (plot on the left) and prediction accuracy (center and right) for the MV-InfoMax and MIB objectives with different amount of training labels and corruption percentage used for data-augmentation. + +# H.2 EFFECT OF $\beta$ + +The hyper-parameter $\beta$ (Equation 5) determines the trade-off between sufficiency and minimality of the representation for the second data view. When $\beta$ is zero, the training objective of MIB is equivalent to the Multi-View InfoMax target, since the representation has no incentive to discard any information. When $0 < \beta \leq 1$ the sufficiency constrain is enforced, while the superfluous information is gradually removed from the representation. Values of $\beta > 1$ can result in representations that violate the sufficiency constraint, since the minimization of $I ( \mathbf { x } ; \mathbf { z } | \mathbf { v } _ { 2 } )$ is prioritized. The trade-off resulting from the choice of different $\beta$ is visualized in Figure 9 and compared against $\beta$ -VAE. Note that in each point of the pareto-front the MIB model results in a better trade-off between $I ( \mathbf { x } ; \mathbf { z } )$ and $I ( \mathbf { y } ; \mathbf { z } )$ when compared to $\beta$ -VAE. The effectiveness of the Multi-View Information Bottleneck model is also justified by the corresponding values of predictive accuracy. + +![](images/94d948e8f83b961c343996b173880467a607ee21dfbe2d76ac46e60c951f5a83.jpg) +Figure 9: Visualization of the coordinates on the Information Plane (plot on the left) and prediction accuracy (center and right) for the $\beta$ -VAE, Multi-View InfoMax and Multi-View Information Bottleneck objectives with different amount of training labels and different values of the respective hyperparameter $\beta$ . \ No newline at end of file diff --git a/parse/train/B1xwcyHFDr/B1xwcyHFDr_content_list.json b/parse/train/B1xwcyHFDr/B1xwcyHFDr_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..aca66468fc6a31d740eb71c749ff9e393d37b148 --- /dev/null +++ b/parse/train/B1xwcyHFDr/B1xwcyHFDr_content_list.json @@ -0,0 +1,4067 @@ +[ + { + "type": "text", + "text": "LEARNING ROBUST REPRESENTATIONS VIA MULTI-VIEW INFORMATION BOTTLENECK ", + "text_level": 1, + "bbox": [ + 173, + 99, + 696, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Marco Federici University of Amsterdam m.federici@uva.nl ", + "bbox": [ + 184, + 170, + 352, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anjan Dutta University of Exeter a.dutta@exeter.ac.uk ", + "bbox": [ + 400, + 170, + 598, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Patrick Forre´ University of Amsterdam p.d.forre@uva.nl ", + "bbox": [ + 643, + 170, + 813, + 212 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Nate Kushmann \nMicrosoft Research \nnkushman@microsoft.com \nZeynep Akata \nUniversity of Tuebingen \nzeynep.akata@uni-tuebingen.de ", + "bbox": [ + 184, + 233, + 401, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 473, + 233, + 759, + 275 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 313, + 544, + 327 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The information bottleneck principle provides an information-theoretic method for representation learning, by training an encoder to retain all information which is relevant for predicting the label while minimizing the amount of other, excess information in the representation. The original formulation, however, requires labeled data to identify the superfluous information. In this work, we extend this ability to the multi-view unsupervised setting, where two views of the same underlying entity are provided but the label is unknown. This enables us to identify superfluous information as that not shared by both views. A theoretical analysis leads to the definition of a new multi-view model that produces state-of-the-art results on the Sketchy dataset and label-limited versions of the MIR-Flickr dataset. We also extend our theory to the single-view setting by taking advantage of standard data augmentation techniques, empirically showing better generalization capabilities when compared to common unsupervised approaches for representation learning. ", + "bbox": [ + 233, + 342, + 764, + 535 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 559, + 336, + 575 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The goal of deep representation learning (LeCun et al., 2015) is to transform a raw observational input, x, into a, typically lower-dimensional, representation, z, that contains the information relevant for a given task or set of tasks. Significant progress has been made in deep learning via supervised representation learning, where the labels, y, for the downstream task are known while $p ( \\mathbf { y } \\vert \\mathbf { x } )$ is learned directly (Sutskever et al., 2012; Hinton et al., 2012). Due to the cost of acquiring large labeled datasets, a recently renewed focus on unsupervised representation learning seeks to generate representations, $\\mathbf { z }$ , that are useful for a wide variety of different tasks where little to no labeled data is available (Devlin et al., 2018; Radford et al., 2019). ", + "bbox": [ + 174, + 589, + 825, + 702 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Our work is based on the information bottleneck principle (Tishby et al., 2000) where a representation becomes less affected by nuisances by discarding all information from the input that is not useful for a given task, resulting in increased robustness. In the supervised setting, one can directly apply the information bottleneck principle by minimizing the mutual information between the data $\\mathbf { x }$ and its representation $\\mathbf { z }$ , $I ( \\mathbf { x } ; \\mathbf { z } )$ , while simultaneously maximizing the mutual information between $\\mathbf { z }$ and the label y (Alemi et al., 2017). In the unsupervised setting, discarding only superfluous information is more challenging, as without labels the model cannot directly identify which information is relevant. Recent literature (Devon Hjelm et al., 2019; van den Oord et al., 2018) has focused on the InfoMax objective maximizing $I ( \\mathbf { x } , \\mathbf { z } )$ instead of minimizing it, to guarantee that all the predictive information is retained by the representation, but doing nothing to discard the irrelevant information. ", + "bbox": [ + 174, + 708, + 823, + 847 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we extend the information bottleneck method to the unsupervised multi-view setting. To do this, we rely on a basic assumption of the multi-view literature – that each view provides the same task-relevant information (Zhao et al., 2017). Hence, one can improve generalization by discarding all the information not shared by both views from the representation. We do this by maximizing the mutual information between the representations of the two views (Multi-View ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "InfoMax objective) while at the same time eliminating the information not shared between them, since it is guaranteed to be superfluous. The resulting representations are more robust for the given task as they have eliminated view specific nuisances. ", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our contributions are three-fold: (1) We extend the information bottleneck principle to the unsupervised multi-view setting and provide a rigorous theoretical analysis of its application. (2) We define a new model 1 that empirically leads to state-of-the-art results in the low-label setting on two standard multi-view datasets, Sketchy and MIR-Flickr. (3) By exploiting data augmentation techniques, we empirically show that the representations learned by our model in single-view settings are more robust than existing unsupervised representation learning methods, connecting our theory to the choice of augmentation strategy. ", + "bbox": [ + 174, + 152, + 825, + 251 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 PRELIMINARIES AND FRAMEWORK ", + "text_level": 1, + "bbox": [ + 174, + 270, + 498, + 285 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The challenge of representation learning can be formulated as finding a distribution $p ( \\mathbf { z } | \\mathbf { x } )$ that maps data observations $\\mathbf { x } \\in \\mathbb { X }$ into a representation $\\mathbf { z } \\in \\mathbb { Z }$ , capturing some desired characteristics. Whenever the end goal involves predicting a label y, we consider only $\\mathbf { z }$ that are discriminative enough to identify y. This requirement can be quantified by considering the amount of label information that remains accessible after encoding the data, and is known as sufficiency of $\\mathbf { z }$ for $\\mathbf { y }$ (Achille & Soatto, 2018): ", + "bbox": [ + 173, + 300, + 825, + 383 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Definition 1. Sufficiency: A representation $\\mathbf { z }$ of $\\mathbf { x }$ is sufficient for $\\mathbf { y }$ if and only if $I ( \\mathbf { x } ; \\mathbf { y } | \\mathbf { z } ) = 0$ ", + "bbox": [ + 169, + 386, + 808, + 401 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Any model that has access to a sufficient representation $\\mathbf { z }$ must be able to predict $\\mathbf { y }$ at least as accurately as if it has access to the original data $\\mathbf { x }$ instead. In fact, $\\mathbf { z }$ is sufficient for $\\mathbf { y }$ if and only if the amount of information regarding the task is unchanged by the encoding procedure (see Proposition B.1 in the Appendix): ", + "bbox": [ + 174, + 410, + 825, + 467 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/5b29bcdc2aff7abcd2c9fc587742d539ac5bbbc8953144fdad55d95e370603a8.jpg", + "text": "$$\nI ( \\mathbf { x } ; \\mathbf { y } | \\mathbf { z } ) = 0 \\iff I ( \\mathbf { x } ; \\mathbf { y } ) = I ( \\mathbf { y } ; \\mathbf { z } ) .\n$$", + "text_format": "latex", + "bbox": [ + 366, + 469, + 630, + 487 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Among sufficient representations, the ones that result in better generalization for unlabeled data instances are particularly appealing. When $\\mathbf { x }$ has higher information content than $\\mathbf { y }$ , some of the information in $\\mathbf { x }$ must be irrelevant for the prediction task. This can be better understood by subdividing $I ( \\mathbf { x } ; \\mathbf { z } )$ into two components by using the chain rule of mutual information (see Appendix A): ", + "bbox": [ + 173, + 496, + 825, + 551 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/a4824f9469b1ec0865c014912dfeb0342d416d01e53f7307c226ada6c66c6510.jpg", + "text": "$$\n\\begin{array} { r } { I ( \\mathbf { x } ; \\mathbf { z } ) = \\underbrace { I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } ) } _ { \\mathrm { s u p e r f l u o u s ~ i n f o r m a t i o n ~ \\mu ~ } } + \\underbrace { I ( \\mathbf { y } ; \\mathbf { z } ) } _ { \\mathrm { p r e d i c t i v e ~ i n f o r m a t i o n ~ } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 554, + 651, + 588 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Conditional mutual information $I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } )$ represents the information in $\\mathbf { z }$ that is not predictive of $\\mathbf { y }$ , i.e. superfluous information. While $I ( \\mathbf { y } ; \\mathbf { z } )$ determines how much label information is accessible from the representation. Note that this last term is independent of the representation as long as $\\mathbf { z }$ is sufficient for $\\mathbf { y }$ (see Equation 1). As a consequence, a sufficient representation contains minimal data information whenever $I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } )$ is minimized. ", + "bbox": [ + 173, + 593, + 823, + 665 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Minimizing the amount of superfluous information can be done directly only in supervised settings. In fact, reducing $I ( \\mathbf { x } ; \\mathbf { z } )$ without violating the sufficiency constraint necessarily requires making some additional assumptions on the predictive task (see Theorem B.1 in the Appendix). In the next section we describe the basis of our technique, a strategy to safely reduce the information content of a representation even when the label $\\mathbf { y }$ is not observed, by exploiting redundant information in the form of an additional view on the data. ", + "bbox": [ + 173, + 670, + 825, + 753 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 MULTI-VIEW INFORMATION BOTTLENECK ", + "text_level": 1, + "bbox": [ + 174, + 773, + 562, + 790 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "As a motivating example, consider $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ to be two images of the same object from different view-points and let $\\mathbf { y }$ be its label. Assuming that the object is clearly distinguishable from both $\\mathbf { v } _ { 1 }$ and let $\\mathbf { v } _ { 2 }$ , any representation $\\mathbf { z }$ containing all information accessible from both views would also contain the necessary label information. Furthermore, if $\\mathbf { z }$ captures only the details that are visible from both pictures, it would eliminate the view-specific details and reduce the sensitivity of the representation to view-changes. The theory to support this intuition is described in the following where $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ are jointly observed and referred to as data-views. ", + "bbox": [ + 174, + 804, + 826, + 901 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3.1 SUFFICIENCY AND ROBUSTNESS IN THE MULTI-VIEW SETTING ", + "text_level": 1, + "bbox": [ + 174, + 103, + 648, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section we extend our analysis of sufficiency and minimality to the multi-view setting. ", + "bbox": [ + 173, + 128, + 784, + 145 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Intuitively, we can guarantee that $\\mathbf { z }$ is sufficient for predicting $\\mathbf { y }$ even without knowing y by ensuring that $\\mathbf { z }$ maintains all information which is shared by $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ . This intuition relies on a basic assumption of the multi-view environment – that the two views provide the same predictive information. To formalize this we define redundancy. ", + "bbox": [ + 173, + 150, + 825, + 207 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 2. Redundancy: $\\mathbf { v } _ { 1 }$ is redundant with respect to $\\mathbf { v } _ { 2 }$ for $\\mathbf { y }$ if and only if $I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } | \\mathbf { v } _ { 2 } ) = 0$ ", + "bbox": [ + 171, + 210, + 820, + 227 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Intuitively, a view $\\mathbf { v } _ { 1 }$ is redundant for a task whenever it is irrelevant for the prediction of $\\mathbf { y }$ if $\\mathbf { v } _ { 2 }$ is already observed. Whenever $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ are mutually redundant $\\mathbf { \\bar { v } } _ { 1 }$ is redundant with respect to $\\mathbf { v } _ { 2 }$ for $\\mathbf { y }$ , and vice-versa), we can show the following: ", + "bbox": [ + 174, + 237, + 825, + 280 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Corollary 1. Let $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ be two mutually redundant views for a target y and let $\\mathbf { z } _ { 1 }$ be a representation of $\\mathbf { v } _ { 1 }$ . If $\\mathbf { \\dot { z } } _ { 1 }$ is sufficient for $v _ { 2 } \\left( I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) = 0 \\right)$ then $\\mathbf { z } _ { 1 }$ is as predictive for y as the joint observation of the two views $( I ( \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ; \\mathbf { y } ) = I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } ) )$ . ", + "bbox": [ + 174, + 284, + 825, + 327 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In other words, whenever it is possible to assume mutual redundancy, any representation which contains all the information shared by both views (the redundant information) is as predictive as their joint observation. ", + "bbox": [ + 174, + 338, + 825, + 380 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "By factorizing the mutual information between $\\mathbf { v } _ { 1 }$ and $\\mathbf { z } _ { 1 }$ analogously to Equation 2, we can identify two components: ", + "bbox": [ + 174, + 387, + 823, + 415 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/16c34669d89995e0d40d834f9b9982c2eff6913c955f4bb5326c42138707aeca.jpg", + "text": "$$\n\\begin{array} { r } { I ( { \\bf v } _ { 1 } ; { \\bf z } _ { 1 } ) = \\underbrace { I ( { \\bf v } _ { 1 } ; { \\bf z } _ { 1 } | { \\bf v } _ { 2 } ) } _ { \\mathrm { s u p e r f l u o u s ~ i n f o r m a t i o n } } + \\underbrace { I ( { \\bf v } _ { 2 } ; { \\bf z } _ { 1 } ) } _ { \\mathrm { p r e d i c t i v e ~ i n f o r m a t i o n ~ f o r ~ } { \\bf v } _ { 2 } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 420, + 676, + 458 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Since $I ( \\mathbf { v } _ { 2 } ; \\mathbf { z } _ { 1 } )$ has to be maximal if we want the representation to be sufficient for the label, we conclude that $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } )$ can be reduced by minimizing $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } )$ . This term intuitively represents the information $\\mathbf { z } _ { 1 }$ contains which is unique to $\\mathbf { v } _ { 1 }$ and is not predictable by observing $\\mathbf { v } _ { 2 }$ . Since we assumed mutual redundancy between the two views, this information must be irrelevant for the predictive task and, therefore, it can be safely discarded. The proofs and formal assertions for the above statements and Corollary 1 can be found in Appendix B. ", + "bbox": [ + 173, + 464, + 825, + 549 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The less the two views have in common, the more $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } )$ can be reduced without violating sufficiency for the label, and consequently, the more robust the resulting representation. At the extreme, $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ share only label information, in which case we can show that $\\mathbf { z } _ { 1 }$ is minimal for y and our method is identical to the supervised information bottleneck method without needing to access the labels. Conversely, if $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ are identical, then our method degenerates to the InfoMax principle since no information can be safely discarded (see Appendix E). ", + "bbox": [ + 173, + 555, + 825, + 640 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 THE MULTI-VIEW INFORMATION BOTTLENECK LOSS FUNCTION ", + "text_level": 1, + "bbox": [ + 176, + 655, + 665, + 671 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ that satisfy the mutual redundancy condition for a label $\\mathbf { y }$ , we would like to define an objective function for the representation $\\mathbf { z } _ { 1 }$ of $\\mathbf { v } _ { 1 }$ that discards as much information as possible without losing any label information. In Section 3.1 we showed that we can obtain sufficiency for $\\mathbf { y }$ by ensuring that the representation $\\mathbf { z } _ { 1 }$ of $\\mathbf { v } _ { 1 }$ is sufficient for $\\mathbf { v } _ { 2 }$ , and that decreasing $I ( \\mathbf { z } _ { 1 } ; \\mathbf { v } _ { 1 } | \\mathbf { v } _ { 2 } )$ will increase the robustness of the representation by discarding irrelevant information. So we can combine these two requirements using a relaxed Lagrangian objective to obtain the minimal sufficient representation $\\mathbf { z } _ { 1 }$ for $\\mathbf { v } _ { 2 }$ : ", + "bbox": [ + 173, + 681, + 825, + 780 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/85644e7d6e87501965b9b2c0326b3de9b83d8eb1eee87e4f8492586b78463ecd.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal L _ { 1 } ( \\boldsymbol \\theta ; \\lambda _ { 1 } ) = I _ { \\boldsymbol \\theta } ( \\mathbf z _ { 1 } ; \\mathbf v _ { 1 } | \\mathbf v _ { 2 } ) - \\lambda _ { 1 } I _ { \\boldsymbol \\theta } ( \\mathbf v _ { 2 } ; \\mathbf z _ { 1 } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 786, + 643, + 804 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\theta$ denotes the dependency on the parameters of the encoder $p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } )$ , and $\\lambda _ { 1 }$ represents the Lagrangian multiplier introduced by the constrained optimization. Symmetrically, we define a loss $\\mathcal { L } _ { 2 }$ to optimize the parameters $\\psi$ of a conditional distribution $p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } )$ that defines a minimal sufficient representation $\\mathbf { z } _ { 2 }$ of the second view $\\mathbf { v } _ { 2 }$ for $\\mathbf { v } _ { 1 }$ : ", + "bbox": [ + 174, + 809, + 825, + 866 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/8839321e7828443186468a2392dc4b7bc67f36e21b4f301d293b9f40158b66f8.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { 2 } ( \\psi ; \\lambda _ { 2 } ) = I _ { \\psi } ( \\mathbf { z } _ { 2 } ; \\mathbf { v } _ { 2 } | \\mathbf { v } _ { 1 } ) - \\lambda _ { 2 } I _ { \\psi } ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 2 } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 872, + 647, + 890 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "By defining $\\mathbf { z } _ { 1 }$ and $\\mathbf { z } _ { 2 }$ on the same domain $\\mathbb { Z }$ and re-parametrizing the Lagrangian multipliers, the average of the two loss functions $\\mathcal { L } _ { 1 }$ and $\\mathcal { L } _ { 2 }$ can be upper bounded as follows: ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/1477c2774fb08cee976c3a6575401d2e64f7ffe2704825b5af1736a3d74f0f68.jpg", + "image_caption": [ + "Figure 1: Visualization our Multi-View Information Bottleneck model for both multi-view and single-view settings, where $\\hat { I } _ { \\xi } ( z _ { 1 } ; z _ { 2 } )$ refers to the sample-based parametric mutual information estimation. Whenever $p ( \\mathbf { v } _ { 1 } )$ and $p ( \\mathbf { v } _ { 2 } )$ have the same distribution, the two encoders can share their parameters. " + ], + "image_footnote": [], + "bbox": [ + 174, + 108, + 828, + 338 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/2ce7cd6d927f40d325f332f502d1ee77fcaa96bc4d3816b01d7ac7f5e179c00b.jpg", + "text": "$$\n\\displaystyle \\mathcal { L } _ { M I B } ( \\theta , \\psi ; \\beta ) = - I _ { \\theta \\psi } ( { \\bf z } _ { 1 } ; { \\bf z } _ { 2 } ) + \\beta D _ { S K L } ( p _ { \\theta } ( { \\bf z } _ { 1 } | { \\bf v } _ { 1 } ) | | p _ { \\psi } ( { \\bf z } _ { 2 } | { \\bf v } _ { 2 } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 274, + 478, + 722, + 496 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $D _ { S K L }$ represents the symmetrized KL divergence obtained by averaging the expected value of $D _ { \\mathrm { K L } } ( p _ { \\boldsymbol \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { \\bar { v } } _ { 1 } ) | | p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) )$ and $D _ { \\mathrm { K L } } ( p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) | | p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) )$ for joint observations of the two views, while the coefficient $\\beta$ defines the trade-off between sufficiency and robustness of the representation, which is a hyper-parameter in this work. The resulting Multi-View Infomation Bottleneck (MIB) model (Equation 5) is visualized in Figure 1, while the batch-based computation of the loss function is summarized in Algorithm 1. ", + "bbox": [ + 174, + 510, + 825, + 594 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The symmetrized KL divergence $D _ { S K L } ( p _ { \\boldsymbol { \\theta } } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) | | p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) )$ can be computed directly whenever $\\dot { p _ { \\theta } } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } )$ and $p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } )$ have a known density, while the mutual information between the two representations $I _ { \\theta \\psi } ( \\mathbf { z } _ { 1 } ; \\mathbf { z } _ { 2 } )$ can be maximized by using any sample-based differentiable mutual information lower bound. We tried the Jensen-Shannon $I _ { \\mathrm { J S } }$ (Devon Hjelm et al., 2019; Poole et al., 2019) and the InfoNCE $I _ { \\mathrm { N C E } }$ (van den Oord et al., 2018) estimators. These both require introducing an auxiliary parameteric model $C _ { \\xi } ( z _ { 1 } , z _ { 2 } )$ which is jointly optimized during the training procedure using re-parametrized samples from $p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } )$ and $p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } )$ . The full derivation for the MIB loss function can be found in Appendix F. ", + "bbox": [ + 173, + 599, + 825, + 713 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 SELF-SUPERVISION AND INVARIANCE ", + "text_level": 1, + "bbox": [ + 176, + 729, + 477, + 743 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our method can also be applied when multiple views are not available by taking advantage of standard data augmentation techniques. This allows learning invariances directly from the augmented data, rather than requiring them to be built into the model architecture. ", + "bbox": [ + 174, + 756, + 825, + 797 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "By picking a class $\\mathbb { T }$ of data augmentation functions $t : \\mathbb { X } \\to \\mathbb { W }$ that do not affect label information, it is possible to artificially build views that satisfy mutual redundancy for $\\mathbf { y }$ . Let $\\mathbf { t } _ { 1 }$ and $\\mathbf { t } _ { 2 }$ be two random variables over $\\mathbb { T }$ , then $\\mathbf { v } _ { 1 } : = \\mathbf { t } _ { 1 } ( \\mathbf { x } )$ and $\\mathbf { v } _ { 2 } : = \\mathbf { t } _ { 2 } ( \\mathbf { x } )$ must be mutually redundant for $\\mathbf { y }$ . Since data augmentation functions in $\\mathbb { T }$ do not affect label information $( I ( { \\bf v } _ { 1 } ; { \\bf \\bar { y } } ) = I ( { \\bf v } _ { 2 } ; { \\bf y } ) =$ $I ( \\mathbf { x } ; \\mathbf { y } ) )$ , a representation $\\mathbf { z } _ { 1 }$ of $\\mathbf { v } _ { 1 }$ that is sufficient for $\\mathbf { v } _ { 2 }$ must contain same amount of predictive information as $\\mathbf { x }$ . Formal proofs for this statement can be found in Appendix B.4. ", + "bbox": [ + 174, + 804, + 825, + 888 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Whenever the two transformations for the same observation are independent $( I ( \\mathbf { t } _ { 1 } ; \\mathbf { t } _ { 2 } | \\mathbf { x } ) = 0 )$ , they introduce uncorrelated variations in the two views, which will be discarded when creating a representation using our training objective. As an example, if $\\mathbb { T }$ represents a set of small translations, the two resulting views will differ by a small shift. Since this information is not shared, any $\\mathbf { z } _ { 1 }$ which is optimal according to the MIB objective must discard fine-grained details regarding the position. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "To enable parameter sharing between the encoders, we generate the two views $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ by independently sampling two functions from the same function class $\\mathbb { T }$ with uniform probability. As a result, $\\mathbf { t } _ { 1 }$ and $\\mathbf { t } _ { 2 }$ will have the same distribution, and so the two generated views will also have the same marginals $( p ( \\mathbf { v } _ { 1 } ) = p ( \\mathbf { v } _ { 2 } ) )$ . For this reason, the two conditional distributions $p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } )$ and $p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } )$ can share their parameters and only one encoder is necessary. Full (or partial) parameter sharing can be also applied in the multi-view settings whenever the two views have the same (or similar) marginal distributions. ", + "bbox": [ + 173, + 152, + 825, + 251 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 272, + 344, + 289 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The relationship between our method and past work on representation learning is best described using the Information Plane (Tishby et al., 2000). In this setting, each representation $\\mathbf { z }$ of $\\mathbf { x }$ for a predictive task y can be characterised by the amount of information regarding the raw observation $I ( \\mathbf { x } ; \\mathbf { z } )$ and the corresponding measure of accessible predictive information $I ( \\mathbf { y } ; \\mathbf { z } )$ $x$ and $y$ axis respectively on Figure 2). Ideally, a good representation would be maximally informative about the label while retaining a minimal amount of information from the observations (top left corner of the parallelogram). Further details on the Information Plane and the bounds visualized in Figure 2 are described in Appendix C. ", + "bbox": [ + 173, + 305, + 825, + 416 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Thanks to recent progress in mutual information estimation (Nguyen et al., 2008; Ishmael Belghazi et al., 2018; Poole et al., 2019), the InfoMax principle (Linsker, 1988) has gained attention for unsupervised representation learning (Devon Hjelm et al., 2019; van den Oord et al., 2018). Since the InfoMax objective involves maximizing $I ( \\mathbf { x } ; \\mathbf { z } )$ , the resulting representation aims to preserve all the information regarding the raw observations (top right corner in Figure 2). ", + "bbox": [ + 174, + 424, + 483, + 577 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Concurrent work has applied the InfoMax principle in the Multi-View setting (Ji et al., 2019; Henaff et al., 2019; Tian et al., 2019; Bachman ´ et al., 2019), aiming to maximize mutual infor", + "bbox": [ + 173, + 583, + 483, + 638 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/8293dec173cf38f592d351c0bf24a9a1218f0f2d6584288a4a2daa3512081767.jpg", + "image_caption": [ + "Figure 2: Information Plane determined by $I ( \\mathbf { x } ; \\mathbf { z } )$ $\\mathbf { \\dot { x } }$ -axis) and $I ( \\mathbf { y } ; \\mathbf { z } )$ (y-axis). Different objectives are compared based on their target. " + ], + "image_footnote": [], + "bbox": [ + 500, + 446, + 816, + 553 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "mation between the representation $\\mathbf { z }$ of a first data-view $\\mathbf { x }$ and a second one $\\mathbf { v } _ { 2 }$ . The target representation for the Multi-View InfoMax (MV-InfoMax) models should contain at least the amount of information in $\\mathbf { x }$ that is predictive for $\\mathbf { v } _ { 2 }$ , targeting the region $I ( \\mathbf { z } ; \\mathbf { x } ) \\ge I ( \\mathbf { x } ; \\mathbf { v } _ { 2 } )$ on the Information Plane (purple dotted line in Figure 2). Since the MV-InfoMax has no incentive to discard any information regarding $\\mathbf { x }$ from z, a representation that is optimal according to the InfoMax principle is also optimal for any MV-InfoMax model. Our model with $\\beta = 0$ (Equation 5) belongs to this family of objectives since the incentive to remove superfluous information is removed. Despite their success, Tschannen et al. (2019) has shown that the effectiveness of the InfoMax models is due to inductive biases introduced by the architecture and estimators rather than the training objective itself, since the InfoMax and MV-InfoMax objectives can be trivially maximized by using invertible encoders. ", + "bbox": [ + 174, + 638, + 825, + 791 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "On the other hand, Variational Autoencoders (VAEs) (Kingma & Welling, 2014) define a training objective that balances compression and reconstruction error (Alemi et al., 2018) through an hyper-parameter $\\beta$ . Whenever $\\beta$ is close to 0, the VAE objective aims for a lossless representation, approaching the same region of the Information Plane as the one targeted by InfoMax (Barber & Agakov, 2003). When $\\beta$ approaches large values, the representation becomes more compressed, showing increased generalization and disentanglement (Higgins et al., 2017; Burgess et al., 2018), and, as $\\beta$ approaches infinity, $I ( \\mathbf { z } ; \\mathbf { x } )$ goes to zero. During this transition from low to high $\\beta$ , however, there are no guarantees that VAEs will retain label information (Theorem B.1 in the Appendix). The path between the two regimes depends on how well the label information aligns with the inductive bias introduced by encoder (Jimenez Rezende & Mohamed, 2015; Kingma et al., 2016), prior (Tomczak & Welling, 2018) and decoder architectures (Gulrajani et al., 2017; Chen et al., 2017). ", + "bbox": [ + 173, + 797, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The idea of discarding irrelevant information was introduced in Tishby et al. (2000) and identified as one of the possible reasons behind the generalization capabilities of deep neural networks by Tishby & Zaslavsky (2015) and Achille & Soatto (2018). Representations based on the information bottleneck principle explicitly minimize the amount of superfluous information in the representation while retaining all the label information from the data (top-left corner of the Information Plane in Figure 2). This direction of research has been explored for both single-view (Alemi et al., 2018) and multi-view settings (Wang et al., 2019), even if explicit label supervision is required to train the representation $\\mathbf { z }$ . ", + "bbox": [ + 174, + 138, + 825, + 250 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In contrast to all of the above, our work is the first to explicitly identify and discard superfluous information from the representation in the unsupervised multi-view setting. This is because unsupervised models based on the $\\beta$ -VAE objective remove information indiscriminately without identifying which part is relevant for teh predictive task, and the InfoMax and Multi-View InfoMax methods do not explicitly try to remove superfluous information at all. The MIB objective, on the other hand, results in the representation with the least superfluous information, i.e. the most robust among the representations that are optimal according to Multi-View InfoMax, without requiring any additional label supervision. ", + "bbox": [ + 174, + 257, + 823, + 368 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 390, + 326, + 406 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section we demonstrate the effectiveness of our model against state-of-the-art baselines in both the multi-view and single-view setting. In the single-view setting, we also estimate the coordinates on the Information Plane for each of the baseline methods as well as our method to validate the theory in Section 3. ", + "bbox": [ + 174, + 424, + 825, + 479 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "The results reported in the following sections are obtained using the Jensen-Shannon $I _ { J S }$ (Devon Hjelm et al., 2019; Poole et al., 2019) estimator, which resulted in better performance for MIB and the other InfoMax-based models (Table 2 in the supplementary material). In order to facilitate the comparison between the effect of the different loss functions, the same estimator is used across the different models. ", + "bbox": [ + 174, + 486, + 825, + 556 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1 MULTI-VIEW TASKS ", + "text_level": 1, + "bbox": [ + 176, + 575, + 357, + 589 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We compare MIB on the sketch-based image retrieval (Sangkloy et al., 2016) and Flickr multiclass image classification (Huiskes & Lew, 2008) tasks with domain specific and prior multi-view learning methods. ", + "bbox": [ + 178, + 602, + 823, + 643 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5.1.1 SKETCH-BASED IMAGE RETRIEVAL ", + "text_level": 1, + "bbox": [ + 176, + 661, + 475, + 676 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Dataset. The Sketchy dataset (Sangkloy et al., 2016) consists of 12,500 images and 75,471 handdrawn sketches of objects from 125 classes. As in Liu et al. (2017), we also include another 60,502 images from the ImageNet (Deng et al., 2009) from the same classes, which results in total 73,002 natural object images. As per the experimental protocol of Zhang et al. (2018), a total of 6,250 sketches (50 sketches per category) are randomly selected and removed from the training set for testing purpose, which leaves 69,221 sketches for training the model. ", + "bbox": [ + 174, + 686, + 825, + 770 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Experimental Setup. The sketch-based image retrieval task is a ranking of 73,002 natural images according to the unseen test (query) sketch. Retrieval is done for our model by generating representations for the query sketch as well as all natural images, and ranking the image by the euclidean distance of their representation from the sketch representation. The baselines use various domain specific ranking methodologies. Model performance is computed based on the class of the ranked pictures corresponding to the query sketch. The training set consists of pairs of image $\\mathbf { v } _ { 1 }$ and sketch $\\mathbf { v } _ { 2 }$ randomly selected from the same class, to ensure that both views contain the equivalent label information (mutual redundancy). ", + "bbox": [ + 174, + 776, + 825, + 888 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Following recent work (Zhang et al., 2018; Dutta & Akata, 2019), we use features extracted from images and sketches by a VGG (Simonyan & Zisserman, 2014) architecture trained for classification on the TU-Berlin dataset (Eitz et al., 2012). The resulting flattened 4096-dimensional feature vectors are fed to our image and sketch encoders to produce a 64-dimensional representation. Both encoders consist of neural networks with hidden layers of 2048 and 1024 units respectively. Size of the representation and regularization strength $\\beta$ are tuned on a validation sub-split. We evaluate MIB on five different train/test splits and report mean and standard deviation in Table 5.1.1. Further details on our training procedure and architecture are in Appendix G. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/726f2e918ff1392fa39ef9bdb9741b3bd6f5d6c84acce49da00d6f18d2bfa125.jpg", + "image_caption": [ + "Table 1: Examples of the two views and class label from the Sketchy dataset (on the left) and comparison between MIB and other popular models in literature on the sketch-based image retrieval task (on the right). ∗ denotes models that use a 64-bits binary representation. The results for MIB corresponds to $\\beta = 1$ . " + ], + "image_footnote": [], + "bbox": [ + 183, + 99, + 826, + 236 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 330, + 825, + 415 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results. Table 5.1.1 shows that the our model achieves strong performance for both mean average precision $( \\mathrm { m A P } @ \\mathrm { a l l } )$ and precision at 200 (Prec $@ 2 0 0 )$ , suggesting that the representation is able to capture the common class information between the paired pictures and sketches. The effectiveness of MIB on the retrieval task can be mostly attributed to the regularization introduced with the symmetrized KL divergence between the two encoded views. In addition to discarding view-private information, this term actively aligns the representations of $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ , making the MIB model especially suitable for retrieval tasks ", + "bbox": [ + 174, + 421, + 825, + 518 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1.2 MIR-FLICKR ", + "text_level": 1, + "bbox": [ + 174, + 536, + 320, + 551 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Dataset. The MIR-Flickr dataset (Huiskes & Lew, 2008) consists of 1M images annotated with 800K distinct user tags. Each image is represented by a vector of 3,857 hand-crafted image features $( \\mathbf { v } _ { 1 } )$ , while the 2,000 most frequent tags are used to produce a 2000-dimensional multi-hot encoding $\\left( \\mathbf { v } _ { 2 } \\right)$ for each picture. The dataset is divided into labeled and unlabeled sets that respectively contain 975K and 25K images, where the labeled set also contains 38 distinct topic classes together with the user tags. Training images with less than two tags are removed, which reduces the total number of training samples to 749,647 pairs (Sohn et al., 2014; Wang et al., 2016). The labeled set contains 5 different splits of train, validation and test sets of size 10K/5K/10K respectively. ", + "bbox": [ + 173, + 563, + 825, + 674 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Experimental Setup. Following standard procedure in the literature (Srivastava & Salakhutdinov, 2014; Wang et al., 2016), we train our model on the unlabeled pairs of images and tags. Then a multi-label logistic classifier is trained from the representation of 10K labeled train images to the corresponding macro-categories. The quality of the representation is assessed based on the performance of the trained logistic classifier on the labeled test set. Each encoder consists of a multi-layer perceptron of 4 hidden layers with ReLU activations learning two 1024-dimensional representations $\\mathbf { z } _ { 1 }$ and $\\mathbf { z } _ { 2 }$ for images $\\mathbf { v } _ { 1 }$ and tags $\\mathbf { v } _ { 2 }$ respectively. Examples of the two views, labels, and further details on the training procedure are in Appendix G. ", + "bbox": [ + 173, + 681, + 825, + 792 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Results. Our MIB model is compared with other popular multi-view learning models in Figure 3 for $\\beta = 0$ (Multi-View InfoMax), $\\beta = 1$ and $\\beta \\stackrel { \\cdot } { = } 1 0 ^ { - 3 }$ (best on validation set). Although the tuned MIB performs similarly to Multi-View InfoMax with a large number of labels, it outperforms it when fewer labels are available. Furthermore, by choosing a larger $\\beta$ the accuracy of our model drastically increases in scarce label regimes, while slightly reducing the accuracy when all the labels are observed (see right side of Figure 3). This effect is likely due to a violation of the mutual redundancy constraint (see Figure 6 in the supplementary material) which can be compensated with smaller values of $\\beta$ for less aggressive compression. ", + "bbox": [ + 173, + 799, + 823, + 883 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/93e7041326b04365403c56c3f7d87401c68d1dcba01f704729dbfa33da116434.jpg", + "image_caption": [ + "Figure 3: Left: mean average precision (mAP) of the classifier trained on different multi-view representations for the MIR-Flickr task. Right: comparing the performance for different values of $\\beta$ and percentages of given labeled examples (from $1 \\%$ up to $100 \\%$ ). Each model uses encoders of comparable size, producing a 1024 dimensional representation. † results from Wang et al. (2016). " + ], + "image_footnote": [], + "bbox": [ + 176, + 113, + 532, + 282 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/546e59c737c14c10bb507eb44771651342c9a72b8a20bd6bb7385c5c68344ff4.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 537, + 114, + 810, + 289 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 393, + 823, + 421 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "A possible reason for the effectiveness of MIB against some of the other baselines may be its ability to use mutual information estimators that do not require reconstruction. Both Multi-View VAE (MVAE) and Deep Variational CCA (VCCA) rely on a reconstruction term to capture cross-modal information, which can introduce bias that decreases performance. ", + "bbox": [ + 174, + 428, + 825, + 484 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5.2 SELF-SUPERVISED SINGLE-VIEW TASK", + "text_level": 1, + "bbox": [ + 176, + 510, + 486, + 523 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we compare the performance of different unsupervised learning models by measuring their data efficiency and empirically estimating the coordinates of their representation on the Information Plane. Since accurate estimation of mutual information is extremely expensive (McAllester & Stratos, 2018), we focus on relatively small experiments that aim to uncover the difference between popular approaches for representation learning. ", + "bbox": [ + 174, + 539, + 823, + 608 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Dataset. The dataset is generated from MNIST by creating the two views, $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ , via the application of data augmentation consisting of small affine transformations and independent pixel corruption to each image. These are kept small enough to ensure that label information is not effected. Each pair of views is generated from the same underlying image, so no label information is used in this process (details in Appendix G). ", + "bbox": [ + 173, + 616, + 825, + 685 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Experimental Setup. To evaluate, we train the encoders using the unlabeled multi-view dataset just described, and then fix the representation model. A logistic regression model is trained using the resulting representations along with a subset of labels for the training set, and we report the accuracy of this model on a disjoint test set as is standard for the unsupervised representation learning literature (Tschannen et al., 2019; Tian et al., 2019; van den Oord et al., 2018). We estimate $I ( \\mathbf { x } ; \\mathbf { z } )$ and $I ( \\mathbf { y } ; \\mathbf { z } )$ using mutual information estimation networks trained from scratch on the final representations using batches of joint samples $\\{ ( { \\pmb x } ^ { ( i ) } , { \\pmb y } ^ { ( i ) } , z ^ { ( i ) } ) \\} _ { i = 1 } ^ { B } \\sim p ( { \\pmb x } , { \\pmb y } ) p _ { \\theta } ( { \\pmb z } | { \\pmb x } )$ . All models are trained using the same encoder architecture consisting of 2 layers of 1024 hidden units with ReLU activations, resulting in 64-dimensional representations. The same data augmentation procedure was also applied for single-view architectures and models were trained for 1 million iterations with batch size $B = 6 4$ . ", + "bbox": [ + 173, + 693, + 825, + 847 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results. Figure 4 summarizes the results. The empirical measurements of mutual information reported on the Information Plane are consistent with the theoretical analysis reported in Section 4: models that retain less information about the data while maintaining the maximal amount of predictive information, result in better classification performance at low-label regimes, confirming the hypothesis that discarding irrelevant information yields robustness and more data-efficient representations. Notably, the MIB model with $\\beta = 1$ retains almost exclusively label information, hardly decreasing the classification performance when only one label is used for each data point. ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/041c04c65ea645bac713211faa7fa73f64ba7d3e4c249ab0746f6c9ab1eefe4f.jpg", + "image_caption": [ + "Figure 4: Comparing the representations obtained with different objectives on MNIST dataset. The empirical estimation of the coordinates on the Information Plane (in nats on the left) is followed by the respective classification accuracy for different number of randomly sampled labels (from 1 example per label up to 6000 examples per label). Representations that discard more observational information tend to perform better in scarce label regimes. The measurements used to produce the two graphs are reported in Appedix G.4.1. " + ], + "image_footnote": [], + "bbox": [ + 186, + 102, + 789, + 220 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 344, + 823, + 372 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUSIONS AND FUTURE WORK ", + "text_level": 1, + "bbox": [ + 174, + 393, + 500, + 409 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this work, we introduce Multi-View Information Bottleneck, a novel method for taking advantage of multiple data-views to produce robust representations for downstream tasks. In our experiments, we compared MIB empirically against other approaches in the literature on three such tasks: sketchbased image retrieval, multi-view and unsupervised representation learning. The strong performance obtained in the different areas show that Multi-View Information Bottleneck can be practically applied to various tasks for which the paired observations are either readily available or artificially produced. Furthermore, the positive results on the MIR-Flickr dataset show that our model can work well in practice even when mutual redundancy holds only approximately. ", + "bbox": [ + 174, + 424, + 825, + 536 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "There are multiple extensions that we would like to explore in future work. One interesting direction would be considering more than two views. In Appendix D we discuss why the mutual redundancy condition cannot be trivially extended to more than two views, but we still believe such an extension is possible. Secondly, we believe that exploring the role played by different choices of data augmentation could bridge the gap between the Information Bottleneck principle and the literature on invariant neural networks (Bloem-Reddy & Whye Teh, 2019), which are able to exploit known symmetries and structure of the data to remove superfluous information. 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On Mutual Information Maximization for Representation Learning. arXiv, 2019. ", + "bbox": [ + 169, + 818, + 823, + 847 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation Learning with Contrastive Predictive Coding. arXiv, 2018. ", + "bbox": [ + 173, + 856, + 821, + 886 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Qi Wang, Claire Boudreau, Qixing Luo, Pang-Ning Tan, and Jiayu Zhou. Deep multi-view information bottleneck. In SIAM, 2019. ", + "bbox": [ + 173, + 895, + 820, + 924 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Weiran Wang, Xinchen Yan, Honglak Lee, and Karen Livescu. Deep Variational Canonical Correlation Analysis. arXiv, 2016. ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Qian Yu, Yongxin Yang, Yi-Zhe Song, Tao Xiang, and Timothy Hospedales. Sketch-a-Net that Beats Humans. IJCV, 2017. ", + "bbox": [ + 173, + 140, + 823, + 170 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Jingyi Zhang, Fumin Shen, Li Liu, Fan Zhu, Mengyang Yu, Ling Shao, Heng Tao Shen, and Luc Van Gool. Generative domain-migration hashing for sketch-to-image retrieval. In ECCV, 2018. ", + "bbox": [ + 171, + 178, + 823, + 208 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Jing Zhao, Xijiong Xie, Xin Xu, and Shiliang Sun. Multi-view learning overview: Recent progress and new challenges. IF, 2017. ", + "bbox": [ + 171, + 217, + 823, + 246 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A PROPERTIES OF MUTUAL INFORMATION AND ENTROPY", + "text_level": 1, + "bbox": [ + 173, + 102, + 671, + 118 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this section we enumerate some of the properties of mutual information that are used to prove the theorems reported in this work. For any random variables $\\mathbf { w } , \\mathbf { x } , \\mathbf { y }$ and $\\mathbf { z }$ : ", + "bbox": [ + 176, + 133, + 825, + 164 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$( P _ { 1 } )$ ) Positivity: ", + "bbox": [ + 194, + 175, + 302, + 191 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/7a5ae0089658e36f5db9a30f0199843384f9a098a2923fe61bc1df7b1d88c5e5.jpg", + "text": "$$\nI ( \\mathbf { x } ; \\mathbf { y } ) \\geq 0 , I ( \\mathbf { x } ; \\mathbf { y } | \\mathbf { z } ) \\geq 0\n$$", + "text_format": "latex", + "bbox": [ + 436, + 200, + 619, + 218 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$( P _ { 2 } )$ Chain rule: ", + "bbox": [ + 194, + 229, + 307, + 246 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/44239778e985528481118dffab24781aeaf3548979e0059d40555127525ef92f.jpg", + "text": "$$\nI ( \\mathbf { x y } ; \\mathbf { z } ) = I ( \\mathbf { y } ; \\mathbf { z } ) + I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } )\n$$", + "text_format": "latex", + "bbox": [ + 423, + 253, + 633, + 271 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$( P _ { 3 } )$ Chain rule (Multivariate Mutual Information): ", + "bbox": [ + 194, + 282, + 535, + 299 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/b9a18089a0d65b65fa64393cad292aea7a77ec8c63dd08c09a5573329801ca35.jpg", + "text": "$$\nI ( \\mathbf { x } ; \\mathbf { y } ; \\mathbf { z } ) = I ( \\mathbf { y } ; \\mathbf { z } ) - I ( \\mathbf { y } ; \\mathbf { z } | \\mathbf { x } )\n$$", + "text_format": "latex", + "bbox": [ + 419, + 308, + 635, + 325 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$( P _ { 4 } )$ Positivity of discrete entropy: For discrete $\\mathbf { x }$ ", + "bbox": [ + 194, + 337, + 428, + 366 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/625fabf3fa0d62108b3c1633d335df52aa576974fa47836c6a0f88e6f71f16dd.jpg", + "text": "$$\nH ( \\mathbf { x } ) \\geq 0 , H ( \\mathbf { x } | \\mathbf { y } ) \\geq 0\n$$", + "text_format": "latex", + "bbox": [ + 447, + 375, + 609, + 392 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$( P _ { 5 } )$ Entropy and Mutual Information ", + "bbox": [ + 194, + 404, + 449, + 420 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/be892a88303715fdbdd8f19611a4a46df54edd2a78f6f13b6e24b48977df0dab.jpg", + "text": "$$\nH ( \\mathbf { x } ) = H ( \\mathbf { x } | \\mathbf { y } ) + I ( \\mathbf { x } ; \\mathbf { y } )\n$$", + "text_format": "latex", + "bbox": [ + 436, + 428, + 619, + 445 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B THEOREMS AND PROOFS ", + "text_level": 1, + "bbox": [ + 176, + 467, + 416, + 483 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In the following section we prove the statements reported in the main text of the paper. Whenever a random variable $\\mathbf { z }$ is defined to be a representation of another random variable $\\mathbf { x }$ , we state that $\\mathbf { z }$ is conditionally independent from any other variable in the system once $\\mathbf { x }$ is observed. This does not imply that $\\mathbf { z }$ must be a deterministic function of $\\mathbf { x }$ , but that the source of stochasticity for $\\mathbf { z }$ is independent of the other random variables. As a result whenever $\\mathbf { z }$ is a representation of $\\mathbf { x }$ : ", + "bbox": [ + 173, + 500, + 826, + 570 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/b4bcc40d0ba68fec65057781e3686997cd78d3a41afea3887556ad3453ea0ae6.jpg", + "text": "$$\nI ( \\mathbf { z } ; \\mathbf { a } | \\mathbf { x b } ) = 0 ,\n$$", + "text_format": "latex", + "bbox": [ + 444, + 579, + 552, + 595 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "for any variable (or groups of variables) a and $\\mathbf { b }$ in the system. ", + "bbox": [ + 173, + 604, + 584, + 621 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.1 ON SUFFICIENCY ", + "text_level": 1, + "bbox": [ + 174, + 638, + 338, + 654 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proposition B.1. Let x and y be random variables with joint distribution $p ( \\mathbf { x } , \\mathbf { y } )$ . Let z be a representation of x, then z is sufficient for y if and only if $I ( \\mathbf { x ; y } ) = I ( \\mathbf { y ; z } )$ ", + "bbox": [ + 173, + 665, + 825, + 695 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Hypothesis: ", + "bbox": [ + 174, + 707, + 253, + 723 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "$( H _ { 1 }$ ) z is a representation of x: $I ( \\mathbf { y } ; \\mathbf { z } | \\mathbf { x } ) = 0$ ", + "bbox": [ + 194, + 734, + 501, + 751 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Thesis: ", + "bbox": [ + 174, + 762, + 223, + 776 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/11879a4b08fab6afdd8fc663fafc43b2c3f7a0507442234bb3edcd8fa10d4efe.jpg", + "text": "$$\n( T _ { 1 } ) \\ I ( \\mathbf { x } ; \\mathbf { y } | \\mathbf { z } ) = 0 \\iff I ( \\mathbf { x } ; \\mathbf { y } ) = I ( \\mathbf { y } ; \\mathbf { z } )\n$$", + "text_format": "latex", + "bbox": [ + 192, + 789, + 491, + 806 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Proof. ", + "bbox": [ + 173, + 828, + 217, + 843 + ], + "page_idx": 12 + }, + { + "type": "equation", + "img_path": "images/1b94e5489458be535edd5b8dbefed717505888d0743b92f99b24850dc8c7e0ed.jpg", + "text": "$$\n\\begin{array} { r l } & { I ( \\mathbf { x } ; \\mathbf { y } | \\mathbf { z } ) \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { x } ; \\mathbf { y } ) - I ( \\mathbf { x } ; \\mathbf { y } ; \\mathbf { z } ) \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { x } ; \\mathbf { y } ) - I ( \\mathbf { y } ; \\mathbf { z } ) - I ( \\mathbf { y } ; \\mathbf { z } | \\mathbf { x } ) } \\\\ & { \\overset { ( H _ { 1 } ) } { = } I ( \\mathbf { x } ; \\mathbf { y } ) - I ( \\mathbf { y } ; \\mathbf { z } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 267, + 852, + 730, + 902 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Since both $I ( \\mathbf { x } ; \\mathbf { y } )$ and $I ( \\mathbf { y } ; \\mathbf { z } )$ are non-negative $P _ { 1 } ) , I ( \\mathbf { x } ; \\mathbf { y } | \\mathbf { z } ) = 0 \\iff I ( \\mathbf { y } ; \\mathbf { z } ) = I ( \\mathbf { x } ; \\mathbf { y } )$ ", + "bbox": [ + 171, + 909, + 784, + 925 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "B.2 NO FREE GENERALIZATION ", + "text_level": 1, + "bbox": [ + 176, + 103, + 411, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Theorem B.1. Let x, z and y be random variables with joint distribution $p ( \\mathbf { x } , \\mathbf { y } , \\mathbf { z } )$ . Let $\\mathbf { z } ^ { \\prime }$ be a representation of x that satisfies $I ( \\mathbf { x } ; \\mathbf { z } ) > I ( \\mathbf { x } ; \\mathbf { z } ^ { \\prime } )$ , then it is always possible to find a label y for which $\\mathbf { z } ^ { \\prime }$ is not predictive for y while z is. ", + "bbox": [ + 174, + 128, + 825, + 172 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Hypothesis: ", + "text_level": 1, + "bbox": [ + 173, + 199, + 253, + 214 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "$\\left( H _ { 1 } \\right) \\textbf { z } ^ { \\prime }$ is a representation of x: $I ( \\mathbf { y } ; \\mathbf { z } ^ { \\prime } | \\mathbf { x } ) = 0$ $( H _ { 2 } ) I ( { \\bf x ; z } ) > I ( { \\bf x ; z ^ { \\prime } } )$ ", + "bbox": [ + 189, + 224, + 511, + 267 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Thesis: ", + "bbox": [ + 174, + 277, + 223, + 292 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/9e2d9d216c2b13221d173a64c0c2545d71c13cade821f7478f651e512d7b6631.jpg", + "text": "$$\n( T _ { 1 } ) \\ I ( \\mathbf { x } ; \\mathbf { z } ^ { \\prime } ) < I ( \\mathbf { x } ; \\mathbf { z } ) \\implies \\exists \\mathbf { y } . I ( \\mathbf { y } ; \\mathbf { z } ) > I ( \\mathbf { y } ; \\mathbf { z } ^ { \\prime } ) = 0\n$$", + "text_format": "latex", + "bbox": [ + 192, + 303, + 573, + 321 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Proof. By construction. ", + "bbox": [ + 174, + 338, + 331, + 354 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "1. We first factorize $\\mathbf { x }$ as a function of two independent random variables (Proposition 2.1 Achille & Soatto (2018)) by picking $\\mathbf { y }$ such that: ", + "bbox": [ + 210, + 371, + 823, + 400 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/1a34df1d2373d3bc1c366833c4a5dcd1eacbd0e4011e33221aa77b5f20c52f71.jpg", + "text": "$$\n\\begin{array} { r } { \\left( C _ { 1 } \\right) I ( \\mathbf { y } ; \\mathbf { z } ^ { \\prime } ) = 0 } \\\\ { \\left( C _ { 2 } \\right) \\mathbf { x } = f ( \\mathbf { z } ^ { \\prime } , \\mathbf { y } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 407, + 361, + 445 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "for some deterministic function $f$ . Note that such $\\mathbf { y }$ always exists. ", + "bbox": [ + 230, + 452, + 665, + 467 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "2. Since $\\mathbf { x }$ is a function of $\\mathbf { y }$ and $\\mathbf { z } ^ { \\prime }$ : ", + "bbox": [ + 212, + 476, + 450, + 491 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/9f23de02c64da6e41b81be850763a987c52862cfa8a5f55a14cf38c05a060320.jpg", + "text": "$$\n\\begin{array} { r l } { \\left( C _ { 4 } \\right) } & { { } I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } \\mathbf { z } ^ { \\prime } ) = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 498, + 380, + 517 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Considering $I ( \\mathbf { y } ; \\mathbf { z } )$ : ", + "bbox": [ + 173, + 532, + 312, + 549 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/3ff873158aee280e5db3e933bfb951dce7de2ac7670bcc04b6f0ee04ff9cc193.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { U } _ { 2 , \\Psi } ^ { \\{ S \\} , \\{ S \\} } ( \\frac { \\mathcal { H } _ { 2 , \\Psi } ^ { \\{ S \\} } } { 2 } \\langle J ( x , x ) | \\phi \\rangle + I ( x , x ) \\langle \\phi \\rangle } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) + I ( x , y ; x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) , } \\\\ & { \\quad \\stackrel { \\mathrm { O D } } { \\underset { \\mathrm { i } \\leq 1 } { \\sum } } I ( x , y ; x ) , } \\\\ & \\quad \\stackrel { \\mathrm { O D } } \\underset \\mathrm { i } \\leq \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 316, + 559, + 681, + 869 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Since $I ( \\mathbf { y } ; \\mathbf { z } ^ { \\prime } ) = 0$ by construction and $I ( \\mathbf { y } ; \\mathbf { z } ) > 0$ , the y built in 1. satisfies the conditions reported in the thesis. □ ", + "bbox": [ + 173, + 895, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Corollary B.1.1. Let $\\mathbf { z } ^ { \\prime }$ be a representation of x that discards observational information. There is always a label y for which a $\\mathbf { z } ^ { \\prime }$ is not predictive, while the original observations are. Hypothesis: ", + "bbox": [ + 173, + 102, + 825, + 146 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$( H _ { 1 }$ ) x is discrete $\\left( H _ { 2 } \\right) \\textbf { z } ^ { \\prime }$ discards information regarding x: $I ( \\mathbf { z } ^ { \\prime } ; \\mathbf { x } ) < H ( \\mathbf { x } )$ ", + "bbox": [ + 189, + 156, + 589, + 198 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Thesis: ", + "bbox": [ + 173, + 208, + 223, + 223 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/0d4186abcd6da63fb8bc1a70e5ea41b4ffa109e418cf0ce86902880f9e60b6bb.jpg", + "text": "$$\n( T _ { 1 } ) \\ \\exists \\mathbf { y } . I ( \\mathbf { y } ; \\mathbf { x } ) > I ( \\mathbf { y } ; \\mathbf { z } ^ { \\prime } ) = 0\n$$", + "text_format": "latex", + "bbox": [ + 192, + 233, + 411, + 251 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. By construction using Theorem B.1. ", + "bbox": [ + 174, + 263, + 460, + 280 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "1. Set $\\mathbf { z } = \\mathbf { x }$ : ", + "bbox": [ + 212, + 294, + 305, + 308 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/226b70f89704b9e2e775a4290be293088fef708ad3ca1363e4bc69363cbee6f2.jpg", + "text": "$$\n( C _ { 1 } ) \\ I ( \\mathbf { x } ; \\mathbf { z } ) \\ { \\stackrel { ( P _ { 5 } ) } { = } } \\ H ( \\mathbf { x } ) - H ( \\mathbf { x } | \\mathbf { z } ) \\ { \\stackrel { ( H _ { 1 } ) } { = } } \\ H ( \\mathbf { x } )\n$$", + "text_format": "latex", + "bbox": [ + 230, + 314, + 537, + 337 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Since the hypothesis are met, we conclude that there exist $\\mathbf { y }$ such that $I ( \\mathbf { y } ; \\mathbf { x } ) > I ( \\mathbf { y } ; \\mathbf { z } ^ { \\prime } ) = 0$ ", + "bbox": [ + 169, + 380, + 784, + 396 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.3 MULTI-VIEW ", + "text_level": 1, + "bbox": [ + 173, + 411, + 312, + 426 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "B.3.1 MULTI-VIEW REDUNDANCY AND SUFFICIENCY ", + "bbox": [ + 173, + 436, + 562, + 453 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proposition B.2. Let $\\mathbf { v } _ { 1 }$ , $\\mathbf { v } _ { 2 }$ , y be random variables with joint distribution $p ( \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { y } )$ . Let $\\mathbf { z } _ { 1 }$ be a representation of $\\mathbf { v } _ { 1 }$ , then: ", + "bbox": [ + 173, + 462, + 823, + 491 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/97b41ba43232d6e1ef237e340d82ec8865f51c28f863161bef060e60f09ff4f7.jpg", + "text": "$$\nI ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) \\leq I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 357, + 497, + 640, + 515 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Hypothesis: ", + "bbox": [ + 173, + 535, + 253, + 550 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "$( H _ { 1 } ) \\ \\mathbf { z } _ { 1 }$ is a representation of $\\mathbf { v } _ { 1 } \\colon I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } \\mathbf { v } _ { 1 } ) = 0$ ", + "bbox": [ + 192, + 560, + 549, + 578 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Thesis: ", + "bbox": [ + 173, + 588, + 223, + 602 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/8d8201334d771bdf2d950ac5a885ad5add9c0cf5bbae3f1b9c6e55e85b865880.jpg", + "text": "$$\n( T _ { 1 } ) \\ I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) \\leq I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 191, + 613, + 514, + 631 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proof. Since $\\mathbf { z } _ { 1 }$ is a representation of $\\mathbf { v } _ { 1 }$ ", + "bbox": [ + 173, + 643, + 446, + 660 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/03e8697c78e2fe5ea4da0e15fd6fd8dfe5c81bf15700dbc7ec9f1b824d683533.jpg", + "text": "$$\n( C _ { 1 } ) \\ I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } \\mathbf { v } _ { 1 } ) = 0\n$$", + "text_format": "latex", + "bbox": [ + 191, + 671, + 359, + 689 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Therefore: ", + "bbox": [ + 173, + 702, + 245, + 717 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/c51ab2c3c5606ec2e146bce43674bea4cc0a33b747b61e16e012c2c5f06b0f0f.jpg", + "text": "$$\n\\begin{array} { r l } { I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) } & { \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } \\mathbf { v } _ { 2 } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) } \\\\ & { \\overset { ( E ) } { = } I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) - I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) } \\\\ & { \\overset { ( E ) } { = } I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) - I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } ) + I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } \\mathbf { y } _ { 1 } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) } \\\\ & { \\overset { ( P _ { 1 } ) } { \\leq } I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) + I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } ) + I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) } \\\\ & { \\overset { ( P _ { 1 } ) } { \\leq } I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) + I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } \\mathbf { y } _ { 1 } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) } \\\\ & { \\overset { ( R ) } { = } I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) } \\\\ & \\overset { ( E ) } { \\leq } I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 720, + 758, + 905 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Proposition B.3. Let $\\mathbf { v } _ { 1 }$ be a redundant view with respect to $\\mathbf { v } _ { 2 }$ for y. Any representation $\\mathbf { z } _ { 1 }$ of $\\mathbf { v } _ { 1 }$ that is sufficient for $\\mathbf { v } _ { 2 }$ is also sufficient for y. ", + "bbox": [ + 171, + 103, + 823, + 133 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Hypothesis: ", + "bbox": [ + 174, + 145, + 253, + 160 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "$' H _ { 1 }$ ) $\\mathbf { z } _ { 1 }$ is a representation of $\\mathbf { v } _ { 1 }$ $\\mathbf { \\sigma } _ { 1 } \\mathbf { : } I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } \\mathbf { v } _ { 1 } ) = 0$ $\\left( H _ { 2 } \\right) \\ \\mathbf { v } _ { 1 }$ is redundant with respect to $\\mathbf { v } _ { 2 }$ for y: $I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } | \\mathbf { v } _ { 2 } ) = 0$ ", + "bbox": [ + 191, + 170, + 616, + 212 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Thesis: ", + "bbox": [ + 173, + 222, + 223, + 237 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/c8c8f54b6fd90ff4518316f1050be531815ae23d1dfc56f2b5b6b77df272b818.jpg", + "text": "$$\n( T _ { 1 } ) \\ I ( { \\bf v } _ { 1 } ; { \\bf v } _ { 2 } | { \\bf z } _ { 1 } ) = 0 \\ \\Longrightarrow \\ I ( { \\bf v } _ { 1 } ; { \\bf y } | { \\bf z } _ { 1 } ) = 0\n$$", + "text_format": "latex", + "bbox": [ + 192, + 247, + 500, + 265 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. Using the results from Theorem B.2: ", + "bbox": [ + 173, + 280, + 465, + 295 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/9684e2080608b8003b7ae1a43bb5304fa149901b752ba1001f05df0047b684ba.jpg", + "text": "$$\nI ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) \\overset { ( T h _ { B . 2 } ) } { \\leq } I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) \\overset { ( H _ { 2 } ) } { = } I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 279, + 303, + 717, + 330 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Therefore $I ( { \\bf v } _ { 1 } ; { \\bf v } _ { 2 } | { \\bf z } _ { 1 } ) = 0 \\implies I ( { \\bf v } _ { 1 } ; { \\bf y } | { \\bf z } _ { 1 } ) = 0$ ", + "bbox": [ + 176, + 335, + 501, + 352 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Theorem B.2. Let $\\mathbf { v } _ { 1 }$ , $\\mathbf { v } _ { 2 }$ and y be random variables with distribution $p ( \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { y } )$ . Let z be $a$ representation of $\\mathbf { v } _ { 1 }$ , then ", + "bbox": [ + 174, + 359, + 825, + 388 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/227ea99f4b73d8f8cc95df89a43e737e7fb1fc8b68b834e128cb5d727ec5a8d2.jpg", + "text": "$$\nI ( \\mathbf { y } ; \\mathbf { z } _ { 1 } ) \\geq I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ) - I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) - I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) - I ( \\mathbf { v } _ { 2 } ; \\mathbf { y } | \\mathbf { v } _ { 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 272, + 395, + 723, + 412 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Hypothesis: ", + "bbox": [ + 173, + 434, + 253, + 449 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "$( H _ { 1 } ) \\ \\mathbf { z } _ { 1 }$ is a representation of $\\mathbf { v } _ { 1 }$ $\\mathbf { \\sigma } _ { 1 } \\mathbf { : } I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ) = 0$ ", + "bbox": [ + 192, + 459, + 549, + 477 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Thesis: ", + "bbox": [ + 174, + 487, + 223, + 501 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/dc4b973e8ccc77cb7e12ec7f6e9ca1d9198e8a7510506b395d2bcfafc113eff4.jpg", + "text": "$$\n( T _ { 1 } ) \\ I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } ) \\geq I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ) - I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) - I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) - I ( \\mathbf { v } _ { 2 } ; \\mathbf { y } | \\mathbf { v } _ { 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 192, + 512, + 681, + 530 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Proof. ", + "bbox": [ + 173, + 545, + 217, + 559 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/a6a3828fd4a253fd4d149e7d3dfe7a360116b2877c9a167170be3c1088559571.jpg", + "text": "$$\n \\begin{array} { r l } { I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } ) } & { ( { \\frac { e ^ { \\mathbf { y } } } { 2 } } ) \\ I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } \\mathbf { y } _ { 2 } ) + I ( \\mathbf { y } ; \\mathbf { y } _ { 1 } \\mathbf { v } _ { 2 } ; \\mathbf { z } _ { 1 } ) } \\\\ & { ( { \\frac { e ^ { \\mathbf { y } } } { 2 } } ) \\ I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { y } _ { 2 } ; \\mathbf { z } _ { 1 } ) } \\\\ & { ( { \\frac { e ^ { \\mathbf { y } } } { 2 } } ) \\ I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ; \\mathbf { z } _ { 1 } ) } \\\\ & { ( \\mathbf { z } ) \\ I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ) - I ( \\mathbf { y } ; \\mathbf { y } _ { 1 } \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) } \\\\ & { ( { \\frac { e ^ { \\mathbf { y } } } { 2 } } ) \\ I ( \\mathbf { y } ; \\mathbf { x } _ { 1 } \\mathbf { v } _ { 2 } ) - I ( \\mathbf { y } ; \\mathbf { y } _ { 1 } \\mathbf { z } _ { 1 } ) - I ( \\mathbf { y } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } \\mathbf { v } _ { 1 } ) } \\\\ & { ( { \\frac { e ^ { \\mathbf { y } } } { 2 } } ) \\ I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ) - I ( \\mathbf { y } ; \\mathbf { x } _ { 1 } | \\mathbf { z } _ { 1 } ) - I ( \\mathbf { y } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) + I ( \\mathbf { y } ; \\mathbf { y } _ { 2 } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) } \\\\ & { ( { \\frac { e ^ { \\mathbf { y } } } { 2 } } ) \\ I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ) - I ( \\mathbf { y } ; \\mathbf { y } _ { 1 } | \\mathbf { z } _ { 1 } ) - I ( \\mathbf { y } ; \\mathbf { v } _ { 2 } | \\mathbf { v } _ { 1 } ) + I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) } \\\\ & ( { \\frac { e ^ { \\mathbf { y } } } { 2 } } ) \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 566, + 785, + 801 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Corollary B.2.1. Let $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ be mutually redundant views for y. Let $\\mathbf { z } _ { 1 }$ be a representation of $\\mathbf { v } _ { 1 }$ that is sufficient for $\\mathbf { v } _ { 2 }$ . Then: ", + "bbox": [ + 173, + 847, + 828, + 876 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/3b43868be1775401e34abd99c070d40108bac0d0c69317da230eb5683b10ab15.jpg", + "text": "$$\nI ( \\mathbf { y } ; \\mathbf { z } _ { 1 } ) = I ( \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ; \\mathbf { y } )\n$$", + "text_format": "latex", + "bbox": [ + 423, + 876, + 575, + 892 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Hypothesis: ", + "bbox": [ + 173, + 909, + 253, + 924 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "$( H _ { 1 }$ ) $\\mathbf { z } _ { 1 }$ is a representation of $\\mathbf { v } _ { 1 } \\colon I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ) = 0$ \n$( H _ { 2 }$ ) $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ are mutually redundant for y: $I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } | \\mathbf { v } _ { 2 } ) + I ( \\mathbf { y } ; \\mathbf { v } _ { 2 } | \\mathbf { v } _ { 1 } ) = 0$ \n$( H _ { 3 } ) \\ \\mathbf { z } _ { 1 }$ is sufficient for $\\mathbf { v } _ { 2 }$ : $I ( \\mathbf { v } _ { 2 } ; \\mathbf { v } _ { 1 } | \\mathbf { z } ) = 0$ ", + "bbox": [ + 191, + 102, + 714, + 167 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Thesis: ", + "bbox": [ + 174, + 179, + 223, + 193 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/27ba1f97595b8aaaa7855f660691190da4359d1b7c890609c261c32c018430ee.jpg", + "text": "$$\n( T _ { 1 } ) \\ I ( { \\bf y } ; { \\bf z } _ { 1 } ) = I ( { \\bf v } _ { 1 } { \\bf v } _ { 2 } ; { \\bf y } )\n$$", + "text_format": "latex", + "bbox": [ + 192, + 204, + 385, + 222 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. Using Theorem B.2 ", + "bbox": [ + 173, + 236, + 354, + 252 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/6ad347d0535f7e3cd25f9507dbaf2c6d245dbfb2501feb56a7568d596c9c7802.jpg", + "text": "$$\n\\begin{array} { r l } & { I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } ) \\overset { ( T h _ { B , 2 } ) } { \\geq } I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ) - I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) - I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) - I ( \\mathbf { y } ; \\mathbf { v } _ { 2 } | \\mathbf { v } _ { 1 } ) } \\\\ & { \\overset { ( H _ { 2 } ) } { = } I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ) - I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) } \\\\ & { \\overset { ( H _ { 3 } ) } { = } I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 258, + 740, + 338 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Since $I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } ) \\leq I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } )$ is a consequence of the data processing inequality, we conclude that $I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } ) = I ( \\mathbf { y } ; \\mathbf { v } _ { 1 } \\mathbf { v } _ { 2 } )$ □ ", + "bbox": [ + 174, + 359, + 825, + 391 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B.4 SUFFICIENCY AND AUGMENTATION ", + "text_level": 1, + "bbox": [ + 176, + 406, + 464, + 421 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Let $\\mathbf { x }$ and $\\mathbf { y }$ be random variables with domain $\\mathbb { X }$ and $\\mathbb { Y }$ respectively. Let $\\mathbb { T }$ be a class of functions $t : \\mathbb { X } \\to \\mathbb { W }$ and let $\\mathbf { t } _ { 1 }$ and $\\mathbf { t } _ { 2 }$ be a random variables over $\\mathbb { T }$ that depends only on x. For the theorems and corollaries discussed in this section, we are going to consider the independence assumption that can be derived from the graphical model $\\mathcal { G }$ reported in Figure 5. ", + "bbox": [ + 173, + 433, + 826, + 489 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/909aaf259ac4731b46cf28b94c79fc401805146a5d9c4d1f118b0bde376f35cf.jpg", + "image_caption": [ + "Figure 5: Visualization of the graphical model $\\mathcal { G }$ that relates the observations $\\mathbf { x }$ , label $\\mathbf { y }$ , functions used for augmentation $\\mathbf { t } _ { 1 }$ , $\\mathbf { t } _ { 2 }$ and the representation $\\mathbf { z } _ { 1 }$ . " + ], + "image_footnote": [], + "bbox": [ + 388, + 501, + 604, + 664 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proposition B.4. Whenever $I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { x } ; \\mathbf { y } )$ the two views $\\mathbf { t } _ { 1 } ( \\mathbf { x } )$ and $\\mathbf { t } _ { 2 } ( \\mathbf { x } )$ must be mutually redundant for y. ", + "bbox": [ + 173, + 724, + 821, + 755 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Hypothesis: ", + "bbox": [ + 173, + 767, + 253, + 782 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "$\\left( H _ { 1 } \\right)$ Independence relations determined by $\\mathcal { G }$ ", + "bbox": [ + 191, + 794, + 498, + 809 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Thesis: ", + "bbox": [ + 173, + 820, + 222, + 834 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/c681514c542df68f03f76c894eb2ff65415b5a9070efca9e55a63e4a926f7abd.jpg", + "text": "$$\n( T _ { 1 } ) \\ I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { x } ; \\mathbf { y } ) \\implies I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } | \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ) + I ( \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ; \\mathbf { y } | \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ) = 0\n$$", + "text_format": "latex", + "bbox": [ + 192, + 845, + 807, + 863 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Proof. ", + "bbox": [ + 173, + 878, + 217, + 893 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "1. Considering $\\mathcal { G }$ we have: ", + "bbox": [ + 214, + 909, + 392, + 924 + ], + "page_idx": 16 + }, + { + "type": "equation", + "img_path": "images/729b2f127bb52f0d9ba0f1623b92799c94e1fcbe684f7b2e11e7d8d112105074.jpg", + "text": "$$\n\\begin{array} { l } { { ( C _ { 1 } ) \\ I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } | \\mathbf { x } \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ) = 0 } } \\\\ { { ( C _ { 2 } ) \\ I ( \\mathbf { y } ; \\mathbf { t } _ { 2 } ( \\mathbf { x } ) | \\mathbf { x } ) = 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 102, + 434, + 138 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "2. Since ${ \\bf t } _ { 2 } ( { \\bf x } )$ is uniquely determined by $\\mathbf { x }$ and $\\mathbf { t } _ { 2 }$ : ", + "bbox": [ + 210, + 145, + 550, + 161 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/a1db9d46d389245495d8073c928494b8b564c81a3ca767e643de8cc72806f99d.jpg", + "text": "$$\n( C _ { 3 } ) I ( \\bf t _ { 2 } ( \\bf x ) ; \\bf y | \\bf x t _ { 2 } ) = 0\n$$", + "text_format": "latex", + "bbox": [ + 232, + 167, + 411, + 186 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "3. Consider $I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } | \\mathbf { t } _ { 2 } ( \\mathbf { x } ) )$ ", + "bbox": [ + 212, + 194, + 415, + 210 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/56ef0934a4ff2a7f8d838026d975ba2e74ed4a2d1fb100bd6fb53964398f432c.jpg", + "text": "$$\n\\begin{array} { r l } { T [ \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } ] \\mathbf { t } _ { 2 } ( \\mathbf { x } ) } & { \\stackrel { \\mathrm { f o r ~ } } { \\Longleftrightarrow } } & { T [ \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } ] \\mathbf { t } _ { 2 } ( \\mathbf { x } ) } \\\\ & { \\stackrel { \\mathrm { f o r ~ } } { \\Longleftrightarrow } } & { \\frac { \\omega } { 2 } \\delta ( t _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } , \\mathbf { x } ; \\mathbf { z } ) [ \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ] } \\\\ & { \\stackrel { \\mathrm { f o r ~ } } { \\Longleftrightarrow } } & { \\frac { \\omega } { 2 } \\delta ( t _ { 2 } ( \\mathbf { x } ) ; \\mathbf { z } , \\mathbf { x } | \\mathbf { z } ) } \\\\ & { \\stackrel { \\mathrm { f o r ~ } } { \\Longleftrightarrow } } & { \\frac { \\omega } { 2 } \\delta ( t _ { 3 } ; \\mathbf { x } | \\mathbf { t } _ { 4 } ( \\mathbf { x } ) ) - f [ \\mathbf { y } ; \\mathbf { x } | \\mathbf { t } _ { 4 } ( \\mathbf { x } ) ; \\mathbf { t } _ { 5 } ( \\mathbf { x } ) ) } \\\\ & { \\stackrel { \\mathrm { f o r ~ } } { \\Longleftrightarrow } } & { \\frac { \\omega } { 2 } \\delta ( t _ { 3 } ; \\mathbf { x } | \\mathbf { t } _ { 5 } ( \\mathbf { x } ) ) } \\\\ & { \\stackrel { \\mathrm { f o r ~ } } { \\Longleftrightarrow } } & { \\frac { \\omega } { 1 } \\delta ( t _ { 5 } ; \\mathbf { x } ) - T [ \\mathbf { y } ; \\mathbf { x } , \\mathbf { t } _ { 6 } ( \\mathbf { x } ) ) } \\\\ & { \\stackrel { \\mathrm { f o r ~ } } { \\Longleftrightarrow } } & { \\frac { \\omega } { 1 } \\delta ( t _ { 5 } ; \\mathbf { x } ) - T [ \\mathbf { y } ; \\mathbf { t } _ { 6 } ( \\mathbf { x } ) ; \\mathbf { t } _ { 7 } ( \\mathbf { x } ) ] } \\\\ & { \\stackrel { \\mathrm { f o r ~ } } { \\Longleftrightarrow } } & { \\frac { \\omega } { 2 } \\delta ( t _ { 5 } ; \\mathbf { x } ) - f [ \\mathbf { y } ; \\mathbf { t } _ { 6 } ( \\mathbf { x } ) ] + T [ \\mathbf { y } ; \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ] \\mathbf { t } _ { 2 } \\delta ( t _ { 5 } ; \\mathbf { t } _ { 6 } ( \\mathbf { x } ) ; \\mathbf { t } _ { 7 } ( \\mathbf { x } ) ] } \\\\ & \\stackrel \\ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 217, + 805, + 506 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Therefore $I ( \\mathbf { y } ; \\mathbf { x } ) = I ( \\mathbf { y } ; \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ) \\implies I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } | \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ) = 0$ ", + "bbox": [ + 232, + 527, + 643, + 546 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "The proof for ${ \\cal I } ( { \\bf y } ; { \\bf x } ) = { \\cal I } ( { \\bf y } ; { \\bf t } _ { 1 } ( { \\bf x } ) ) \\Longrightarrow { \\cal I } ( { \\bf t } _ { 2 } ( { \\bf x } ) ; { \\bf y } | { \\bf t } _ { 1 } ( { \\bf x } ) ) = 0$ is symmetric, therefore we conclude I ${ \\bf \\bar { \\Psi } ( t _ { 1 } ( x ) ; y ) } = I ( { \\bf t } _ { 2 } ( { \\bf x } ) ; { \\bf y } ) = I ( { \\bf x } ; { \\bf y } ) \\Longrightarrow I ( { \\bf t } _ { 1 } ( { \\bf x } ) ; { \\bf y } | { \\bf t } _ { 2 } ( { \\bf x } ) ) + I ( { \\bf t } _ { 2 } ( { \\bf x } ) ; { \\bf y } | { \\bf t } _ { 1 } ( { \\bf x } ) ) = I ( { \\bf t } _ { 1 } ( { \\bf x } ) ; { \\bf y } ) ,$ 0 ", + "bbox": [ + 173, + 560, + 826, + 604 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Theorem B.3. Let $I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { x } ; \\mathbf { y } )$ . Let $\\mathbf { z } _ { 1 }$ be a representation of $\\mathbf { t } _ { 1 } ( \\mathbf { x } )$ . I f $\\mathbf { z } _ { 1 }$ is sufficient for ${ \\bf t } _ { 2 } ( { \\bf x } )$ then $I ( \\mathbf { x } ; \\mathbf { y } ) = I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } )$ . ", + "bbox": [ + 171, + 616, + 826, + 647 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Hypothesis: ", + "bbox": [ + 173, + 659, + 253, + 674 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "$\\left( H _ { 1 } \\right)$ Independence relations determined by $\\mathcal { G }$ ", + "bbox": [ + 192, + 685, + 500, + 700 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/496a7fb0073d42ff48f075ed75200d9b8840e0f16727d4ddf591401882b7c99f.jpg", + "text": "$$\n( H _ { 2 } ) \\ I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { x } ; \\mathbf { y } )\n$$", + "text_format": "latex", + "bbox": [ + 189, + 709, + 478, + 727 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Thesis: ", + "bbox": [ + 174, + 737, + 223, + 751 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/d25e845bdad7db36abf406c26684a05415e5bc27b71d32929b6b629be07ef8cc.jpg", + "text": "$$\n( T _ { 1 } ) \\ I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { t } _ { 2 } ( \\mathbf { x } ) | \\mathbf { z } _ { 1 } ) = 0 \\ \\Longrightarrow \\ I ( \\mathbf { x } ; \\mathbf { y } ) = I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 192, + 762, + 558, + 781 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Proof. Since $\\mathbf { t } _ { 1 } ( \\mathbf { x } )$ is redundant for ${ \\bf t } _ { 2 } ( { \\bf x } )$ (Proposition B.4) any representation $\\mathbf { z } _ { 1 }$ of $\\mathbf { t } _ { 1 } ( \\mathbf { x } )$ that is sufficient for ${ \\bf t } _ { 2 } ( { \\bf x } )$ must also be sufficient for $\\mathbf { y }$ (Theorem B.2). Using Proposition B.1 we have ${ \\cal I } ( { \\bf y } ; { \\bf z } _ { 1 } ) = { \\cal I } ( { \\bf y } ; { \\bf t } _ { 1 } ( { \\bf x } ) )$ . Since $I ( \\mathbf { y } ; \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ) ~ = ~ I ( \\mathbf { y } ; \\mathbf { x } )$ by hypothesis, we conclude $I ( \\mathbf { x } ; \\mathbf { y } ) \\ =$ $I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } )$ ", + "bbox": [ + 173, + 797, + 826, + 857 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C INFORMATION PLANE ", + "text_level": 1, + "bbox": [ + 174, + 876, + 392, + 893 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Every representation $\\mathbf { z }$ of $\\mathbf { x }$ must satisfy the following constraints: ", + "bbox": [ + 173, + 909, + 607, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "• $0 \\leq I ( \\mathbf { y } ; \\mathbf { z } ) \\leq I ( \\mathbf { x } ; \\mathbf { y } )$ : The amount of label information ranges from 0 to the total predictive information accessible from the raw observations $I ( \\mathbf { x } ; \\mathbf { y } )$ . ", + "bbox": [ + 212, + 103, + 823, + 132 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• $I ( \\mathbf { y } ; \\mathbf { z } ) \\leq I ( \\mathbf { x } ; \\mathbf { z } ) \\leq I ( \\mathbf { y } ; \\mathbf { z } ) + H ( \\mathbf { x } | \\mathbf { y } )$ : The representation must contain more information about the observations than about the label. When $\\mathbf { x }$ is discrete, the amount of discarded label information $I ( \\mathbf { x } ; \\mathbf { y } ) - I ( \\mathbf { y } ; \\mathbf { z } )$ must be smaller than the amount of discarded observational information $H ( \\mathbf { x } ) - I ( \\mathbf { x } ; \\mathbf { z } )$ , which implies $I ( \\mathbf { x } ; \\mathbf { z } ) \\leq I ( \\mathbf { y } ; \\mathbf { z } ) + H ( \\mathbf { x } | \\mathbf { y } )$ . ", + "bbox": [ + 217, + 136, + 823, + 193 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Proof. Since $\\mathbf { z }$ is a representation of $\\mathbf { x }$ ", + "bbox": [ + 174, + 207, + 431, + 223 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/e1b50b53bccffcf6f86453b2b13be247d7e3521ddb08e7ca2440206162e340ca.jpg", + "text": "$$\n( C _ { 1 } ) \\ I ( { \\bf y } ; { \\bf z } | { \\bf x } ) = 0\n$$", + "text_format": "latex", + "bbox": [ + 191, + 234, + 328, + 252 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Considering the four bounds separately: ", + "bbox": [ + 173, + 265, + 436, + 280 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "1. $I ( \\mathbf { y } ; \\mathbf { z } ) \\geq 0$ : Follows from $P _ { 1 }$ ", + "bbox": [ + 214, + 294, + 429, + 310 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "2. $I ( \\mathbf { x ; z } ) \\geq I ( \\mathbf { y ; z } )$ : Follows from: ", + "bbox": [ + 212, + 318, + 452, + 333 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/8a74a40a8af0f7bca47aee8941fb84b3958bba406997fb1bc36de731889595e2.jpg", + "text": "$$\n\\begin{array} { l } { I ( \\mathbf { x } ; \\mathbf { z } ) \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } ) + I ( \\mathbf { x } ; \\mathbf { y } ; \\mathbf { z } ) } \\\\ { \\begin{array} { l } { ( P _ { 1 } ) } \\\\ { \\geq } \\\\ { \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { y } ; \\mathbf { z } ) - I ( \\mathbf { y } ; \\mathbf { z } | \\mathbf { x } ) } \\end{array} } \\\\ { \\begin{array} { l } { ( \\underline { { C _ { 1 } } } ) } \\\\ { = } \\end{array} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 411, + 339, + 643, + 443 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "3. $I ( \\mathbf { y } ; \\mathbf { z } ) \\leq I ( \\mathbf { y } ; \\mathbf { x } )$ : Data processing inequality ", + "bbox": [ + 212, + 454, + 534, + 469 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/32c30199c5c89002da312b32f7c5304dc76186d9cbc73efe436ef3ea99291fa5.jpg", + "text": "$$\n\\begin{array} { r l } & { I ( \\mathbf { y } ; \\mathbf { z } ) \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { y } ; \\mathbf { z } | \\mathbf { x } ) + I ( \\mathbf { y } ; \\mathbf { z } ; \\mathbf { x } ) } \\\\ & { \\overset { ( C _ { 1 } ) } { = } I ( \\mathbf { y } ; \\mathbf { z } ; \\mathbf { x } ) } \\\\ & { \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { x } ; \\mathbf { y } ) - I ( \\mathbf { x } ; \\mathbf { y } | \\mathbf { z } ) } \\\\ & { \\overset { ( P _ { 1 } ) } { \\leq } I ( \\mathbf { x } ; \\mathbf { y } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 411, + 474, + 642, + 579 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "4. $I ( \\mathbf { x } ; \\mathbf { z } ) \\leq I ( \\mathbf { y } ; \\mathbf { z } ) + H ( \\mathbf { x } | \\mathbf { y } )$ : For discrete $\\mathbf { x }$ : ", + "bbox": [ + 210, + 588, + 527, + 606 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/fc41af205fea0bb14c942a51d21a2986b54bb110ee55fca5576323ea8927a35c.jpg", + "text": "$$\n\\begin{array} { r l } { I ( \\mathbf { x } ; \\mathbf { z } ) \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } ) + I ( \\mathbf { x } ; \\mathbf { y } ; \\mathbf { z } ) } & { } \\\\ & { \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } ) + I ( \\mathbf { y } ; \\mathbf { z } ) - I ( \\mathbf { y } ; \\mathbf { z } | \\mathbf { x } ) } \\\\ & { \\overset { ( C _ { 1 } ) } { = } I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } ) + I ( \\mathbf { y } ; \\mathbf { z } ) } \\\\ & { \\overset { ( P _ { 4 } ) } { \\leq } I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } ) + H ( \\mathbf { x } | \\mathbf { y } \\mathbf { z } ) + I ( \\mathbf { y } ; \\mathbf { z } ) } \\\\ & { \\overset { ( P _ { 5 } ) } { = } I ( \\mathbf { x } | \\mathbf { y } ) + I ( \\mathbf { y } ; \\mathbf { z } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 380, + 609, + 676, + 739 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Note that the discreetness of $\\mathbf { x }$ is required only to prove bound 4. For continuous $\\mathbf { x }$ bounds $1 , 2$ and 3 still hold. ", + "bbox": [ + 173, + 780, + 825, + 809 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "D NON-TRANSITIVITY OF MUTUAL REDUNDANCY ", + "text_level": 1, + "bbox": [ + 174, + 828, + 611, + 845 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The mutual redundancy condition between two views $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ for a label y can not be trivially extended to an arbitrary number of views, as the relation is not transitive because of some higher order interaction between the different views and the label. This can be shown with a simple example. ", + "bbox": [ + 174, + 859, + 825, + 902 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Given three views $\\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 }$ and $\\mathbf { v } _ { 3 }$ and a task y such that: ", + "bbox": [ + 174, + 909, + 540, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ are mutually redundant for y • $\\mathbf { v } _ { 2 }$ and $\\mathbf { v } _ { 3 }$ are mutually redundant for y ", + "bbox": [ + 215, + 103, + 495, + 138 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Then, we show that $\\mathbf { v } _ { 1 }$ is not necessarily mutually redundant with respect to $\\mathbf { v } _ { 3 }$ for y. ", + "bbox": [ + 173, + 148, + 735, + 165 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Let $\\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 }$ and $\\mathbf { v } _ { 3 }$ be fair and independent binary random variables. Defining y as the exclusive or operator applied to $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 3 }$ ( $\\mathbf { y } : = \\mathbf { v } _ { 1 } \\mathrm { X O R } \\mathbf { v } _ { 3 } )$ ), we have that $I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } ) = I ( \\mathbf { v } _ { 3 } ; \\mathbf { y } ) = 0$ . In this settings, $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ are mutually redundant for y: ", + "bbox": [ + 173, + 170, + 825, + 213 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/b0ac5cb8ee03110b277ec47a98a34515439f6888bb3199ee36248bc33519a67a.jpg", + "text": "$$\n\\begin{array} { r } { I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) = H ( \\mathbf { v } _ { 1 } | \\mathbf { v } _ { 2 } ) - H ( \\mathbf { v } _ { 1 } | \\mathbf { v } _ { 2 } \\mathbf { y } ) = H ( \\mathbf { v } _ { 1 } ) - H ( \\mathbf { v } _ { 1 } ) = 0 } \\\\ { I ( \\mathbf { v } _ { 2 } ; \\mathbf { y } | \\mathbf { v } _ { 1 } ) = H ( \\mathbf { v } _ { 2 } | \\mathbf { v } _ { 1 } ) - H ( \\mathbf { v } _ { 2 } | \\mathbf { v } _ { 1 } \\mathbf { y } ) = H ( \\mathbf { v } _ { 2 } ) - H ( \\mathbf { v } _ { 2 } ) = 0 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 219, + 715, + 257 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Analogously, $\\mathbf { v } _ { 2 }$ and $\\mathbf { v } _ { 3 }$ are also mutually redundant for $\\mathbf { y }$ as the three random variables are not predictive for each other. Nevertheless, $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 3 }$ are not mutually redundant for $\\mathbf { y }$ : ", + "bbox": [ + 174, + 262, + 821, + 291 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/ff442a2a82df6202e2e3634a26f64ee994844bd2b1668c71806aefc0001255e4.jpg", + "text": "$$\n\\begin{array} { r l } & { I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 3 } ) = H ( \\mathbf { v } _ { 1 } | \\mathbf { v } _ { 3 } ) - \\underbrace { H ( \\mathbf { v } _ { 1 } | \\mathbf { v } _ { 3 } \\mathbf { y } ) } _ { 0 } = H ( \\mathbf { v } _ { 1 } ) = 1 } \\\\ & { I ( \\mathbf { v } _ { 3 } ; \\mathbf { y } | \\mathbf { v } _ { 1 } ) = H ( \\mathbf { v } _ { 3 } | \\mathbf { v } _ { 1 } ) - \\underbrace { H ( \\mathbf { v } _ { 3 } | \\mathbf { v } _ { 1 } \\mathbf { y } ) } _ { 0 } = H ( \\mathbf { v } _ { 3 } ) = 1 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 315, + 296, + 681, + 369 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Where $H ( \\mathbf { v } _ { 1 } | \\mathbf { v } _ { 3 } \\mathbf { y } ) = H ( \\mathbf { v } _ { 3 } | \\mathbf { v } _ { 1 } \\mathbf { y } ) = 0$ follows from $\\mathbf { v } _ { 1 } = \\mathbf { v } _ { 3 }$ XOR y and $\\mathbf { v } _ { 3 } = \\mathbf { v } _ { 1 }$ XOR y, while $H ( \\mathbf { v } _ { 1 } ) = H ( \\mathbf { v } _ { 3 } ) = 1$ holds by construction. ", + "bbox": [ + 174, + 376, + 823, + 405 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "This counter-intuitive higher order interaction between multiple views makes our theory non-trivial to generalize to more than two views. ", + "bbox": [ + 173, + 411, + 823, + 440 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E EQUIVALENCES OF DIFFERENT OBJECTIVES ", + "text_level": 1, + "bbox": [ + 173, + 460, + 570, + 477 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Different objectives in literature can be seen as a special case of the Multi-View Information Bottleneck principle. In this section we show that the supervised version of Information Bottleneck is equivalent to the corresponding Multi-View version whenever the two redundant views have only label information in common. A second subsection show equivalence between InfoMax and MultiView Information Bottleneck whenever the two views are identical. ", + "bbox": [ + 173, + 492, + 825, + 563 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E.1 MULTI-VIEW INFORMATION BOTTLENECK AND SUPERVISED INFORMATION BOTTLENECK ", + "text_level": 1, + "bbox": [ + 176, + 579, + 740, + 608 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Whenever the two mutually redundant views $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ have only label information in common (or when one of the two views is the label itself) the Multi-View Information Bottleneck objective is equivalent to the respective supervised version. This can be shown by proving that $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } ) =$ $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { y } )$ , i.e. a representation $\\mathbf { z } _ { 1 }$ of $\\mathbf { v } _ { 1 }$ that is sufficient and minimal for $\\mathbf { v } _ { 2 }$ is also sufficient and minimal for y. ", + "bbox": [ + 173, + 619, + 825, + 690 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proposition E.1. Let $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ be mutually redundant views for a label y that share only label information. Then a sufficient representation $\\mathbf { z } _ { 1 }$ of $\\mathbf { v } _ { 1 } f o r \\mathbf { v } _ { 2 }$ that is minimal for $\\mathbf { v } _ { 2 }$ is also a minimal representation for y. ", + "bbox": [ + 174, + 694, + 826, + 737 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Hypothesis: ", + "text_level": 1, + "bbox": [ + 173, + 750, + 253, + 763 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "$( H _ { 1 }$ ) $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ are mutually redundant for y: $I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { v } _ { 2 } ) + I ( \\mathbf { v } _ { 2 } ; \\mathbf { y } | \\mathbf { v } _ { 1 } ) = 0$ ", + "bbox": [ + 192, + 775, + 712, + 792 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "$\\left( H _ { 2 } \\right) \\ \\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ share only label information: $I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } ) = I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } )$ ", + "bbox": [ + 191, + 799, + 638, + 816 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "$( H _ { 3 } ) \\ \\mathbf { z } _ { 1 }$ is sufficient for $\\mathbf { v } _ { 2 }$ : $I ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) = 0$ ", + "bbox": [ + 194, + 824, + 496, + 840 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Thesis: ", + "bbox": [ + 173, + 851, + 223, + 864 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/1078c390f226bcd149285b576463f7f99403c0fdfc9da2485ad6230d9281dd15.jpg", + "text": "$$\n( T _ { 1 } ) \\ I ( { \\bf v } _ { 1 } ; { \\bf z } _ { 1 } | { \\bf v } _ { 2 } ) = I ( { \\bf v } _ { 1 } ; { \\bf z } _ { 1 } | { \\bf y } )\n$$", + "text_format": "latex", + "bbox": [ + 192, + 876, + 416, + 893 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Proof. ", + "bbox": [ + 174, + 910, + 217, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "1. Consider $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } )$ : ", + "bbox": [ + 212, + 103, + 356, + 118 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/b25b079fda8e5726bf4cffcf50fa2d3b02a25d4f32c94844824ddea070fffd72.jpg", + "text": "$$\n\\begin{array} { l } { { { I } ( { \\bf { v } } _ { 1 } ; { \\bf { z } } _ { 1 } ) \\stackrel { { ( P _ { 3 } ) } } { = } I ( { \\bf { v } } _ { 1 } ; { \\bf { z } } _ { 1 } | { \\bf { v } } _ { 2 } ) + I ( { \\bf { v } } _ { 1 } ; { \\bf { v } } _ { 2 } ; { \\bf { z } } _ { 1 } ) } \\ ~ } \\\\ { { \\displaystyle ~ \\stackrel { { ( P _ { 3 } ) } } { = } I ( { \\bf { v } } _ { 1 } ; { \\bf { z } } _ { 1 } | { \\bf { v } } _ { 2 } ) + I ( { \\bf { v } } _ { 1 } ; { \\bf { v } } _ { 2 } ) - I ( { \\bf { v } } _ { 1 } ; { \\bf { v } } _ { 2 } | { \\bf { z } } _ { 1 } ) } \\ ~ } \\\\ { { \\displaystyle ~ \\stackrel { { ( H _ { 3 } ) } } { = } I ( { \\bf { v } } _ { 1 } ; { \\bf { z } } _ { 1 } | { \\bf { v } } _ { 2 } ) + I ( { \\bf { v } } _ { 1 } ; { \\bf { v } } _ { 2 } ) } \\ ~ } \\\\ { { \\displaystyle ~ \\stackrel { { ( H _ { 1 } ) } } { = } I ( { \\bf { v } } _ { 1 } ; { \\bf { z } } _ { 1 } | { \\bf { v } } _ { 2 } ) + I ( { \\bf { v } } _ { 1 } ; { \\bf { y } } ) } \\ ~ } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 121, + 710, + 222 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "2. Using Corollary 1, from $\\left( H _ { 2 } \\right)$ and $\\left( H _ { 3 } \\right)$ follows $I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) = 0$ ", + "bbox": [ + 210, + 246, + 658, + 262 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "3. $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } )$ can be alternatively expressed as: ", + "bbox": [ + 212, + 270, + 509, + 285 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/21cd859786e0697ae98c10c37c1250b1ec5136052fa4358a0050c903faa01e69.jpg", + "text": "$$\n\\begin{array} { r l } & { I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } ) \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { y } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } ; \\mathbf { z } _ { 1 } ) } \\\\ & { \\overset { ( P _ { 3 } ) } { = } I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { y } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } ) - I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } | \\mathbf { z } _ { 1 } ) } \\\\ & { \\overset { ( C o r _ { 1 } ) } { = } I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { y } ) + I ( \\mathbf { v } _ { 1 } ; \\mathbf { y } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 287, + 702, + 363 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Equating 1 and 3, we conclude $I ( { \\bf v } _ { 1 } ; { \\bf z } _ { 1 } | { \\bf v } _ { 2 } ) = I ( { \\bf v } _ { 1 } ; { \\bf z } _ { 1 } | { \\bf y } )$ , therefore $\\mathbf { z } _ { 1 }$ which minimizes $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } )$ is also minimizing $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { y } )$ . When $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { y } )$ is minimal, $I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } )$ is also minimal (see equation 2). □ ", + "bbox": [ + 174, + 387, + 823, + 430 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "E.2 MULTI-VIEW INFORMATION BOTTLENECK AND INFOMAX", + "text_level": 1, + "bbox": [ + 174, + 445, + 619, + 462 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Whenever $\\mathbf { v } _ { 1 } = \\mathbf { v } _ { 2 }$ , a representation $\\mathbf { z } _ { 1 }$ of $\\mathbf { v } _ { 1 }$ that is sufficient for $\\mathbf { v } _ { 2 }$ must contain all the original information regarding $\\mathbf { v } _ { 1 }$ . Furthermore since $I ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } ) = 0$ for every representation, no superfluous information can be identified and removed. As a consequence, a minimal sufficient representation $\\mathbf { z } _ { 1 }$ of $\\mathbf { v } _ { 1 }$ for $\\mathbf { v } _ { 2 }$ is any representation for which mutual information is maximal, hence InfoMax. ", + "bbox": [ + 173, + 472, + 825, + 542 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "F LOSS COMPUTATION ", + "text_level": 1, + "bbox": [ + 174, + 561, + 380, + 578 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Starting from Equation 3, we consider the average of the losses $\\mathcal { L } _ { 1 } ( \\theta ; \\lambda _ { 1 } )$ and $\\mathcal { L } _ { 2 } ( \\psi ; \\lambda _ { 2 } )$ that aim to create the minimal sufficient representations $\\mathbf { z } _ { 1 }$ and $\\mathbf { z } _ { 2 }$ respectively: ", + "bbox": [ + 171, + 590, + 823, + 621 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/c9ca92669770bc28d762d239d3a6d389cf60761e4937e0584458424c5a0413d3.jpg", + "text": "$$\n\\mathscr { L } _ { \\frac { 1 + 2 } { 2 } } ( \\theta , \\psi ; \\lambda _ { 1 } , \\lambda _ { 2 } ) = \\frac { I _ { \\theta } ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } ) + I _ { \\psi } ( \\mathbf { v } _ { 2 } ; \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 1 } ) } { 2 } + \\frac { \\lambda _ { 1 } I _ { \\theta } ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) + \\lambda _ { 2 } I _ { \\psi } ( \\mathbf { v } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 1 } ) } { 2 }\n$$", + "text_format": "latex", + "bbox": [ + 183, + 622, + 794, + 654 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Considering $\\mathbf { z } _ { 1 }$ and $\\mathbf { z } _ { 2 }$ on the same domain $\\mathbb { Z }$ , $I _ { \\theta } ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } )$ can be expressed as: ", + "bbox": [ + 174, + 655, + 707, + 670 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/6c2d64759b7e2905d118de7083cb77a7c2f6b56484a36b991c7b40f94c9d0235.jpg", + "text": "$$\n\\begin{array} { r l } & { I _ { \\theta } ( \\mathbf { v } _ { 1 } ; \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } ) = \\mathbb { E } _ { v _ { 1 } , v _ { 2 } \\sim p ( \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } ) } \\mathbb { E } _ { z \\sim p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) } \\left[ \\log \\frac { p _ { \\theta } ( \\mathbf { z } _ { 1 } = z | \\mathbf { v } _ { 1 } = v _ { 1 } ) } { p _ { \\theta } ( \\mathbf { z } _ { 1 } = z | \\mathbf { v } _ { 2 } = v _ { 2 } ) } \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { v _ { 1 } , v _ { 2 } \\sim p ( \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } ) } \\mathbb { E } _ { z \\sim p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) } \\left[ \\log \\frac { p _ { \\theta } ( \\mathbf { z } _ { 1 } = z | \\mathbf { v } _ { 1 } = v _ { 1 } ) } { p _ { \\psi } ( \\mathbf { z } _ { 2 } = z | \\mathbf { v } _ { 2 } = v _ { 2 } ) } \\frac { p _ { \\psi } ( \\mathbf { z } _ { 2 } = z | \\mathbf { v } _ { 2 } = v _ { 2 } ) } { p _ { \\theta } ( \\mathbf { z } _ { 1 } = z | \\mathbf { v } _ { 2 } = v _ { 2 } ) } \\right] } \\\\ & { \\qquad = D _ { \\mathrm { K L } } ( p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) | | p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) ) - D _ { \\mathrm { K L } } ( p _ { \\theta } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 1 } ) | | p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) ) } \\\\ & { \\qquad \\leq D _ { \\mathrm { K L } } ( p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) | | p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 671, + 810, + 779 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Note that the bound is tight whenever $p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } )$ coincides with $p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 2 } )$ . This happens whenever $\\mathbf { z } _ { 1 }$ and $\\mathbf { z } _ { 2 }$ produce a consistent encoding. Analogously $I _ { \\psi } ( \\mathbf { v } _ { 2 } ; \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 1 } )$ is upper bounded by $D _ { \\mathrm { K L } } ( p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) | | p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) )$ . ", + "bbox": [ + 176, + 779, + 823, + 823 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "$I _ { \\theta } ( \\mathbf { v } _ { 2 } ; \\mathbf { z } _ { 1 } )$ can be rephrased as: ", + "bbox": [ + 174, + 828, + 380, + 842 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/430940024ffa072349800fafa1a414ab6fbb15422c6e4d58cddfc6eaac739775.jpg", + "text": "$$\n\\begin{array} { r l } & { I _ { \\boldsymbol { \\theta } } ( \\mathbf { z } _ { 1 } ; \\mathbf { v } _ { 2 } ) \\overset { ( P _ { 2 } ) } { = } I _ { \\boldsymbol { \\theta } \\boldsymbol { \\psi } } ( \\mathbf { z } _ { 1 } ; \\mathbf { z } _ { 2 } \\mathbf { v } _ { 2 } ) - I _ { \\boldsymbol { \\theta } \\boldsymbol { \\psi } } ( \\mathbf { z } _ { 1 } ; \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) } \\\\ & { = ^ { \\ast } I _ { \\boldsymbol { \\theta } \\boldsymbol { \\psi } } ( \\mathbf { z } _ { 1 } ; \\mathbf { z } _ { 2 } \\mathbf { v } _ { 2 } ) } \\\\ & { = I _ { \\boldsymbol { \\theta } \\boldsymbol { \\psi } } ( \\mathbf { z } _ { 1 } ; \\mathbf { z } _ { 2 } ) + I _ { \\boldsymbol { \\theta } \\boldsymbol { \\psi } } ( \\mathbf { z } _ { 1 } ; \\mathbf { v } _ { 2 } | \\mathbf { z } _ { 2 } ) } \\\\ & { \\geq I _ { \\boldsymbol { \\theta } \\boldsymbol { \\psi } } ( \\mathbf { z } _ { 1 } ; \\mathbf { z } _ { 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 845, + 656, + 924 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Where ∗ follows from $\\mathbf { z } _ { 2 }$ representation of $\\mathbf { v } _ { 2 }$ . The bound reported in this equation is tight whenever $\\mathbf { z } _ { 2 }$ is sufficient for ${ \\bf z } _ { 1 } ( { \\cal I } _ { \\theta \\psi } ( { \\bf z } _ { 1 } ; { \\bf v } _ { 2 } | { \\bf z } _ { 2 } ) = 0 )$ . This happens whenever $\\mathbf { z } _ { 2 }$ contains all the information regarding $\\mathbf { z } _ { 1 }$ (and therefore $\\mathbf { v } _ { 1 }$ ). Once again, the same bound can symmetrically be used to show $I _ { \\theta } ( \\mathbf { z } _ { 2 } ; \\mathbf { v } _ { 1 } ) \\ge I _ { \\theta \\psi } ( \\mathbf { z } _ { 1 } ; \\mathbf { z } _ { 2 } )$ . Therefore, the loss function in Equation 6 can be upper-bounded with: ", + "bbox": [ + 173, + 103, + 825, + 160 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/dd12f85641badda22a468df1f3ee439268015ef48a0d81da48545af69090e94f.jpg", + "text": "$$\n\\mathcal { L } _ { \\frac { 1 + 2 } { 2 } } ( \\theta , \\psi ; \\lambda _ { 1 } , \\lambda _ { 2 } ) \\leq D _ { S K L } ( p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) | | p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) ) - \\frac { \\lambda _ { 1 } + \\lambda _ { 2 } } { 2 } I _ { \\theta \\psi } ( \\mathbf { z } _ { 1 } ; \\mathbf { z } _ { 2 } )\n$$", + "text_format": "latex", + "bbox": [ + 248, + 166, + 748, + 196 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Where: ", + "bbox": [ + 174, + 202, + 225, + 217 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/d2001f968341eed09fd220d2a1dcbb54232841681f7cc4459adf72c23b0089d7.jpg", + "text": "$$\n\\partial _ { S K L } ( p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) | | p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) ) : = \\frac { 1 } { 2 } D _ { \\mathrm { K L } } ( p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) | | p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) ) + \\frac { 1 } { 2 } D _ { \\mathrm { K L } } ( p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) | | p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 219, + 823, + 251 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Lastly, multiplying both terms with $\\begin{array} { r } { \\beta : = \\frac { 2 } { \\lambda _ { 1 } + \\lambda _ { 2 } } } \\end{array}$ and re-parametrizing the objective, we obtain: ", + "bbox": [ + 169, + 256, + 795, + 275 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/f868efdc1315bc3efc20ad0037c907d23d708c92dd0b112b71708a85378d05fe.jpg", + "text": "$$\n\\mathcal { L } _ { M I B } ( \\theta , \\psi ; \\beta ) = - I _ { \\theta \\psi } ( \\mathbf { z } _ { 1 } ; \\mathbf { z } _ { 2 } ) + \\beta D _ { S K L } ( p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) | | p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) )\n$$", + "text_format": "latex", + "bbox": [ + 276, + 280, + 722, + 299 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "G EXPERIMENTAL PROCEDURE AND DETAILS ", + "text_level": 1, + "bbox": [ + 173, + 318, + 566, + 333 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "G.1 MODELING ", + "text_level": 1, + "bbox": [ + 174, + 348, + 299, + 363 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "The two stochastic encoders $p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } )$ and $p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } )$ are modeled by Normal distributions parametrized with neural networks $( \\mu _ { \\theta } , \\sigma _ { \\theta } ^ { 2 } )$ and $( \\mu _ { \\psi } , \\sigma _ { \\psi } ^ { 2 } )$ respectively: ", + "bbox": [ + 174, + 373, + 823, + 406 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/c8baaeab2400be044cd4068f80b6d25d498bf6e363eadd827f6256997fcb4738.jpg", + "text": "$$\n\\begin{array} { r } { p _ { \\theta } ( \\mathbf { z } _ { 1 } | \\mathbf { v } _ { 1 } ) : = \\mathcal { N } \\left( \\mathbf { z } _ { 1 } | \\pmb { \\mu } _ { \\theta } ( \\mathbf { v } _ { 1 } ) , \\pmb { \\sigma } _ { \\theta } ^ { 2 } ( \\mathbf { v } _ { 1 } ) \\right) } \\\\ { p _ { \\psi } ( \\mathbf { z } _ { 2 } | \\mathbf { v } _ { 2 } ) : = \\mathcal { N } \\left( \\mathbf { z } _ { 2 } | \\pmb { \\mu } _ { \\psi } ( \\mathbf { v } _ { 2 } ) , \\pmb { \\sigma } _ { \\psi } ^ { 2 } ( \\mathbf { v } _ { 2 } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 411, + 629, + 455 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Since the density of the two encoders can be evaluated, the symmetrized KL-divergence in equation 4 can be directly computed. On the other hand, $I _ { \\theta \\psi } ( \\mathbf { z } _ { 1 } ; \\mathbf { z } _ { 2 } )$ requires the use of a mutual information estimator. ", + "bbox": [ + 174, + 457, + 823, + 500 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "To facilitate the optimization, the hyper-parameter $\\beta$ is slowly increased during training, starting from a small value $\\approx 1 0 ^ { - 4 }$ to its final value with an exponential schedule. This is because the mutual information estimator is trained together with the other architectures and, since it starts from a random initialization, it requires an initial warm-up. Starting with bigger $\\beta$ results in the encoder collapsing into a fixed representation. The update policy for the hyper-parameter during training has not shown strong influence on the representation, as long as the mutual information estimator network has reached full capacity. ", + "bbox": [ + 173, + 506, + 825, + 604 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "All the experiments have been performed using the Adam optimizer with a learning rate of $1 0 ^ { - 4 }$ for both encoders and the estimation network. Higher learning rate can result in instabilities in the training procedure. The results reported in the main text relied on the Jensen-Shannon mutual information estimator (Devon Hjelm et al., 2019) since the InfoNCE counterpart (van den Oord et al., 2018) generally resulted in worse performance that could be explained by the effect of the factorization of the critic network (Poole et al., 2019). ", + "bbox": [ + 174, + 611, + 825, + 695 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "G.2 SKETCHY EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 712, + 392, + 726 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "• Input: The two views for the sketch-based classification task consist of 4096 dimensional sketch and image features extracted from two distinct VGG-16 network models which were pre-trained on images and sketches from the TU-Berlin dataset Eitz et al. (2012) for endto-end classification. The feature extractors are frozen during the training procedure of for the two representations. Each training iteration used batches of size $B = 1 2 8$ . Encoder and Critic architectures: Both sketch and image encoders consist of multi-layer perceptrons of 2 hidden ReLU units of size 2,048 and 1,024 respectively with an output of size $2 \\mathrm { x } 6 4$ that parametrizes mean and variance for the two Gaussian posteriors. The critic architecture also consists of a multi layer perceptron of 2 hidden ReLU units of size 512. $\\beta$ update policy: The initial value of $\\beta$ is set to $1 0 ^ { - 4 }$ . Starting from the $1 0 { , } 0 0 0 ^ { \\mathrm { t h } }$ training iteration, the value of $\\beta$ is exponentially increased up to 1.0 during the following 250,000 training iterations. The value of $\\beta$ is then kept fixed to one until the end of the training procedure (500,000 iterations). ", + "bbox": [ + 215, + 733, + 825, + 924 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "• Evaluation: All natural images are used as both training sets and retrieval galleries. The 64 dimensional real outputs of sketch and image representation are compared using Euclidean distance. For having a fair comparison other methods that rely on binary hashing (Liu et al., 2017; Zhang et al., 2018), we used Hamming distance on a binarized representation (obtained by applying iterative quantization Gong et al. (2013) on our real valued representation). We report the mean average precision $( \\mathrm { m A P } @ \\mathrm { a l l } )$ and precision at toprank 200 (Prec $@ 2 0 0$ ) Su et al. (2015) on both the real and binary representation to evaluate our method and compare it with prior works. ", + "bbox": [ + 217, + 103, + 825, + 215 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "G.3 MIR-FLICKR EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 231, + 415, + 246 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Figure 6: Examples of pictures $\\mathbf { v } _ { 1 }$ , tags $\\mathbf { v } _ { 2 }$ and category labels y for the MIR-Flickr dataset (Srivastava & Salakhutdinov, 2014). As visualized is the second row, the tags are not always predictive of the label. For this reason, the mutual redundancy assumption holds only approximately. ", + "bbox": [ + 174, + 272, + 825, + 314 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/0d8f12fdddf514eb4f72b6b35413f89d2358eeac502f25d9701f440c26b770c1.jpg", + "image_caption": [], + "image_footnote": [], + "bbox": [ + 334, + 313, + 663, + 464 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• Input: Whitening is applied to the handcrafted image features. Batches of size $B = 1 2 8$ are used for each update step. ", + "bbox": [ + 215, + 491, + 823, + 520 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• Encoders and Critic architectures: The two encoders consists of a multi layer perceptron of 4 hidden ReLU units of size 1,024, which exactly resemble the architecture used in Wang et al. (2016). Both representations $\\mathbf { z } _ { 1 }$ and $\\mathbf { z } _ { 2 }$ have a size of 1,024, therefore the two architecture output a total of $^ { 2 \\mathrm { x } 1 , 0 2 4 }$ parameters that define mean and variance of the respective factorized Gaussian posterior. Similarly to the Sketchy experiments, the critic is consists of a multi-layer perceptron of 2 hidden ReLU units of size 512. ", + "bbox": [ + 217, + 523, + 825, + 607 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• $\\beta$ update policy: The initial value of $\\beta$ is set to $1 0 ^ { - 8 }$ . Starting from $1 5 0 0 0 0 ^ { \\mathrm { t h } }$ iteration, $\\beta$ is set to exponentially increase up to 1.0 (and $1 0 ^ { - 3 }$ ) during the following 150,000 iterations. ", + "bbox": [ + 217, + 611, + 823, + 640 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• Evaluation: Once the models are trained on the unlabeled set, the representation of the 25,000 labeled images is computed. The resulting vectors are used for training and evaluating a multi-label logistic regression classifier on the respective splits. The optimal parameters (such as $\\beta$ ) for our model are chosen based on the performance on the validation set. In Table 3, we report the aggregated mean of the 5 test splits as the final value mean average precision value. ", + "bbox": [ + 217, + 645, + 823, + 727 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "G.4 MNIST EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 174, + 744, + 379, + 758 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• Input: The two views $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ for the MNIST dataset are generated by applying small translation $( [ 0 - 1 0 ] \\% )$ , rotation ([-15,15] degrees), scale $( [ 9 0 , 1 1 0 ] \\% )$ , shear ([-15,15] degrees) and pixel corruption $( 2 0 \\% )$ . Batches of size $B = 6 4$ samples are used during training. ", + "bbox": [ + 215, + 766, + 823, + 823 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• Encoders, Decoders and Critic architectures: All the encoders used for the MNIST experiments consist of neural networks with two hidden layers of 1,024 units and ReLU activations, producing a $2 \\mathrm { x } 6 4$ -dimensional parameter vector that is used to parameterize mean and variance for the Gaussian posteriors. The decoders used for the VAE experiments also consist of the networks of the same size. Similarly, the critic architecture used for mutual information estimation consists of two hidden layers of 1,204 units each and ReLU activations. ", + "bbox": [ + 217, + 827, + 823, + 922 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "• $\\beta$ update policy: The initial value of $\\beta$ is set to $1 0 ^ { - 3 }$ , which is increased with an exponential schedule starting from the $5 0 { , } 0 0 0 ^ { \\mathrm { t h } }$ until 1the $5 0 { , } 0 0 0 ^ { \\mathrm { t h } }$ iteration. The value of $\\beta$ is then kept constant until the $1 , 0 0 0 { , } 0 0 0 ^ { \\mathrm { t h } }$ iteration. The same annealing policy is used to trained the different $\\beta$ -VAEs reported in this work. ", + "bbox": [ + 217, + 103, + 823, + 159 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "• Evaluation: The trained representation are evaluated following the well-known protocol described in Tschannen et al. (2019); Tian et al. (2019); Bachman et al. (2019); van den Oord et al. (2018). Each logistic regression is trained 5 different balanced splits of the training set for different percentages of training examples, ranging from 1 example per label to the whole training set. The accuracy reported in this work has been computed on the disjoint test set. Mean and standard deviation are computed according to the 5 different subsets used for training the logistic regression. Mean and variance for the mutual information estimation reported on the Information Plane (Figure 4) are computed by training two estimation networks from scratch on the final representation of the non-augmented train set. The two estimation architectures consist of 2 hidden layers of 2048 and 1024 units each, and have been trained with batches of size $B = 2 5 6$ for a total of approximately 25,000 iterations. The Jensen-Shannon mutual information lower bound is maximized during training, while the numerical estimation are computed using an energy-based bound (Poole et al., 2019; Devon Hjelm et al., 2019). The final values for $I ( \\mathbf { x } ; \\mathbf { z } )$ and $I ( \\mathbf { y } ; \\mathbf { z } )$ are computed by averaging the mutual information estimation on the whole dataset. In order to reduce the variance of the estimator, the lowest and highest $5 \\%$ are removed before averaging. This practical detail makes the estimation more consistent and less susceptible to numerical instabilities. ", + "bbox": [ + 217, + 164, + 825, + 415 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "G.4.1 RESULTS AND VISUALIZATION ", + "text_level": 1, + "bbox": [ + 178, + 431, + 444, + 445 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In this section we include additional quantitative results and visualizations which refer to the singleview MNIST experiments reported in section 5.2. ", + "bbox": [ + 176, + 457, + 821, + 484 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Table 2 reports the quantitative results used for to produce the visualizations reported in Figure 4, including the comparison between the performance resulting from different mutual information estimators. As the Jensen-Shannon estimator generally resulted in better performance for the InfoMax, MV-InfoMax and MIB models, all the experiments reported on the main text make use of this estimator. Note that the InfoMax model with the $I _ { \\mathrm { J S } }$ estimator is equivalent to the global model reported in Devon Hjelm et al. (2019), while MV-InfoMax with the $I _ { \\mathrm { N C E } }$ estimator results in a similar architecture to the one introduced in Tian et al. (2019). ", + "bbox": [ + 173, + 491, + 825, + 590 + ], + "page_idx": 23 + }, + { + "type": "table", + "img_path": "images/3f9838f8ab1b01dd3a39b4e191fcdccf15003a6bb800cda44561738c238f925f.jpg", + "table_caption": [ + "Table 2: Comparison of the amount of input information $I ( \\mathbf { x } ; \\mathbf { z } )$ , label information $I ( \\mathbf { z } ; \\mathbf { y } )$ , and accuracy of a linear classifier trained with different amount of labeled Examples $\\left( \\operatorname { E x } \\right)$ for the models reported in Figure 4. Both the results obtained using the Jensen-Shannon $I _ { \\mathrm { J S D } }$ (Devon Hjelm et al., 2019; Poole et al., 2019) and the InfoNCE $I _ { \\mathrm { N C E } }$ (van den Oord et al., 2018) estimators are reported. " + ], + "table_footnote": [], + "table_body": "
ModelI(x; z) [nats]I(z;y)[nats]Test Accuracy [%]
10 Ex50Ex3750Ex60000 Ex
VAE (beta=0)12.5 ± 0.72.3± 0.243.8±1.665.6±3.389.0± 0.491.3 ±0.1
VAE (beta=4)7.5 ±1.02.0±0.255.9 ± 2.681.4 ± 4.094.2 ± 0.396.0 ± 0.2
VAE (beta=8)3.0 ± 0.51.0 ± 0.143.8 ±2.861.1 ± 4.881.9 ± 1.187.2 ± 0.6
InfoMax (INCE)12.8 ± 0.52.3 ±0.225.4 ± 1.939.6 ± 3.369.2 ± 0.774.6 ± 0.6
InfoMax(IJs)13.7 ± 0.72.2 ±0.235.0± 2.852.5 ± 2.874.4 ± 1.178.2 ±1.2
MV-InfoMax(INCE)12.2 ± 0.72.3 ±0.250.2 ± 3.675.8±3.894.6 ± 0.496.5 ± 0.1
MV-InfoMax (IJs)11.1 ± 1.02.3 ± 0.254.0 ± 6.178.3 ± 4.494.1 ± 0.395.90 ± 0.08
MIB (β=1,INCE)4.6 ± 0.72.1 ± 0.281.8 ± 5.092.7 ± 0.997.19 ± 0.0897.75 ± 0.05
MIB(β=1,Is)2.4 ± 0.22.1 ± 0.297.1 ± 0.297.2 ± 0.297.70 ± 0.0697.82 ± 0.01
", + "bbox": [ + 174, + 603, + 870, + 746 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Figure 7 reports the linear projection of the embedding obtained using the MIB model. The latent space appears to roughly consists of ten clusters which corresponds to the different digits. This observation is consistent with the empirical measurement of input and label information $\\bar { I } ( { \\bf { x } } ; { \\bf { z } } ) \\approx$ $I ( \\mathbf { z } ; \\mathbf { y } ) \\approx \\log 1 0$ , and the performance of the linear classifier in scarce label regimes. As the cluster are distinct and concentrated around the respective centroids, 10 labeled examples are sufficient to align the centroid coordinates with the digit labels. ", + "bbox": [ + 173, + 839, + 825, + 924 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/f76dfca9ce31325a34acae4c0ad7ac8778fd112016ade2e04378631dbe9ba589.jpg", + "image_caption": [ + "Figure 7: Linear projection of the embedding obtained by applying the MIB encoder to the MNIST test set. The 64 dimensional representation is projected onto the two principal components. Different colors are used to represent the 10 digit classes. " + ], + "image_footnote": [], + "bbox": [ + 356, + 130, + 630, + 273 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "H ABLATION STUDIES ", + "text_level": 1, + "bbox": [ + 174, + 381, + 375, + 396 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "H.1 DIFFERENT RANGES OF DATA AUGMENTATION ", + "text_level": 1, + "bbox": [ + 174, + 412, + 537, + 426 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Figure 8 visualizes the effect of different ranges of corruption probabily as data augmentation strategy to produce the two views $\\mathbf { v } _ { 1 }$ and $\\mathbf { v } _ { 2 }$ . The MV-InfoMax Model does not seem to get any advantage from the use increasing amount of corruption, and it representation remains approximately in the same region of the information plane. On the other hand, the models trained with the MIB objective are able to take advantage of the augmentation to remove irrelevant data information and the representation transitions from the top right corner of the Information Plane (no-augmentation) to the top-left. When the amount of corruption approaches $100 \\%$ , the mutual redundancy assumption is clearly violated, and the performances of MIB deteriorate. In the initial part of the transitions between the two regimes (which corresponds to extremely low probability of corruption) the MIB models drops some label information that is quickly re-gained when pixel corruption becomes more frequent. We hypothesize that this behavior is due to a problem with the optimization procedure, since the corruption are extremely unlikely, the Monte-Carlo estimation for the symmetrized Kullback-Leibler divergence is more biased. Using more examples of views produced from the same data-point within the same batch could mitigate this issue. ", + "bbox": [ + 173, + 438, + 825, + 633 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/33faa873a6686b66ec62de391891ff78f26f3fa6c9429a7ee8cc067cef8b2108.jpg", + "image_caption": [ + "Figure 8: Visualization of the coordinates on the Information Plane (plot on the left) and prediction accuracy (center and right) for the MV-InfoMax and MIB objectives with different amount of training labels and corruption percentage used for data-augmentation. " + ], + "image_footnote": [], + "bbox": [ + 181, + 650, + 815, + 780 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "H.2 EFFECT OF $\\beta$ ", + "text_level": 1, + "bbox": [ + 174, + 868, + 308, + 883 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "The hyper-parameter $\\beta$ (Equation 5) determines the trade-off between sufficiency and minimality of the representation for the second data view. When $\\beta$ is zero, the training objective of MIB is equivalent to the Multi-View InfoMax target, since the representation has no incentive to discard any information. When $0 < \\beta \\leq 1$ the sufficiency constrain is enforced, while the superfluous information is gradually removed from the representation. Values of $\\beta > 1$ can result in representations that violate the sufficiency constraint, since the minimization of $I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { v } _ { 2 } )$ is prioritized. The trade-off resulting from the choice of different $\\beta$ is visualized in Figure 9 and compared against $\\beta$ -VAE. Note that in each point of the pareto-front the MIB model results in a better trade-off between $I ( \\mathbf { x } ; \\mathbf { z } )$ and $I ( \\mathbf { y } ; \\mathbf { z } )$ when compared to $\\beta$ -VAE. The effectiveness of the Multi-View Information Bottleneck model is also justified by the corresponding values of predictive accuracy. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 215 + ], + "page_idx": 25 + }, + { + "type": "image", + "img_path": "images/94d948e8f83b961c343996b173880467a607ee21dfbe2d76ac46e60c951f5a83.jpg", + "image_caption": [ + "Figure 9: Visualization of the coordinates on the Information Plane (plot on the left) and prediction accuracy (center and right) for the $\\beta$ -VAE, Multi-View InfoMax and Multi-View Information Bottleneck objectives with different amount of training labels and different values of the respective hyperparameter $\\beta$ . 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A theoretical analysis", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 358, + 469, + 370 + ], + "spans": [ + { + "bbox": [ + 141, + 358, + 469, + 370 + ], + "score": 1.0, + "content": "leads to the definition of a new multi-view model that produces state-of-the-art re-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 370, + 469, + 381 + ], + "spans": [ + { + "bbox": [ + 142, + 370, + 469, + 381 + ], + "score": 1.0, + "content": "sults on the Sketchy dataset and label-limited versions of the MIR-Flickr dataset.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 380, + 469, + 392 + ], + "spans": [ + { + "bbox": [ + 141, + 380, + 469, + 392 + ], + "score": 1.0, + "content": "We also extend our theory to the single-view setting by taking advantage of stan-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 391, + 469, + 404 + ], + "spans": [ + { + "bbox": [ + 141, + 391, + 469, + 404 + ], + "score": 1.0, + "content": "dard data augmentation techniques, empirically showing better generalization ca-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 402, + 469, + 415 + ], + "spans": [ + { + "bbox": [ + 141, + 402, + 469, + 415 + ], + "score": 1.0, + "content": "pabilities when compared to common unsupervised approaches for representation", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 141, + 412, + 180, + 426 + ], + "spans": [ + { + "bbox": [ + 141, + 412, + 180, + 426 + ], + "score": 1.0, + "content": "learning.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 24.5, + "bbox_fs": [ + 141, + 271, + 470, + 426 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 443, + 206, + 456 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 208, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 208, + 459 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 467, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "The goal of deep representation learning (LeCun et al., 2015) is to transform a raw observational", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "input, x, into a, typically lower-dimensional, representation, z, that contains the information relevant", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "for a given task or set of tasks. Significant progress has been made in deep learning via supervised", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 464, + 514 + ], + "score": 1.0, + "content": "representation learning, where the labels, y, for the downstream task are known while", + "type": "text" + }, + { + "bbox": [ + 465, + 500, + 493, + 513 + ], + "score": 0.92, + "content": "p ( \\mathbf { y } \\vert \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 500, + 506, + 514 + ], + "score": 1.0, + "content": "is", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 510, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 104, + 510, + 505, + 525 + ], + "score": 1.0, + "content": "learned directly (Sutskever et al., 2012; Hinton et al., 2012). Due to the cost of acquiring large", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "labeled datasets, a recently renewed focus on unsupervised representation learning seeks to generate", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 171, + 546 + ], + "score": 1.0, + "content": "representations,", + "type": "text" + }, + { + "bbox": [ + 172, + 536, + 178, + 543 + ], + "score": 0.47, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 533, + 505, + 546 + ], + "score": 1.0, + "content": ", that are useful for a wide variety of different tasks where little to no labeled data", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 545, + 323, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 323, + 556 + ], + "score": 1.0, + "content": "is available (Devlin et al., 2018; Radford et al., 2019).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 467, + 506, + 556 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 504, + 671 + ], + "lines": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "Our work is based on the information bottleneck principle (Tishby et al., 2000) where a represen-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "tation becomes less affected by nuisances by discarding all information from the input that is not", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "useful for a given task, resulting in increased robustness. 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When-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 260, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 364, + 273 + ], + "score": 1.0, + "content": "ever the end goal involves predicting a label y, we consider only", + "type": "text" + }, + { + "bbox": [ + 365, + 262, + 371, + 270 + ], + "score": 0.56, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 260, + 506, + 273 + ], + "score": 1.0, + "content": "that are discriminative enough to", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 272, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 283 + ], + "score": 1.0, + "content": "identify y. 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(2) We", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "define a new model 1 that empirically leads to state-of-the-art results in the low-label setting on two", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "standard multi-view datasets, Sketchy and MIR-Flickr. 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As a consequence, a sufficient representation contains minimal", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 514, + 313, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 216, + 527 + ], + "score": 1.0, + "content": "data information whenever", + "type": "text" + }, + { + "bbox": [ + 216, + 514, + 254, + 527 + ], + "score": 0.93, + "content": "I ( \\mathbf { x } ; \\mathbf { z } | \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 514, + 313, + 527 + ], + "score": 1.0, + "content": "is minimized.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 469, + 506, + 527 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 531, + 505, + 597 + ], + "lines": [ + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 545 + ], + "score": 1.0, + "content": "Minimizing the amount of superfluous information can be done directly only in supervised settings.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 541, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 178, + 555 + ], + "score": 1.0, + "content": "In fact, reducing", + "type": "text" + }, + { + "bbox": [ + 178, + 542, + 208, + 554 + ], + "score": 0.92, + "content": "I ( \\mathbf { x } ; \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 541, + 505, + 555 + ], + "score": 1.0, + "content": "without violating the sufficiency constraint necessarily requires making", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 554, + 506, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 554, + 506, + 565 + ], + "score": 1.0, + "content": "some additional assumptions on the predictive task (see Theorem B.1 in the Appendix). In the next", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "score": 1.0, + "content": "section we describe the basis of our technique, a strategy to safely reduce the information content of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 575, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 104, + 575, + 255, + 588 + ], + "score": 1.0, + "content": "a representation even when the label", + "type": "text" + }, + { + "bbox": [ + 256, + 577, + 263, + 586 + ], + "score": 0.48, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 575, + 506, + 588 + ], + "score": 1.0, + "content": "is not observed, by exploiting redundant information in the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 586, + 264, + 598 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 264, + 598 + ], + "score": 1.0, + "content": "form of an additional view on the data.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 36.5, + "bbox_fs": [ + 104, + 530, + 506, + 598 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 613, + 344, + 626 + ], + "lines": [ + { + "bbox": [ + 104, + 611, + 345, + 627 + ], + "spans": [ + { + "bbox": [ + 104, + 611, + 345, + 627 + ], + "score": 1.0, + "content": "3 MULTI-VIEW INFORMATION BOTTLENECK", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 506, + 714 + ], + "lines": [ + { + "bbox": [ + 105, + 636, + 506, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 250, + 650 + ], + "score": 1.0, + "content": "As a motivating example, consider", + "type": "text" + }, + { + "bbox": [ + 250, + 639, + 262, + 648 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 636, + 281, + 650 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 281, + 639, + 293, + 648 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 636, + 506, + 650 + ], + "score": 1.0, + "content": "to be two images of the same object from different", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 648, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 187, + 660 + ], + "score": 1.0, + "content": "view-points and let", + "type": "text" + }, + { + "bbox": [ + 188, + 650, + 196, + 660 + ], + "score": 0.5, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 648, + 505, + 660 + ], + "score": 1.0, + "content": "be its label. Assuming that the object is clearly distinguishable from both", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 660, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 118, + 670 + ], + "score": 0.84, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 660, + 151, + 671 + ], + "score": 1.0, + "content": "and let", + "type": "text" + }, + { + "bbox": [ + 151, + 661, + 163, + 670 + ], + "score": 0.83, + "content": "\\mathbf { v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 660, + 245, + 671 + ], + "score": 1.0, + "content": ", any representation", + "type": "text" + }, + { + "bbox": [ + 245, + 661, + 253, + 669 + ], + "score": 0.3, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 660, + 506, + 671 + ], + "score": 1.0, + "content": "containing all information accessible from both views would", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 360, + 682 + ], + "score": 1.0, + "content": "also contain the necessary label information. Furthermore, if", + "type": "text" + }, + { + "bbox": [ + 360, + 672, + 367, + 680 + ], + "score": 0.45, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "captures only the details that are", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 506, + 693 + ], + "score": 1.0, + "content": "visible from both pictures, it would eliminate the view-specific details and reduce the sensitivity of", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 705 + ], + "score": 1.0, + "content": "the representation to view-changes. The theory to support this intuition is described in the following", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 704, + 378, + 715 + ], + "spans": [ + { + "bbox": [ + 106, + 704, + 133, + 715 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 704, + 145, + 714 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 145, + 704, + 163, + 715 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 163, + 704, + 175, + 714 + ], + "score": 0.83, + "content": "\\mathbf { v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 704, + 378, + 715 + ], + "score": 1.0, + "content": "are jointly observed and referred to as data-views.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 636, + 506, + 715 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 397, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 398, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 398, + 95 + ], + "score": 1.0, + "content": "3.1 SUFFICIENCY AND ROBUSTNESS IN THE MULTI-VIEW SETTING", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 102, + 480, + 115 + ], + "lines": [ + { + "bbox": [ + 104, + 100, + 481, + 118 + ], + "spans": [ + { + "bbox": [ + 104, + 100, + 481, + 118 + ], + "score": 1.0, + "content": "In this section we extend our analysis of sufficiency and minimality to the multi-view setting.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 106, + 119, + 505, + 164 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 245, + 133 + ], + "score": 1.0, + "content": "Intuitively, we can guarantee that", + "type": "text" + }, + { + "bbox": [ + 246, + 122, + 253, + 130 + ], + "score": 0.51, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 119, + 363, + 133 + ], + "score": 1.0, + "content": "is sufficient for predicting", + "type": "text" + }, + { + "bbox": [ + 363, + 122, + 371, + 131 + ], + "score": 0.34, + "content": "\\mathbf { y }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 119, + 505, + 133 + ], + "score": 1.0, + "content": "even without knowing y by en-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 130, + 506, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 153, + 144 + ], + "score": 1.0, + "content": "suring that", + "type": "text" + }, + { + "bbox": [ + 153, + 133, + 160, + 141 + ], + "score": 0.27, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 130, + 349, + 144 + ], + "score": 1.0, + "content": "maintains all information which is shared by", + "type": "text" + }, + { + "bbox": [ + 349, + 133, + 361, + 142 + ], + "score": 0.86, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 130, + 381, + 144 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 381, + 132, + 393, + 142 + ], + "score": 0.85, + "content": "\\mathbf { v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 130, + 506, + 144 + ], + "score": 1.0, + "content": ". This intuition relies on a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 142, + 505, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 154 + ], + "score": 1.0, + "content": "basic assumption of the multi-view environment – that the two views provide the same predictive", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 151, + 326, + 166 + ], + "spans": [ + { + "bbox": [ + 105, + 151, + 326, + 166 + ], + "score": 1.0, + "content": "information. To formalize this we define redundancy.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 105, + 167, + 502, + 180 + ], + "lines": [ + { + "bbox": [ + 105, + 165, + 502, + 181 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 218, + 181 + ], + "score": 1.0, + "content": "Definition 2. 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Let", + "type": "text" + }, + { + "bbox": [ + 179, + 226, + 190, + 236 + ], + "score": 0.87, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 225, + 209, + 237 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 210, + 226, + 222, + 236 + ], + "score": 0.84, + "content": "\\mathbf { v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 225, + 446, + 237 + ], + "score": 1.0, + "content": "be two mutually redundant views for a target y and let", + "type": "text" + }, + { + "bbox": [ + 446, + 226, + 457, + 236 + ], + "score": 0.83, + "content": "\\mathbf { z } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 225, + 505, + 237 + ], + "score": 1.0, + "content": "be a repre-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 236, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 155, + 250 + ], + "score": 1.0, + "content": "sentation of", + "type": "text" + }, + { + "bbox": [ + 155, + 237, + 167, + 248 + ], + "score": 0.82, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 236, + 179, + 250 + ], + "score": 1.0, + "content": ". 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This direction of research has been explored for both single-view (Alemi et al., 2018)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "and multi-view settings (Wang et al., 2019), even if explicit label supervision is required to train the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 176, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 165, + 199 + ], + "score": 1.0, + "content": "representation", + "type": "text" + }, + { + "bbox": [ + 165, + 190, + 172, + 197 + ], + "score": 0.37, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 188, + 176, + 199 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 504, + 292 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "In contrast to all of the above, our work is the first to explicitly identify and discard superfluous", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "information from the representation in the unsupervised multi-view setting. This is because unsu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 228, + 239 + ], + "score": 1.0, + "content": "pervised models based on the", + "type": "text" + }, + { + "bbox": [ + 228, + 226, + 236, + 237 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "-VAE objective remove information indiscriminately without iden-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "tifying which part is relevant for teh predictive task, and the InfoMax and Multi-View InfoMax", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "methods do not explicitly try to remove superfluous information at all. The MIB objective, on the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "other hand, results in the representation with the least superfluous information, i.e. the most robust", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "score": 1.0, + "content": "among the representations that are optimal according to Multi-View InfoMax, without requiring any", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 222, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 222, + 293 + ], + "score": 1.0, + "content": "additional label supervision.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 108, + 309, + 200, + 322 + ], + "lines": [ + { + "bbox": [ + 104, + 308, + 202, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 202, + 324 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "In this section we demonstrate the effectiveness of our model against state-of-the-art baselines in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "both the multi-view and single-view setting. In the single-view setting, we also estimate the coor-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 369 + ], + "score": 1.0, + "content": "dinates on the Information Plane for each of the baseline methods as well as our method to validate", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 369, + 202, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 202, + 380 + ], + "score": 1.0, + "content": "the theory in Section 3.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 454, + 398 + ], + "score": 1.0, + "content": "The results reported in the following sections are obtained using the Jensen-Shannon", + "type": "text" + }, + { + "bbox": [ + 455, + 386, + 471, + 397 + ], + "score": 0.89, + "content": "I _ { J S }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "(Devon", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "Hjelm et al., 2019; Poole et al., 2019) estimator, which resulted in better performance for MIB and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "the other InfoMax-based models (Table 2 in the supplementary material). In order to facilitate the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "comparison between the effect of the different loss functions, the same estimator is used across the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 429, + 177, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 177, + 441 + ], + "score": 1.0, + "content": "different models.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "title", + "bbox": [ + 108, + 456, + 219, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 220, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 220, + 469 + ], + "score": 1.0, + "content": "5.1 MULTI-VIEW TASKS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 109, + 477, + 504, + 510 + ], + "lines": [ + { + "bbox": [ + 107, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 107, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "We compare MIB on the sketch-based image retrieval (Sangkloy et al., 2016) and Flickr multiclass", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 487, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 502 + ], + "score": 1.0, + "content": "image classification (Huiskes & Lew, 2008) tasks with domain specific and prior multi-view learning", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 498, + 146, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 146, + 511 + ], + "score": 1.0, + "content": "methods.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 524, + 291, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 293, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 293, + 537 + ], + "score": 1.0, + "content": "5.1.1 SKETCH-BASED IMAGE RETRIEVAL", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "Dataset. The Sketchy dataset (Sangkloy et al., 2016) consists of 12,500 images and 75,471 hand-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "drawn sketches of objects from 125 classes. As in Liu et al. (2017), we also include another 60,502", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "images from the ImageNet (Deng et al., 2009) from the same classes, which results in total 73,002", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "natural object images. As per the experimental protocol of Zhang et al. (2018), a total of 6,250", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "score": 1.0, + "content": "sketches (50 sketches per category) are randomly selected and removed from the training set for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 599, + 384, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 384, + 612 + ], + "score": 1.0, + "content": "testing purpose, which leaves 69,221 sketches for training the model.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5 + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "Experimental Setup. The sketch-based image retrieval task is a ranking of 73,002 natural images", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "according to the unseen test (query) sketch. Retrieval is done for our model by generating represen-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "tations for the query sketch as well as all natural images, and ranking the image by the euclidean", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "distance of their representation from the sketch representation. The baselines use various domain", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "specific ranking methodologies. Model performance is computed based on the class of the ranked", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 447, + 683 + ], + "score": 1.0, + "content": "pictures corresponding to the query sketch. 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(2000) and identified", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 134 + ], + "score": 1.0, + "content": "as one of the possible reasons behind the generalization capabilities of deep neural networks by", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "Tishby & Zaslavsky (2015) and Achille & Soatto (2018). Representations based on the information", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "bottleneck principle explicitly minimize the amount of superfluous information in the representation", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 155, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 106, + 155, + 505, + 165 + ], + "score": 1.0, + "content": "while retaining all the label information from the data (top-left corner of the Information Plane in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "Figure 2). This direction of research has been explored for both single-view (Alemi et al., 2018)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "and multi-view settings (Wang et al., 2019), even if explicit label supervision is required to train the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 176, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 165, + 199 + ], + "score": 1.0, + "content": "representation", + "type": "text" + }, + { + "bbox": [ + 165, + 190, + 172, + 197 + ], + "score": 0.37, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 188, + 176, + 199 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 110, + 505, + 199 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 504, + 292 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 506, + 217 + ], + "score": 1.0, + "content": "In contrast to all of the above, our work is the first to explicitly identify and discard superfluous", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 505, + 227 + ], + "score": 1.0, + "content": "information from the representation in the unsupervised multi-view setting. This is because unsu-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 226, + 505, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 226, + 228, + 239 + ], + "score": 1.0, + "content": "pervised models based on the", + "type": "text" + }, + { + "bbox": [ + 228, + 226, + 236, + 237 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 236, + 226, + 505, + 239 + ], + "score": 1.0, + "content": "-VAE objective remove information indiscriminately without iden-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 506, + 249 + ], + "score": 1.0, + "content": "tifying which part is relevant for teh predictive task, and the InfoMax and Multi-View InfoMax", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 259 + ], + "score": 1.0, + "content": "methods do not explicitly try to remove superfluous information at all. The MIB objective, on the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 506, + 271 + ], + "score": 1.0, + "content": "other hand, results in the representation with the least superfluous information, i.e. the most robust", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 283 + ], + "score": 1.0, + "content": "among the representations that are optimal according to Multi-View InfoMax, without requiring any", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 281, + 222, + 293 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 222, + 293 + ], + "score": 1.0, + "content": "additional label supervision.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 204, + 506, + 293 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 309, + 200, + 322 + ], + "lines": [ + { + "bbox": [ + 104, + 308, + 202, + 324 + ], + "spans": [ + { + "bbox": [ + 104, + 308, + 202, + 324 + ], + "score": 1.0, + "content": "5 EXPERIMENTS", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "In this section we demonstrate the effectiveness of our model against state-of-the-art baselines in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "both the multi-view and single-view setting. In the single-view setting, we also estimate the coor-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 505, + 369 + ], + "score": 1.0, + "content": "dinates on the Information Plane for each of the baseline methods as well as our method to validate", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 369, + 202, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 202, + 380 + ], + "score": 1.0, + "content": "the theory in Section 3.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 336, + 505, + 380 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 454, + 398 + ], + "score": 1.0, + "content": "The results reported in the following sections are obtained using the Jensen-Shannon", + "type": "text" + }, + { + "bbox": [ + 455, + 386, + 471, + 397 + ], + "score": 0.89, + "content": "I _ { J S }", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "(Devon", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "Hjelm et al., 2019; Poole et al., 2019) estimator, which resulted in better performance for MIB and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 407, + 505, + 419 + ], + "score": 1.0, + "content": "the other InfoMax-based models (Table 2 in the supplementary material). In order to facilitate the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 505, + 431 + ], + "score": 1.0, + "content": "comparison between the effect of the different loss functions, the same estimator is used across the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 429, + 177, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 177, + 441 + ], + "score": 1.0, + "content": "different models.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 385, + 505, + 441 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 456, + 219, + 467 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 220, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 220, + 469 + ], + "score": 1.0, + "content": "5.1 MULTI-VIEW TASKS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 109, + 477, + 504, + 510 + ], + "lines": [ + { + "bbox": [ + 107, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 107, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "We compare MIB on the sketch-based image retrieval (Sangkloy et al., 2016) and Flickr multiclass", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 487, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 502 + ], + "score": 1.0, + "content": "image classification (Huiskes & Lew, 2008) tasks with domain specific and prior multi-view learning", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 498, + 146, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 146, + 511 + ], + "score": 1.0, + "content": "methods.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 477, + 505, + 511 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 524, + 291, + 536 + ], + "lines": [ + { + "bbox": [ + 105, + 523, + 293, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 293, + 537 + ], + "score": 1.0, + "content": "5.1.1 SKETCH-BASED IMAGE RETRIEVAL", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 505, + 610 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 505, + 556 + ], + "score": 1.0, + "content": "Dataset. The Sketchy dataset (Sangkloy et al., 2016) consists of 12,500 images and 75,471 hand-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "drawn sketches of objects from 125 classes. As in Liu et al. (2017), we also include another 60,502", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "images from the ImageNet (Deng et al., 2009) from the same classes, which results in total 73,002", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 589 + ], + "score": 1.0, + "content": "natural object images. As per the experimental protocol of Zhang et al. (2018), a total of 6,250", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 506, + 601 + ], + "score": 1.0, + "content": "sketches (50 sketches per category) are randomly selected and removed from the training set for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 599, + 384, + 612 + ], + "spans": [ + { + "bbox": [ + 106, + 599, + 384, + 612 + ], + "score": 1.0, + "content": "testing purpose, which leaves 69,221 sketches for training the model.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 545, + 506, + 612 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 615, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "Experimental Setup. The sketch-based image retrieval task is a ranking of 73,002 natural images", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "according to the unseen test (query) sketch. Retrieval is done for our model by generating represen-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "tations for the query sketch as well as all natural images, and ranking the image by the euclidean", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 505, + 661 + ], + "score": 1.0, + "content": "distance of their representation from the sketch representation. The baselines use various domain", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "specific ranking methodologies. Model performance is computed based on the class of the ranked", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 447, + 683 + ], + "score": 1.0, + "content": "pictures corresponding to the query sketch. The training set consists of pairs of image", + "type": "text" + }, + { + "bbox": [ + 447, + 672, + 459, + 682 + ], + "score": 0.84, + "content": "\\mathbf { v } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "and sketch", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 683, + 118, + 693 + ], + "score": 0.84, + "content": "\\mathbf { v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "randomly selected from the same class, to ensure that both views contain the equivalent label", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 692, + 243, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 243, + 705 + ], + "score": 1.0, + "content": "information (mutual redundancy).", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 615, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Following recent work (Zhang et al., 2018; Dutta & Akata, 2019), we use features extracted from", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "images and sketches by a VGG (Simonyan & Zisserman, 2014) architecture trained for classification", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "on the TU-Berlin dataset (Eitz et al., 2012). The resulting flattened 4096-dimensional feature vectors", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "are fed to our image and sketch encoders to produce a 64-dimensional representation. Both encoders", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "consist of neural networks with hidden layers of 2048 and 1024 units respectively. Size of the", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 273, + 308 + ], + "score": 1.0, + "content": "representation and regularization strength", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 273, + 296, + 281, + 307 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 282, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "are tuned on a validation sub-split. We evaluate MIB on", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "five different train/test splits and report mean and standard deviation in Table 5.1.1. Further details", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 318, + 355, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 355, + 330 + ], + "score": 1.0, + "content": "on our training procedure and architecture are in Appendix G.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + } + ], + "index": 47.5, + "bbox_fs": [ + 106, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 79, + 506, + 187 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 79, + 506, + 187 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 79, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 112, + 79, + 506, + 187 + ], + "score": 0.709, + "type": "image", + "image_path": "726f2e918ff1392fa39ef9bdb9741b3bd6f5d6c84acce49da00d6f18d2bfa125.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 79, + 506, + 115.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 115.0, + 506, + 151.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 151.0, + 506, + 187.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 196, + 505, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "Table 1: Examples of the two views and class label from the Sketchy dataset (on the left) and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "comparison between MIB and other popular models in literature on the sketch-based image retrieval", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "task (on the right). ∗ denotes models that use a 64-bits binary representation. The results for MIB", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 229, + 198, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 167, + 241 + ], + "score": 1.0, + "content": "corresponds to", + "type": "text" + }, + { + "bbox": [ + 168, + 230, + 193, + 240 + ], + "score": 0.92, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 229, + 198, + 241 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 107, + 262, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 505, + 276 + ], + "score": 1.0, + "content": "on the TU-Berlin dataset (Eitz et al., 2012). The resulting flattened 4096-dimensional feature vectors", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "are fed to our image and sketch encoders to produce a 64-dimensional representation. Both encoders", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 298 + ], + "score": 1.0, + "content": "consist of neural networks with hidden layers of 2048 and 1024 units respectively. Size of the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 273, + 308 + ], + "score": 1.0, + "content": "representation and regularization strength", + "type": "text" + }, + { + "bbox": [ + 273, + 296, + 281, + 307 + ], + "score": 0.84, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "are tuned on a validation sub-split. We evaluate MIB on", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 318 + ], + "score": 1.0, + "content": "five different train/test splits and report mean and standard deviation in Table 5.1.1. Further details", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 318, + 355, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 355, + 330 + ], + "score": 1.0, + "content": "on our training procedure and architecture are in Appendix G.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 107, + 334, + 505, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 348 + ], + "score": 1.0, + "content": "Results. 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The effective-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 368, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 368, + 505, + 379 + ], + "score": 1.0, + "content": "ness of MIB on the retrieval task can be mostly attributed to the regularization introduced with the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "symmetrized KL divergence between the two encoded views. 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The dataset is divided into labeled and unlabeled sets that respectively contain", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 502 + ], + "score": 1.0, + "content": "975K and 25K images, where the labeled set also contains 38 distinct topic classes together with the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 501, + 506, + 513 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 506, + 513 + ], + "score": 1.0, + "content": "user tags. Training images with less than two tags are removed, which reduces the total number of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "training samples to 749,647 pairs (Sohn et al., 2014; Wang et al., 2016). The labeled set contains 5", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 522, + 429, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 429, + 536 + ], + "score": 1.0, + "content": "different splits of train, validation and test sets of size 10K/5K/10K respectively.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 540, + 505, + 628 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "Experimental Setup. Following standard procedure in the literature (Srivastava & Salakhutdinov,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "2014; Wang et al., 2016), we train our model on the unlabeled pairs of images and tags. 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Although the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "tuned MIB performs similarly to Multi-View InfoMax with a large number of labels, it outperforms", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 666, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 388, + 679 + ], + "score": 1.0, + "content": "it when fewer labels are available. 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This effect is likely due to a violation of the mutual", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 117, + 709, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 117, + 709, + 506, + 724 + ], + "score": 1.0, + "content": "2These results are included only for completeness, as the Multi-View InfoMax objective does not produce", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 468, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 468, + 734 + ], + "score": 1.0, + "content": "consistent representations for the two views so there is no straight-forward way to use it for ranking.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 79, + 506, + 187 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 79, + 506, + 187 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 79, + 506, + 187 + ], + "spans": [ + { + "bbox": [ + 112, + 79, + 506, + 187 + ], + "score": 0.709, + "type": "image", + "image_path": "726f2e918ff1392fa39ef9bdb9741b3bd6f5d6c84acce49da00d6f18d2bfa125.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 79, + 506, + 115.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 115.0, + 506, + 151.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 151.0, + 506, + 187.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 196, + 505, + 240 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "Table 1: Examples of the two views and class label from the Sketchy dataset (on the left) and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 106, + 207, + 505, + 219 + ], + "score": 1.0, + "content": "comparison between MIB and other popular models in literature on the sketch-based image retrieval", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 230 + ], + "score": 1.0, + "content": "task (on the right). ∗ denotes models that use a 64-bits binary representation. 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These are kept small enough to ensure that label information is not", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 534 + ], + "score": 1.0, + "content": "effected. Each pair of views is generated from the same underlying image, so no label information", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 532, + 294, + 544 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 294, + 544 + ], + "score": 1.0, + "content": "is used in this process (details in Appendix G).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 671 + ], + "lines": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "Experimental Setup. 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Figure 4 summarizes the results. 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Both Multi-View VAE", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 375 + ], + "score": 1.0, + "content": "(MVAE) and Deep Variational CCA (VCCA) rely on a reconstruction term to capture cross-modal", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 372, + 374, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 372, + 374, + 385 + ], + "score": 1.0, + "content": "information, which can introduce bias that decreases performance.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 340, + 506, + 385 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 404, + 298, + 415 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 300, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 300, + 417 + ], + "score": 1.0, + "content": "5.2 SELF-SUPERVISED SINGLE-VIEW TASK", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 504, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 506, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 506, + 442 + ], + "score": 1.0, + "content": "In this section, we compare the performance of different unsupervised learning models by measuring", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 505, + 451 + ], + "score": 1.0, + "content": "their data efficiency and empirically estimating the coordinates of their representation on the Infor-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "mation Plane. 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Figure 4 summarizes the results. 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The", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 197, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 505, + 209 + ], + "score": 1.0, + "content": "empirical estimation of the coordinates on the Information Plane (in nats on the left) is followed", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 208, + 505, + 220 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 220 + ], + "score": 1.0, + "content": "by the respective classification accuracy for different number of randomly sampled labels (from 1", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "example per label up to 6000 examples per label). 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The strong performance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 379, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 394 + ], + "score": 1.0, + "content": "obtained in the different areas show that Multi-View Information Bottleneck can be practically ap-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "plied to various tasks for which the paired observations are either readily available or artificially", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "produced. Furthermore, the positive results on the MIR-Flickr dataset show that our model can", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 413, + 423, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 423, + 427 + ], + "score": 1.0, + "content": "work well in practice even when mutual redundancy holds only approximately.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "There are multiple extensions that we would like to explore in future work. One interesting direction", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "would be considering more than two views. 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Secondly, we believe that exploring the role played by different choices of data", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 488 + ], + "score": 1.0, + "content": "augmentation could bridge the gap between the Information Bottleneck principle and the literature", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 505, + 497 + ], + "score": 1.0, + "content": "on invariant neural networks (Bloem-Reddy & Whye Teh, 2019), which are able to exploit known", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 497, + 397, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 397, + 509 + ], + "score": 1.0, + "content": "symmetries and structure of the data to remove superfluous information.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "title", + "bbox": [ + 108, + 521, + 200, + 531 + ], + "lines": [ + { + "bbox": [ + 107, + 522, + 200, + 532 + ], + "spans": [ + { + "bbox": [ + 107, + 522, + 200, + 532 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 539, + 505, + 583 + ], + "lines": [ + { + "bbox": [ + 106, + 539, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 504, + 551 + ], + "score": 1.0, + "content": "We thank Andy Keller, Karen Ullrich, Maximillian Ilse and the anonymous reviewers for their feed-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 551, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 562 + ], + "score": 1.0, + "content": "back and insightful comments. This work has received funding from the ERC under the Horizon", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 560, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 575 + ], + "score": 1.0, + "content": "2020 program (grant agreement No. 853489). The Titan Xp and Titan V used for this research were", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 572, + 258, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 258, + 585 + ], + "score": 1.0, + "content": "donated by the NVIDIA Corporation.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "title", + "bbox": [ + 108, + 601, + 175, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 600, + 177, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 177, + 614 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 619, + 504, + 641 + ], + "lines": [ + { + "bbox": [ + 105, + 616, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 634 + ], + "score": 1.0, + "content": "Alessandro Achille and Stefano Soatto. 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The strong performance", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 379, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 394 + ], + "score": 1.0, + "content": "obtained in the different areas show that Multi-View Information Bottleneck can be practically ap-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 391, + 505, + 405 + ], + "score": 1.0, + "content": "plied to various tasks for which the paired observations are either readily available or artificially", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "produced. Furthermore, the positive results on the MIR-Flickr dataset show that our model can", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 413, + 423, + 427 + ], + "spans": [ + { + "bbox": [ + 106, + 413, + 423, + 427 + ], + "score": 1.0, + "content": "work well in practice even when mutual redundancy holds only approximately.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 335, + 506, + 427 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 508 + ], + "lines": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "There are multiple extensions that we would like to explore in future work. One interesting direction", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 455 + ], + "score": 1.0, + "content": "would be considering more than two views. In Appendix D we discuss why the mutual redundancy", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "condition cannot be trivially extended to more than two views, but we still believe such an exten-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "score": 1.0, + "content": "sion is possible. 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y ) } = I ( { \\bf t } _ { 2 } ( { \\bf x } ) ; { \\bf y } ) = I ( { \\bf x } ; { \\bf y } ) \\Longrightarrow I ( { \\bf t } _ { 1 } ( { \\bf x } ) ; { \\bf y } | { \\bf t } _ { 2 } ( { \\bf x } ) ) + I ( { \\bf t } _ { 2 } ( { \\bf x } ) ; { \\bf y } | { \\bf t } _ { 1 } ( { \\bf x } ) ) = I ( { \\bf t } _ { 1 } ( { \\bf x } ) ; { \\bf y } ) ,", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 468, + 115, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 115, + 479 + ], + "score": 1.0, + "content": "0", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 105, + 488, + 506, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 507, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 185, + 503 + ], + "score": 1.0, + "content": "Theorem B.3. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 489, + 339, + 501 + ], + "score": 0.9, + "content": "I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { x } ; \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 487, + 360, + 503 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 360, + 491, + 371, + 500 + ], + "score": 0.86, + "content": "\\mathbf { z } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 487, + 463, + 503 + ], + "score": 1.0, + "content": "be a representation of", + "type": "text" + }, + { + "bbox": [ + 464, + 489, + 487, + 501 + ], + "score": 0.77, + "content": "\\mathbf { t } _ { 1 } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 487, + 507, + 503 + ], + "score": 1.0, + "content": ". 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\\mathbf { x } ) = I ( \\mathbf { y } ; \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ) \\implies I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } | \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ) = 0", + "type": "inline_equation" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 142, + 418, + 393, + 433 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 444, + 506, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 165, + 459 + ], + "score": 1.0, + "content": "The proof for", + "type": "text" + }, + { + "bbox": [ + 166, + 445, + 390, + 457 + ], + "score": 0.88, + "content": "{ \\cal I } ( { \\bf y } ; { \\bf x } ) = { \\cal I } ( { \\bf y } ; { \\bf t } _ { 1 } ( { \\bf x } ) ) \\Longrightarrow { \\cal I } ( { \\bf t } _ { 2 } ( { \\bf x } ) ; { \\bf y } | { \\bf t } _ { 1 } ( { \\bf x } ) ) = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 444, + 506, + 459 + ], + "score": 1.0, + "content": "is symmetric, therefore we", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 455, + 504, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 455, + 150, + 471 + ], + "score": 1.0, + "content": "conclude I", + "type": "text" + }, + { + "bbox": [ + 150, + 457, + 504, + 469 + ], + "score": 0.71, + "content": "{ \\bf \\bar { \\Psi } ( t _ { 1 } ( x ) ; y ) } = I ( { \\bf t } _ { 2 } ( { \\bf x } ) ; { \\bf y } ) = I ( { \\bf x } ; { \\bf y } ) \\Longrightarrow I ( { \\bf t } _ { 1 } ( { \\bf x } ) ; { \\bf y } | { \\bf t } _ { 2 } ( { \\bf x } ) ) + I ( { \\bf t } _ { 2 } ( { \\bf x } ) ; { \\bf y } | { \\bf t } _ { 1 } ( { \\bf x } ) ) = I ( { \\bf t } _ { 1 } ( { \\bf x } ) ; { \\bf y } ) ,", + "type": "inline_equation" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 468, + 115, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 115, + 479 + ], + "score": 1.0, + "content": "0", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 104, + 444, + 506, + 479 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 488, + 506, + 513 + ], + "lines": [ + { + "bbox": [ + 105, + 487, + 507, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 185, + 503 + ], + "score": 1.0, + "content": "Theorem B.3. Let", + "type": "text" + }, + { + "bbox": [ + 186, + 489, + 339, + 501 + ], + "score": 0.9, + "content": "I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { x } ; \\mathbf { y } )", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 487, + 360, + 503 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 360, + 491, + 371, + 500 + ], + "score": 0.86, + "content": "\\mathbf { z } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 487, + 463, + 503 + ], + "score": 1.0, + "content": "be a representation of", + "type": "text" + }, + { + "bbox": [ + 464, + 489, + 487, + 501 + ], + "score": 0.77, + "content": "\\mathbf { t } _ { 1 } ( \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 487, + 507, + 503 + ], + "score": 1.0, + "content": ". I f", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 499, + 307, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 117, + 511 + ], + "score": 0.83, + "content": "\\mathbf { z } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 499, + 180, + 514 + ], + "score": 1.0, + "content": "is sufficient for", + "type": "text" + }, + { + "bbox": [ + 180, + 501, + 204, + 512 + ], + "score": 0.91, + "content": "{ \\bf t } _ { 2 } ( { \\bf x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 499, + 225, + 514 + ], + "score": 1.0, + "content": "then", + "type": "text" + }, + { + "bbox": [ + 226, + 501, + 302, + 512 + ], + "score": 0.91, + "content": "I ( \\mathbf { x } ; \\mathbf { y } ) = I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 499, + 307, + 514 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 487, + 507, + 514 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 522, + 155, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 520, + 157, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 157, + 535 + ], + "score": 1.0, + "content": "Hypothesis:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 520, + 157, + 535 + ] + }, + { + "type": "text", + "bbox": [ + 118, + 543, + 306, + 555 + ], + "lines": [ + { + "bbox": [ + 118, + 542, + 305, + 557 + ], + "spans": [ + { + "bbox": [ + 118, + 543, + 137, + 555 + ], + "score": 0.74, + "content": "\\left( H _ { 1 } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 542, + 297, + 557 + ], + "score": 1.0, + "content": "Independence relations determined by", + "type": "text" + }, + { + "bbox": [ + 297, + 544, + 305, + 554 + ], + "score": 0.81, + "content": "\\mathcal { G }", + "type": "inline_equation" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 118, + 542, + 305, + 557 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 116, + 562, + 293, + 576 + ], + "lines": [ + { + "bbox": [ + 116, + 562, + 293, + 576 + ], + "spans": [ + { + "bbox": [ + 116, + 562, + 293, + 576 + ], + "score": 0.88, + "content": "( H _ { 2 } ) \\ I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { t } _ { 2 } ( \\mathbf { x } ) ; \\mathbf { y } ) = I ( \\mathbf { x } ; \\mathbf { y } )", + "type": "interline_equation", + "image_path": "496a7fb0073d42ff48f075ed75200d9b8840e0f16727d4ddf591401882b7c99f.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 116, + 562, + 293, + 576 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 584, + 137, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 138, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 138, + 597 + ], + "score": 1.0, + "content": "Thesis:", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16, + "bbox_fs": [ + 106, + 583, + 138, + 597 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 118, + 604, + 342, + 619 + ], + "lines": [ + { + "bbox": [ + 118, + 604, + 342, + 619 + ], + "spans": [ + { + "bbox": [ + 118, + 604, + 342, + 619 + ], + "score": 0.83, + "content": "( T _ { 1 } ) \\ I ( \\mathbf { t } _ { 1 } ( \\mathbf { x } ) ; \\mathbf { t } _ { 2 } ( \\mathbf { x } ) | \\mathbf { z } _ { 1 } ) = 0 \\ \\Longrightarrow \\ I ( \\mathbf { x } ; \\mathbf { y } ) = I ( \\mathbf { y } ; \\mathbf { z } _ { 1 } )", + "type": "interline_equation", + "image_path": "d25e845bdad7db36abf406c26684a05415e5bc27b71d32929b6b629be07ef8cc.jpg" + } + ] + } + ], + "index": 17, + "virtual_lines": [ + { + "bbox": [ + 118, + 604, + 342, + 619 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 632, + 506, + 679 + ], + "lines": [ + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 161, + 646 + ], + "score": 1.0, + "content": "Proof. 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The value of", + "type": "text" + }, + { + "bbox": [ + 277, + 711, + 284, + 721 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "is then kept fixed to one until the end of the training", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 141, + 721, + 268, + 732 + ], + "spans": [ + { + "bbox": [ + 141, + 721, + 268, + 732 + ], + "score": 1.0, + "content": "procedure (500,000 iterations).", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38, + "bbox_fs": [ + 134, + 581, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 133, + 82, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 132, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 132, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "• Evaluation: All natural images are used as both training sets and retrieval galleries. 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For having a fair comparison other methods that rely on binary hashing", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 141, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "(Liu et al., 2017; Zhang et al., 2018), we used Hamming distance on a binarized represen-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 141, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 141, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "tation (obtained by applying iterative quantization Gong et al. (2013) on our real valued", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 140, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 140, + 137, + 370, + 151 + ], + "score": 1.0, + "content": "representation). We report the mean average precision", + "type": "text" + }, + { + "bbox": [ + 370, + 138, + 416, + 148 + ], + "score": 0.57, + "content": "( \\mathrm { m A P } @ \\mathrm { a l l } )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "and precision at top-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 148, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 141, + 148, + 201, + 161 + ], + "score": 1.0, + "content": "rank 200 (Prec", + "type": "text" + }, + { + "bbox": [ + 202, + 149, + 226, + 159 + ], + "score": 0.4, + "content": "@ 2 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 148, + 505, + 161 + ], + "score": 1.0, + "content": ") Su et al. 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The decoders used for the VAE experiments", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 142, + 699, + 505, + 710 + ], + "spans": [ + { + "bbox": [ + 142, + 699, + 505, + 710 + ], + "score": 1.0, + "content": "also consist of the networks of the same size. 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The value of", + "type": "text" + }, + { + "bbox": [ + 468, + 94, + 475, + 105 + ], + "score": 0.85, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "is then", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 140, + 102, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 140, + 102, + 235, + 118 + ], + "score": 1.0, + "content": "kept constant until the", + "type": "text" + }, + { + "bbox": [ + 235, + 104, + 281, + 115 + ], + "score": 0.88, + "content": "1 , 0 0 0 { , } 0 0 0 ^ { \\mathrm { t h } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 102, + 506, + 118 + ], + "score": 1.0, + "content": "iteration. 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(2019); Tian et al. (2019); Bachman et al. (2019); van den", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 142, + 154, + 505, + 165 + ], + "spans": [ + { + "bbox": [ + 142, + 154, + 505, + 165 + ], + "score": 1.0, + "content": "Oord et al. (2018). Each logistic regression is trained 5 different balanced splits of the", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 164, + 506, + 177 + ], + "spans": [ + { + "bbox": [ + 141, + 164, + 506, + 177 + ], + "score": 1.0, + "content": "training set for different percentages of training examples, ranging from 1 example per la-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 175, + 505, + 188 + ], + "spans": [ + { + "bbox": [ + 141, + 175, + 505, + 188 + ], + "score": 1.0, + "content": "bel to the whole training set. The accuracy reported in this work has been computed on", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 186, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 141, + 186, + 505, + 198 + ], + "score": 1.0, + "content": "the disjoint test set. Mean and standard deviation are computed according to the 5 different", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 197, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 141, + 197, + 505, + 209 + ], + "score": 1.0, + "content": "subsets used for training the logistic regression. 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The final values for", + "type": "text" + }, + { + "bbox": [ + 411, + 274, + 441, + 286 + ], + "score": 0.93, + "content": "I ( \\mathbf { x } ; \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 274, + 459, + 286 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 459, + 274, + 489, + 286 + ], + "score": 0.93, + "content": "I ( \\mathbf { y } ; \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 274, + 505, + 286 + ], + "score": 1.0, + "content": "are", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 285, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 141, + 285, + 505, + 296 + ], + "score": 1.0, + "content": "computed by averaging the mutual information estimation on the whole dataset. In order", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 296, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 141, + 296, + 393, + 308 + ], + "score": 1.0, + "content": "to reduce the variance of the estimator, the lowest and highest", + "type": "text" + }, + { + "bbox": [ + 393, + 296, + 408, + 306 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 296, + 505, + 308 + ], + "score": 1.0, + "content": "are removed before av-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 307, + 506, + 320 + ], + "spans": [ + { + "bbox": [ + 141, + 307, + 506, + 320 + ], + "score": 1.0, + "content": "eraging. This practical detail makes the estimation more consistent and less susceptible to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 318, + 235, + 329 + ], + "spans": [ + { + "bbox": [ + 141, + 318, + 235, + 329 + ], + "score": 1.0, + "content": "numerical instabilities.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 109, + 342, + 272, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 340, + 275, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 275, + 356 + ], + "score": 1.0, + "content": "G.4.1 RESULTS AND VISUALIZATION", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 108, + 362, + 503, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 505, + 375 + ], + "score": 1.0, + "content": "In this section we include additional quantitative results and visualizations which refer to the single-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 373, + 307, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 373, + 307, + 385 + ], + "score": 1.0, + "content": "view MNIST experiments reported in section 5.2.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 389, + 505, + 468 + ], + "lines": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 106, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "Table 2 reports the quantitative results used for to produce the visualizations reported in Figure 4,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 402, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 505, + 413 + ], + "score": 1.0, + "content": "including the comparison between the performance resulting from different mutual information esti-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 412, + 504, + 424 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 504, + 424 + ], + "score": 1.0, + "content": "mators. 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ModelI(x; z) [nats]I(z;y)[nats]Test Accuracy [%]
10 Ex50Ex3750Ex60000 Ex
VAE (beta=0)12.5 ± 0.72.3± 0.243.8±1.665.6±3.389.0± 0.491.3 ±0.1
VAE (beta=4)7.5 ±1.02.0±0.255.9 ± 2.681.4 ± 4.094.2 ± 0.396.0 ± 0.2
VAE (beta=8)3.0 ± 0.51.0 ± 0.143.8 ±2.861.1 ± 4.881.9 ± 1.187.2 ± 0.6
InfoMax (INCE)12.8 ± 0.52.3 ±0.225.4 ± 1.939.6 ± 3.369.2 ± 0.774.6 ± 0.6
InfoMax(IJs)13.7 ± 0.72.2 ±0.235.0± 2.852.5 ± 2.874.4 ± 1.178.2 ±1.2
MV-InfoMax(INCE)12.2 ± 0.72.3 ±0.250.2 ± 3.675.8±3.894.6 ± 0.496.5 ± 0.1
MV-InfoMax (IJs)11.1 ± 1.02.3 ± 0.254.0 ± 6.178.3 ± 4.494.1 ± 0.395.90 ± 0.08
MIB (β=1,INCE)4.6 ± 0.72.1 ± 0.281.8 ± 5.092.7 ± 0.997.19 ± 0.0897.75 ± 0.05
MIB(β=1,Is)2.4 ± 0.22.1 ± 0.297.1 ± 0.297.2 ± 0.297.70 ± 0.0697.82 ± 0.01
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ModelI(x; z) [nats]I(z;y)[nats]Test Accuracy [%]
10 Ex50Ex3750Ex60000 Ex
VAE (beta=0)12.5 ± 0.72.3± 0.243.8±1.665.6±3.389.0± 0.491.3 ±0.1
VAE (beta=4)7.5 ±1.02.0±0.255.9 ± 2.681.4 ± 4.094.2 ± 0.396.0 ± 0.2
VAE (beta=8)3.0 ± 0.51.0 ± 0.143.8 ±2.861.1 ± 4.881.9 ± 1.187.2 ± 0.6
InfoMax (INCE)12.8 ± 0.52.3 ±0.225.4 ± 1.939.6 ± 3.369.2 ± 0.774.6 ± 0.6
InfoMax(IJs)13.7 ± 0.72.2 ±0.235.0± 2.852.5 ± 2.874.4 ± 1.178.2 ±1.2
MV-InfoMax(INCE)12.2 ± 0.72.3 ±0.250.2 ± 3.675.8±3.894.6 ± 0.496.5 ± 0.1
MV-InfoMax (IJs)11.1 ± 1.02.3 ± 0.254.0 ± 6.178.3 ± 4.494.1 ± 0.395.90 ± 0.08
MIB (β=1,INCE)4.6 ± 0.72.1 ± 0.281.8 ± 5.092.7 ± 0.997.19 ± 0.0897.75 ± 0.05
MIB(β=1,Is)2.4 ± 0.22.1 ± 0.297.1 ± 0.297.2 ± 0.297.70 ± 0.0697.82 ± 0.01
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When the amount of corruption approaches", + "type": "text" + }, + { + "bbox": [ + 346, + 414, + 370, + 424 + ], + "score": 0.87, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 412, + 505, + 427 + ], + "score": 1.0, + "content": ", the mutual redundancy assump-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "tion is clearly violated, and the performances of MIB deteriorate. In the initial part of the transi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 434, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 104, + 434, + 506, + 449 + ], + "score": 1.0, + "content": "tions between the two regimes (which corresponds to extremely low probability of corruption) the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 446, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 505, + 460 + ], + "score": 1.0, + "content": "MIB models drops some label information that is quickly re-gained when pixel corruption becomes", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 470 + ], + "score": 1.0, + "content": "more frequent. 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V1∈R4096V2 ∈R4096y∈[125]MethodmAP@allPrec@200
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ModelI(x; z) [nats]I(z;y)[nats]Test Accuracy [%]
10 Ex50Ex3750Ex60000 Ex
VAE (beta=0)12.5 ± 0.72.3± 0.243.8±1.665.6±3.389.0± 0.491.3 ±0.1
VAE (beta=4)7.5 ±1.02.0±0.255.9 ± 2.681.4 ± 4.094.2 ± 0.396.0 ± 0.2
VAE (beta=8)3.0 ± 0.51.0 ± 0.143.8 ±2.861.1 ± 4.881.9 ± 1.187.2 ± 0.6
InfoMax (INCE)12.8 ± 0.52.3 ±0.225.4 ± 1.939.6 ± 3.369.2 ± 0.774.6 ± 0.6
InfoMax(IJs)13.7 ± 0.72.2 ±0.235.0± 2.852.5 ± 2.874.4 ± 1.178.2 ±1.2
MV-InfoMax(INCE)12.2 ± 0.72.3 ±0.250.2 ± 3.675.8±3.894.6 ± 0.496.5 ± 0.1
MV-InfoMax (IJs)11.1 ± 1.02.3 ± 0.254.0 ± 6.178.3 ± 4.494.1 ± 0.395.90 ± 0.08
MIB (β=1,INCE)4.6 ± 0.72.1 ± 0.281.8 ± 5.092.7 ± 0.997.19 ± 0.0897.75 ± 0.05
MIB(β=1,Is)2.4 ± 0.22.1 ± 0.297.1 ± 0.297.2 ± 0.297.70 ± 0.0697.82 ± 0.01
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Modern neural networks are known to suffer under this setting, as they quickly forget previously acquired knowledge. To address such catastrophic forgetting, many continual learning methods implement different types of experience replay, re-learning on past data stored in a small buffer known as episodic memory. In this work, we complement experience replay with a meta-learning technique that we call “anchoring”: the learner updates its knowledge on the current task, while keeping predictions on some anchor points of past tasks intact. These anchor points are learned using gradient-based optimization as to maximize forgetting of the current task, in hindsight, when the learner is fine-tuned on the episodic memory of past tasks. Experiments on several supervised learning benchmarks for continual learning demonstrate that our approach improves the standard experience replay in terms of both accuracy and forgetting metrics and for various sizes of episodic memories. + +# 1 INTRODUCTION + +This manuscript studies the problem of continual learning, where a machine learning model experiences a sequence of tasks. Each of these tasks is presented as a stream of input-output pairs, where each pair is drawn from the corresponding task probability distribution. Since the length of the learning experience is not specified a-priori, the learner can only assume a single pass over the data, and store nothing but a few examples into a fixed-size episodic memory. At all times during the lifetime of the model, predictions on examples from all tasks can be requested. Addressing continual learning is an important research problem, since it would waive the long-obsolete assumption of “identically and independently distributed data” impeding progress towards artificial intelligence, and allow the deployment of models learning in-the-wild. However, continual learning presents one major challenge, catastrophic forgetting (McCloskey & Cohen, 1989). That is, as the learner experiences new tasks, it quickly forgets previously acquired knowledge. This is a hindrance for state-of-the-art deep learning models, where all parameters are updated after observing each example. + +Continual learning has received increasing attention from the scientific community during the last decade. The state of the art in algorithms for continual learning fall into three categories. First, regularization-based approaches reduce forgetting by restricting the updates in parameters that were important for previous tasks (Kirkpatrick et al., 2016; Aljundi et al., 2018; Chaudhry et al., 2018; Nguyen et al., 2018). However, when the number of tasks are large, the regularization of past tasks becomes obsolete, leading to representation drift (Titsias et al., 2019). Second, modular approaches (Rusu et al., 2016; Lee et al., 2017) add new modules to the learner as new tasks are learned. While modular architectures overcome forgetting by design, the memory complexity of these approaches scales with the number of tasks. Third, memory-based methods (Lopez-Paz & Ranzato, 2017; Hayes et al., 2018; Isele & Cosgun, 2018; Riemer et al., 2019; Chaudhry et al., 2019a), store a few examples from past tasks in a so-called episodic memory, to be revisited when training for a new task. Memory-based methods are the reigning state of the art, but remain a far-cry from the achievable performance by a simple multi-task learning baseline accessing all data at once. Despite intense research efforts, such gap in performance renders the problem of continual learning an open research question. + +Contribution We propose Hindsight Anchor Learning (HAL), a continual learning approach to improve the performance of memory-based continual learning algorithms. HAL leverages metalearning to regularize the training objective with one representational point per class per task, called anchor. Since it is desirable to preserve the performance of a learner during its lifetime, HAL constructs anchors as to maximize forgetting on the current task throughout the learning experience. We estimate the amount of forgetting that the learner would suffer on those anchors when learning future tasks in hindsight: that is, by measuring forgetting on a temporary predictor that has been fine-tuned on the episodic memories of past tasks. Then, the main parameter update of HAL is to minimize the loss on the currently observed mini-batch, while keeping the predictions at all anchors invariant. + +Results We compare HAL to EWC (Kirkpatrick et al., 2016), AGEM (Chaudhry et al., 2019a), experience replay (Hayes et al., 2018; Riemer et al., 2019), and MER (Riemer et al., 2019), across four standard benchmarks in continual learning (MNIST permutations, MNIST rotations, split CIFAR100, and split miniImageNet). In these experiments, HAL achieves state-of-the-art performance, improving accuracy by $5 \%$ and reducing forgetting by $20 \%$ . We show that these results hold for various sizes of episodic memories (between 1 and 5 examples per class per task). Finally, we perform an ablation study to show that learning anchors in hindsight is a critical factor to achieve state-of-the-art performance. + +We now begin our exposition by reviewing the continual learning setup. The rest of the manuscript then presents our new algorithm HAL (Section 3), showcases its empirical performance (Section 4), surveys the related literature (Section 5), and offers some concluding remarks (Section 6). + +# 2 CONTINUAL LEARNING SETUP + +In continual learning, we experience a stream of data triplets $( x _ { i } , y _ { i } , t _ { i } )$ , each containing an input $x _ { i }$ , a target $y _ { i }$ , and a task identifier $t _ { i } \in \mathcal { T } = \{ 1 , . . . , T \}$ . Each input-target pair $( x _ { i } , y _ { i } ) \in \mathcal { X } \times \mathcal { Y } _ { t _ { i } }$ is an identically and independently distributed example drawn from some unknown distribution $P ( X _ { t _ { i } } , Y _ { t _ { i } } )$ , representing the $t _ { i }$ -th learning task. Without loss of generality, we assume that the tasks are experienced in order $( t _ { i } \leq t _ { j }$ for all $i \leq j$ ), and that the number of total tasks $T$ is not known a priori. Under this setup, our goal is to estimate a predictor $f = ( w \circ \phi ) : \mathcal { X } \times \mathcal { T } \mathcal { Y }$ , composed by a featurizer $\phi : \mathcal { X } \to \mathcal { H }$ and a classifier $w : \mathcal { H } \to \mathcal { V }$ , that minimizes the multi-task error + +$$ +\frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { E } _ { ( x , y ) \sim P _ { t } } \left[ \ell ( f ( x , t ) , y ) \right] , +$$ + +where $\mathcal { V } = \cup _ { t \in T } \mathcal { V } _ { t }$ , and $\ell : \mathcal { V } \times \mathcal { V } \mathbb { R }$ is a suitable loss function. + +In the sequel, and similarly to prior literature in continual learning (Lopez-Paz & Ranzato, 2017; Hayes et al., 2018; Riemer et al., 2019; Chaudhry et al., 2019a), we consider streams of data, which are experienced only once. Therefore, the learner cannot store or revisit any but a small amount of data triplets chosen to be stored in its episodic memory $\mathcal { M }$ . More specifically, we consider tiny “ring” episodic memories, which contain the last $m$ observed examples per class for each of the experienced tasks, where $m \in \{ 1 , 3 , 5 \}$ . That is, considering as variables the number of experienced tasks $t$ and examples $n$ , we study continual learning algorithms with a $O ( t )$ memory footprint. + +Following Lopez-Paz & Ranzato (2017), we monitor two statistics to evaluate the quality of continual learning algorithms: final average accuracy, and final maximum forgetting. First, the final average accuracy of a predictor is defined as + +$$ +\mathrm { A c c u r a c y } = \frac { 1 } { T } \sum _ { j = 1 } ^ { T } a _ { T , j } , +$$ + +where $a _ { i , j }$ denotes the test accuracy on task $j$ after the model finished experiencing task $i$ . That is, the final average accuracy measures the test performance of the predictor at each of the tasks after the continual learning experience has finished. Second, the final maximum forgetting is defined as + +$$ +\mathrm { F o r g e t t i n g } = \frac { 1 } { T - 1 } \sum _ { j = 1 } ^ { T - 1 } \operatorname* { m a x } _ { l \in \{ 1 , \dots , T - 1 \} } ( a _ { l , j } - a _ { T , j } ) , +$$ + +that is, the decrease in performance at each of the tasks between their peak accuracy and their accuracy after the continual learning experience has finished. + +Finally, following Chaudhry et al. (2019a), we use the first $k < T$ tasks to cross-validate the hyperparameters of each of the considered continual learning algorithms. These first $k$ tasks are not considered when computing the final average accuracy and maximum forgetting metrics. + +# 3 HINSDIGHT ANCHOR LEARNING (HAL) + +Our starting point to describe our proposed continual learning algorithm is the current state of the art methods based on experience replay (Hayes et al., 2018; Riemer et al., 2019; Chaudhry et al., 2019b). These are predictors $f _ { \theta }$ that, during training, store a small amount of past observed triplets into an episodic memory $\boldsymbol { \mathcal { M } } = \{ ( x ^ { \prime } , y ^ { \prime } , t ^ { \prime } ) \bar \}$ . When facing a new mini-batch of observations $\tilde { \boldsymbol { B } } : \bar { = } \left\{ ( x , y , t ) \right\}$ from task $t$ , they employ the rule $\Dot { \theta ^ { * } } \theta - \alpha \cdot \nabla _ { \theta } \ell ( B \cup B _ { \mathcal { M } } )$ to update their parameters, where + +$$ +\ell ( A ) = { \frac { 1 } { | { \mathcal { A } } | } } \sum _ { ( x , y , t ) \in A } \ell ( f _ { \theta } ( x , t ) , y ) +$$ + +denotes the average loss across a collection of triplets $\mathcal { A }$ , and $B _ { { \mathcal { M } } }$ is a mini-batch constructed by sampling triplets of previous tasks from $\mathcal { M }$ at random. In general, $p _ { { \mathcal { M } } }$ is constructed to have the same size as $\boldsymbol { B }$ , but can be smaller if the episodic memory $\mathcal { M }$ does not yet contain enough samples. + +In the previous, the episodic memory $\mathcal { M }$ reminds the predictor about how to perform at past tasks using only a very small amount of datum. As such, the behaviour of the predictor outside the data stored in $\mathcal { M }$ is not guaranteed, and subject to worsen. Because of this reason, we would like to be more conservative, and propose to further fix the behaviour of the predictor at a collection of carefully constructed anchor points $e _ { t ^ { \prime } }$ , one per class per past task $t ^ { \prime }$ , after a parameter update1. + +To implement this, we propose a two-step parameter update rule: + +$$ +\begin{array} { r l } & { \tilde { \theta } \gets \theta - \alpha \cdot \nabla _ { \theta } \ell ( \mathcal { B } \cup \mathcal { B } _ { \mathcal { M } } ) , } \\ & { \theta \gets \theta - \alpha \cdot \nabla _ { \theta } \left( \ell ( \mathcal { B } \cup \mathcal { B } _ { \mathcal { M } } ) + \lambda \cdot \displaystyle \sum _ { t ^ { \prime } < t } \big ( f _ { \theta } ( e _ { t ^ { \prime } } , t ^ { \prime } ) - f _ { \tilde { \theta } } ( e _ { t ^ { \prime } } , t ^ { \prime } ) \big ) ^ { 2 } \right) . } \end{array} +$$ + +The first step computes a temporary parameter vector $\tilde { \theta }$ by minimizing the loss at a minibatch from the current task $t$ , and the episodic memory of past tasks (this is the usual experience replay parameter update). The second step employs nested optimization to perform the final update of the parameter $\theta$ , which trades-off the minimization of $( a )$ the loss value at the current minibatch and the episodic memory, as well as $( b )$ the change in predictions at the anchor points for all past tasks. We use L2 over the unnormalized logits as a distance measure in Eq. 5, as we experimentally find it to be superior than cosine similarity or L1 distance. This two-step nested optimization bears a similarity to gradient-based meta-learning approach (Finn et al., 2017) where we restrict the inner update to only a single gradient step. + +Next, let us discuss how to choose the anchor points. Recall that our objective is to preserve the performance of the current task throughout the entire continual learning experience. Ideally, we would like to choose each $e _ { t }$ as a tool to maximally suppress the amount of forgetting about current task throughout the entire learning experience. For this, we are interested in letting $e _ { t }$ be the example from task $t$ on which the loss would increase maximally had training with future tasks been performed. Such $e _ { t }$ would capture instances where the current task would be forgotten the most in future training. Then, requiring the predictions to remain invariant at such $e _ { t }$ , by using Eq. 5, would minimize forgetting on the current task. Mathematically, the desirable $e _ { t }$ and its label $y _ { t }$ is given by: + +$$ +\left( e _ { t } , y _ { t } \right) \gets \underset { ( x , y ) \sim P _ { t } } { \arg \operatorname* { m a x } } \ \ell ( f _ { \theta _ { T } } ( x , t ) , y ) - \ell ( f _ { \theta _ { t } } ( x , t ) , y ) , +$$ + +where $\theta _ { t }$ is the parameter vector after training on task $t$ and $\theta _ { T }$ is the final parameter vector after the entire learning experience. Thus, keeping predictions intact on the pair $( e _ { t } , y _ { t } )$ above would + +maximally preserve the performance of task $t$ . However, the idealistic Eq. 6 requires access to $( a )$ the entire distribution $P _ { t }$ as to compute the maximization, and (b) access to all future distributions $t ^ { \prime } > t$ as to compute the final parameter vector $\theta _ { T }$ . Both are unrealistic assumptions. + +To circumvent $( a )$ , we can recast Eq. 6 as to learn $e _ { t }$ by initializing it at random and using $k$ gradient ascent updates for a given label $y _ { t }$ : + +$$ +\begin{array} { r } { e _ { t } \gets e _ { t } + \alpha \cdot \nabla _ { e _ { t } } \left( \underbrace { \ell ( f _ { \theta _ { T } } ( e _ { t } , t ) , y _ { t } ) - \ell ( f _ { \theta _ { t } } ( e _ { t } , t ) , y _ { t } ) } _ { \mathrm { F o r g e t i n g l o s s } } - \gamma \underbrace { ( \phi ( e _ { t } ) - \phi _ { t } ) ^ { 2 } } _ { \mathrm { M e a n e m b e d d i n g l o s s } } \right) , } \end{array} +$$ + +where we recall that $\phi$ denotes the feature extractor of the predictor, and $\phi _ { t }$ is the neural mean embedding (Smola et al., 2007) of all observed examples from task $t$ which regularizes the constructed $e _ { t }$ against the outliers. Since the feature extractor is updated after experiencing each data point, the mean embeddings $\phi _ { t }$ are computed as running averages. That is, after observing a minibatch $\boldsymbol { B } = \{ ( x , y , t ) \}$ of task $t$ , we update: + +$$ +\phi _ { t } \beta \cdot \phi _ { t } + ( 1 - \beta ) \frac { 1 } { | \mathcal { B } | } \sum _ { x \in \mathcal { B } } \phi ( x ) , +$$ + +where $\phi _ { t }$ is initialized to zero at the beginning of the learning experience. In our experiments, we learn one $e _ { t }$ per class for each task. We fix the $y _ { t }$ to the corresponding class label, and discard $\phi _ { t }$ after training on task $t$ . Learning $e _ { t }$ in this manner circumvents the requirement of storing the entire distribution $P _ { t }$ for a task. + +Still, Eq. 7 requires the parameter vector $\theta _ { T }$ , to be obtained in the distant future after all learning tasks have been experienced. To waive this impossible requirement, we will approximate the future by simulating the past. That is, instead of measuring the forgetting that would happen after the model is trained at future tasks, we will measure the forgetting that happens when the model is fine-tuned at past tasks. In this way, we say that forgetting is estimated in hindsight, using past experiences. More concretely, after training on task $t$ and obtaining the parameter vector $\theta _ { t }$ , we minimize the loss during one epoch on the episodic memory $\mathcal { M }$ to obtain the temporary parameter vector $\theta _ { \mathcal { M } }$ , and recast Eq. 7 as: + +$$ +e _ { t } \gets e _ { t } + \alpha \cdot \nabla _ { e _ { t } } \left( \ell ( f _ { \theta _ { M } } ( e _ { t } , t ) , y _ { t } ) - \ell ( f _ { \theta _ { t } } ( e _ { t } , t ) , y _ { t } ) - \gamma ( \phi ( e _ { t } ) - \phi _ { t } ) ^ { 2 } \right) . +$$ + +Our experiments show that the proposed approximation of future by replaying past data in hindsight reduces forgetting by $20 \%$ when compared to a standard experience replay baseline. + +This completes the description of our proposed algorithm for continual learning, which combines experience replay with anchors learned in hindsight. We call our approach Hindsight Anchor Learning (HAL) and summarize the entire learning process as follows: + +# Hindsight Anchor Learning (HAL) + +• Initialize $\theta \sim P ( \theta )$ and $\{ e _ { t } \sim P ( e ) \} _ { t = 1 } ^ { T }$ from normal distributions $P ( \theta )$ and $P ( e )$ + +• Initialize $\mathcal { M } = \{ \}$ • For each task $t = 1 , \dots , T$ : + +Initialize $\phi _ { t } = 0$ +– For each minibatch $\boldsymbol { B }$ from task $t$ : $^ *$ Sample $B _ { { \mathcal { M } } }$ from $\mathcal { M }$ $^ *$ Update $\theta$ using Eq. 5 $^ *$ Update $\phi _ { t }$ using Eq. 8 $^ *$ Update $\mathcal { M }$ by adding $\boldsymbol { B }$ in a first-in first-out (FIFO) ring buffer +Fine-tune on $\mathcal { M }$ to obtain $\theta _ { \mathcal { M } }$ +Build $e _ { t }$ using Eq. $9 k$ times +Discard $\phi _ { t }$ + +• Return θ. + +# 4 EXPERIMENTS + +We now evaluate the performance of HAL against a variety of state-of-the-art methods, across four popular continual learning benchmarks. + +# 4.1 DATESETS AND TASKS + +We perform experiments on four popular benchmarks for continual learning. + +• Permuted MNIST is a variant of the MNIST dataset of handwritten digits (LeCun, 1998) where each task applies a fixed random pixel permutation to the original dataset. This benchmark contains 23 tasks, each with 1000 samples from 10 different classes. +• Rotated MNIST is another variant of MNIST, where each task applies a fixed random image rotation (between 0 and 180 degrees) to the original dataset. This benchmark contains 23 tasks, each with 1000 samples from 10 different classes. +• Split CIFAR is a variant of the CIFAR-100 dataset (Krizhevsky & Hinton, 2009; Zenke et al., 2017), where each task contains the data pertaining 5 random classes (without replacement) out of the total 100 classes. This benchmark contains 20 tasks, each with 250 samples per each of the 5 classes. +• Split miniImageNet is a variant of the ImageNet dataset (Russakovsky et al., 2015; Vinyals et al., 2016), containing a subset of images and classes from the original dataset. This benchmark contains 20 tasks, each with 250 samples per each of the 5 classes. + +For all datasets, the first 3 tasks are used for hyper-parameter optimization (grids available in Appendix B). The learners can perform multiple epochs on these three initial tasks, that are later discarded for evaluation. + +# 4.2 BASELINES + +We compare our proposed model HAL to the following baselines. + +• Finetune is a single model trained continually on the stream of data, without any regularization or episodic memory. +• EWC (Kirkpatrick et al., 2016) is a continual learning method that limits changes to parameters critical to past tasks, as measured by the Fisher information matrix. +• AGEM (Chaudhry et al., 2019a) is a continual learning method improving on (Lopez-Paz & Ranzato, 2017), which uses an episodic memory of parameter gradients to limit forgetting. +• MER (Riemer et al., 2019) is a continual learning method that combines episodic memories with meta-learning to limit forgetting. +• ER-Ring (Chaudhry et al., 2019b) is a continual learning method that uses a ring buffer as episodic memory. +• Multitask is an oracle baseline that has access to all data to optimize Eq. 1, useful to estimate an upper bound on the obtainable Accuracy (Eq. 2). +• Clone-and-finetune is an oracle baseline training one independent model per task, where the model for task $t ^ { \prime }$ is initialized by cloning the parameters of the model for task $t ^ { \prime } - 1$ . + +All baselines use the same neural network architectures: a perceptron with two hidden layers of 256 ReLU neurons in the MNIST experiments, and a ResNet18, with three times less feature maps across all layers, similar to Lopez-Paz & Ranzato (2017), their CIFAR and ImageNet experiments. The task identifiers are used to select the output head in the CIFAR and ImageNet experiments, while ignored in the MNIST experiments. Batch size is set to 10 for both the stream of data and episodic memories, across experiments and models. The size of episodic memories is set between 1 and 5 examples per class per task. The same unified code base runs all experiments, and is available at https://bit.ly/2mw8bsE.2 + +Table 1: Accuracy (Eq. 2) and Forgetting (Eq. 3) results of continual learning experiments. Averages and standard deviations are computed over five random seeds. When used, episodic memories contain up to one example per class per task. Last two rows are oracle baselines. + +
MethodPermuted MNISTRotated MNIST
AccuracyForgettingAccuracyForgetting
Finetune53.5 (±1.46)0.29 (±0.01)41.9 (±1.37)0.50 (±0.01)
EWC63.1 (±1.40)0.18 (±0.01)44.1 (±0.99)0.47 (±0.01)
AGEM62.1 (±1.39)0.21 (±0.01)50.9 (±0.92)0.40 (±0.01)
MER69.9 (±0.40)0.14 (±0.01)66.0 (±2.04)0.23 (±0.01)
ER-Ring70.2 (±0.56)0.12 (±0.01)65.9 (±0.41)0.24 (±0.01)
HAL (ours)73.6 (±0.31)0.09 (±0.01)68.4 (±0.72)0.21 (±0.01)
Clone-and-finetune Multitask81.4 (±0.35)0.087.5 (±0.11)0.0
83.00.083.30.0
MethodSplit CIFARSplit miniImageNet
AccuracyForgettingAccuracyForgetting
Finetune42.9 (±2.07)0.25 (±0.03)34.7 (±2.69)0.26 (±0.03)
EWC42.4 (±3.02)0.26 (±0.02)37.7 (±3.29)0.21 (±0.03)
AGEM54.9 (±2.92)0.14 (±0.03)48.2 (±2.49)0.13 (±0.02)
MER49.7 (±2.97)0.19 (±0.03)45.5 (±1.49)0.15 (±0.01)
ER-Ring56.2 (±1.93)0.13 (±0.01)49.0 (±2.61)0.12 (±0.02)
HAL (ours)60.4 (±0.54)0.10 (±0.01)51.6 (±2.02)0.10 (±0.01)
Clone-and-finetune60.3 (±0.55)0.050.3 (±1.00)0.0
Multitask68.30.063.50.0
+ +# 4.3 RESULTS + +Table 1 summarizes the main results of our experiments. First, our proposed HAL is the method achieving maximum Accuracy (Eq. 2) and minimal Forgetting (Eq. 3) at all benchmarks. This does not include oracle baselines Multitask (which has access to all data simultaneously) and Clone-andfinetune (which trains a separate model per task). Second, the relative gains from the second-best method ER-Ring to HAL are significant, confirming that the anchoring objective (Eq. 5) allows experience-replay methods to generalize better from the same amount of episodic memory. + +Figure 1 shows a more fine grained analysis of average accuracy as new tasks are learned on Permuted MNIST and Split CIFAR. HAL preserves the performance of a predictor more effectively than other baselines. + +Table 2 shows the Accuracy of methods that employ an episodic memory, when we allow the size of this memory to increase; 3 or 5 examples per class per task, resulting in a total memory size of 600 or 1000 for MNIST experiments, and 255 or 425 for CIFAR and ImageNet experiments. The corresponding numbers for Forgetting are given in Appendix Table 4. HAL outperforms all competitors at all benchmarks. + +Figure 2 provides the training time of all the continual learning baselines on MNIST benchmarks. Although HAL adds a overhead on top of experience replay baseline, it is significantly faster than, MER, another meta-learning approach to reduce forgetting. + +Finally, Table 3 summarizes the results of an ablation study to better understand the impact of anchor selection. We compare selecting anchors as random noise (Random-Anchors), as a random example for the associated class and task (Data-Anchors), or our proposed optimization in hindsight procedure (HAL). Our results confirm that optimizing anchors in hindsight is the most effective strategy in terms of both accuracy and forgetting metrics. + +![](images/f14896312758d6b8490d35a8d890359edf983ee590185c2c8239e80306f2822c.jpg) +Figure 1: Evolution of Accuracy (Eq. 2) as new tasks are learned. When used, episodic memories contain up to one example per class per task. + +Table 2: Accuracy (Eq. 2) results for large (3 to 5 examples per class per task) episodic memory sizes. Here we only compare methods that use an episodic memory. Averages and standard deviations are computed over five random seeds. + +
MethodPermuted MNISTRotated MNIST
|M|= 600|M|= 1000|M|= 600|M|= 1000
AGEM63.2 (±1.47)64.1 (±0.74)49.9 (±1.49)53.0 (±1.52)
MER74.9 (±0.49)78.3 (±0.19)76.5 (±0.30)77.3 (±1.13)
ER-Ring73.5 (±0.43)75.8 (±0.24)74.7 (±0.56)76.5 (±0.48)
HAL (ours)76.2 (±0.52)78.4 (±0.27)77.0 (±0.66)78.7 (±0.97)
MethodSplit CIFARSplit miniImageNet
|M| = 255|M| = 425M|= 255[M|= 425
AGEM56.9 (±3.45)59.9 (±2.64)51.6 (±2.69)54.3 (±1.56)
MER57.7 (±2.59)60.6 (±2.09)49.4 (±3.43)54.8 (±1.79)
ER-Ring60.9 (±1.44)62.6 (±1.77)53.5 (±1.42)54.2 (±3.23)
HAL (ours)62.9 (±1.49)64.4 (±2.15)56.5 (±0.87)57.2 (±1.54)
+ +# 5 RELATED WORK + +In continual learning (Ring, 1997), also called lifelong learning (Thrun, 1998), a learner addresses a sequence of changing tasks without storing the complete datasets of these tasks. This is in contrast to multitask learning (Caruana, 1997), where the learner assumes simultaneous access to data from all tasks. The main challenge in continual learning is to avoid catastrophic interference or forgetting (McCloskey & Cohen, 1989; McClelland et al., 1995; Goodfellow et al., 2013), that is, the learner forgetting previously acquired knowledge when learning new tasks. The state-of-the art methods in continual learning can be categorized into three classes. + +First, regularization approaches discourage updating parameters important for past tasks (Kirkpatrick et al., 2016; Aljundi et al., 2018; Nguyen et al., 2018; Zenke et al., 2017). While efficient in terms of memory and computation, these ap + +![](images/42ba534765d99259f9f9127f5c62fef49a83981dae81575a5750156689ee6117.jpg) +Figure 2: Training time (s) of MNIST experiments for the entire continual learning experience. MER and HAL both use meta-learning objectives to reduce forgetting. + +proaches suffer from brittleness due to feature drift as the number of tasks increases (Titsias et al., + +Table 3: Impact of anchor selection, where we compare random-noise anchors (Random-), one random example-per-class anchors (Data-), and our optimized anchor selection (HAL). + +
Anchor typePermuted MNISTSplit CIFAR
AccuracyForgettingAccuracyForgetting
ER-Ring70.20.1256.20.13
Random-Anchors72.70.1057.80.13
Data-Anchors73.20.1059.00.12
HAL (ours)73.60.0960.40.10
+ +2019). Additionally, these approaches are only effective when we can perform multiple passes over each dataset (Chaudhry et al., 2019a), a case deemed unrealistic in this work. + +Second, modular approaches use different parts of the prediction function for each new task (Fernando et al., 2017; Aljundi et al., 2017; Rosenbaum et al., 2018; Chang et al., 2018; Xu & Zhu, 2018; Ferran Alet, 2018). Modular approaches do not scale to a large number of tasks, as they require searching over combinatorial space of module architectures. Another modular approach (Rusu et al., 2016; Lee et al., 2017) adds new parts to the prediction function as new tasks are learned. By construction, modular approaches have zero forgetting, but their memory requirements increase with the number of tasks. + +Third, episodic memory approaches maintain and revisit a small episodic memory of datum from past tasks. In some of these methods (Li & Hoiem, 2016; Rebuffi et al., 2017), examples in the episodic memory are replayed and predictions are kept invariant by means of distillation (Hinton et al., 2014). In other approaches (Lopez-Paz & Ranzato, 2017; Chaudhry et al., 2019a; Aljundi et al., 2019b) the episodic memory is used as an optimization constraint that discourages increases in loss at past tasks. More recently, several works (Hayes et al., 2018; Riemer et al., 2019; Rolnick et al., 2018; Chaudhry et al., 2019b) have shown that directly optimizing the loss on the episodic memory, also known as experience replay, is cheaper than constraint-based approaches and improves prediction performance. Our contribution in this paper has been to improve experience replay methods with task anchors learned in hindsight. + +There are other definitions of continual learning, such as the one of task-free continual learning. The task-free formulation does not consider the notion of tasks, and instead works on undivided data streams (Aljundi et al., 2019a,b). We have focused on the task-based definition of continual learning and, similar to many recent works (Lopez-Paz & Ranzato, 2017; Hayes et al., 2018; Riemer et al., 2019; Chaudhry et al., 2019a), assumed that only a single pass through the data was possible. + +Finally, our gradient-based learning of anchors bears a similarity to (Simonyan et al., 2014) and (Wang et al., 2018). In Simonyan et al. (2014), the authors use gradient ascent on class scores to find saliency maps of a classification model. Contrary to them, our proposed hindsight learning objective optimizes for the forgetting metric, as reducing it is necessary while learning continually. Dataset distillation (Wang et al., 2018) proposes to encode the entire dataset in a few synthetic points at a given parameter vector by a gradient-based optimization process. Their method requires access to the entire dataset of a task for optimization purposes. We, instead, learn anchors in hindsight from the replay buffer of past tasks after training for current task. While Wang et al. (2018) aim to replicate the performance of the entire dataset from the synthetic points, we focus on reducing forgetting of an already learned task. + +# 6 CONCLUSION + +We introduced HAL, a meta-learning objective for continual learning. In our approach, we learn one “anchor point” per class per task, where predictions are requested to remain invariant. These anchors are learned using gradient-based optimization, and represent points that would maximize the forgetting of the current task throughout the entire learning experience. We simulate the forgetting that would happen during the learning of future tasks in hindsight, that is, by taking temporary gradient steps across a small episodic memory of past tasks. As shown in our experiments, anchoring in hindsight complements and improves the performance of continual learning methods based on experience replay, achieving a new state of the art on four standard continual learning benchmarks. + +# REFERENCES + +Rahaf Aljundi, Punarjay Chakravarty, and Tinne Tuytelaars. Expert gate: Lifelong learning with a network of experts. In CVPR, pp. 7120–7129, 2017. + +Rahaf Aljundi, Francesca Babiloni, Mohamed Elhoseiny, Marcus Rohrbach, and Tinne Tuytelaars. Memory aware synapses: Learning what (not) to forget. In ECCV, 2018. + +Rahaf Aljundi, Klaas Kelchtermans, and Tinne Tuytelaars. Task-free continual learning. In CVPR, pp. 11254–11263, 2019a. + +Rahaf Aljundi, Min Lin, Baptiste Goujaud, and Yoshua Bengio. Online continual learning with no task boundaries. arXiv preprint arXiv:1903.08671, 2019b. + +Rich Caruana. Multitask learning. Machine learning, 28(1):41–75, 1997. + +Michael Chang, Abhishek Gupta, Sergey Levine, and Thomas L. Griffiths. 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Continual learning through synaptic intelligence. In ICML, 2017. + +# APPENDIX + +In Section A, we are report the forgetting 3 metric when large episodic memories are used. Section B provides the grid considered for hyper-parameters. Section C provides pseudo-code for HAL. + +# A MORE RESULTS + +Table 4: Forgetting (Eq. 3) results for large (3 to 5 examples per class per task) episodic memory sizes. Here we only compare methods that use an episodic memory. Averages and standard deviations are computed over five random seeds. + +
MethodPermuted MNISTRotated MNIST
|M| = 600|M| = 1000|M|= 600|M|= 1000
AGEM0.20 (±0.01)0.19(±0.01)0.41 1 (±0.01)0.38 (±0.01)
MER0.14 (±0.01)0.09(±0.01)0.12(±0.01)0.11 (±0.01)
ER-Ring0.09 (±0.01)0.07 (±0.01)0.15 (±0.01)0.13 (±0.01)
HAL (ours)0.07 (±0.01)0.05 (±0.01)0.12 (±0.01)0.11 (±0.01)
+ +
MethodSplit CIFARSplit miniImageNet
M|= 255|M|= 425|M| = 255|M| = 425
AGEM0.13 (±0.03)0.10 (±0.02)0.10 (±0.02)0.08 (±0.01)
MER0.11 1(±0.01)0.09 (±0.02)0.12 (±0.02)0.07 (±0.01)
ER-Ring0.09 (±0.01)0.06 (±0.01)0.07 (±0.02)0.08 (±0.02)
HAL (ours)0.08 (±0.01)0.06 (±0.01)0.06 (±0.01)0.06 (±0.01)
+ +# B HYPER-PARAMETER SELECTION + +In this section, we report the hyper-parameters grid considered for experiments. The best values for different benchmarks are given in parenthesis. + +– learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] + +– learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] + +• Finetune – learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] + +• EWC – learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] – regularization: [0.1, 1, 10 (MNIST perm, rot, CIFAR, miniImageNet), 100, 1000] + +• AGEM – learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] +• MER +– learning rate: [0.003, 0.01, 0.03 (MNIST, CIFAR, miniImageNet), 0.1, 0.3, 1.0] +– within batch meta-learning rate: [0.01, 0.03, 0.1 (MNIST, CIFAR, miniImageNet), 0.3, 1.0] +– current batch learning rate multiplier: [1, 2, 5 (CIFAR, miniImageNet), 10 (MNIST)] + +# • ER-Ring + +– learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] + +• HAL + +– learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] +– regularization (λ): [0.01, 0.03, 0.1 (MNIST perm, rot), 0.3 (miniImageNet), 1 (CIFAR), 3, 10] +– mean embedding strength (γ): [0.01, 0.03, 0.1 (MNIST perm, rot, CIFAR, miniImageNet), 0.3, 1, 3, 10] + +# C HAL ALGORITHM + +Algorithm 1 provides pseudo-code for HAL. + +Algorithm 1 Training of HAL on sequential data $\mathcal { D } = \{ \mathcal { D } _ { 1 } , \cdot \cdot \cdot , \mathcal { D } _ { T } \}$ , with total replay buffer size ‘mem sz’, learning rate $\cdot _ { \alpha } ,$ , regularization strength $^ { \bullet } \lambda ^ { \bullet }$ , mean embedding decay $\cdot _ { \beta } ,$ , mean embedding strength $\cdot _ { \eta } ,$ . + +1: procedure $\mathrm { H A L } ( { \mathcal { D } } , \mathrm { m e m . s z } , \alpha , \lambda , \beta )$ +2: M ← {} ∗ mem sz +3: {e1, · · · , eT } ← {} +4: for t ∈ {1, · · · , T } do +5: φt ← \~0 +6: for B ∼ Dt do $\triangleright$ Sample a batch from current task +7: BM ∼ M $\triangleright$ Sample a batch from episodic memory +8: ˜θ ← θ − α · ∇θ \`(B ∪ BM) . Temporary parameter update +9: θ ← θ − α · ∇θ \`(B ∪ BM) + λ · Pt0 t$ as to compute the final parameter vector $\\theta _ { T }$ . Both are unrealistic assumptions. ", + "bbox": [ + 173, + 103, + 825, + 146 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "To circumvent $( a )$ , we can recast Eq. 6 as to learn $e _ { t }$ by initializing it at random and using $k$ gradient ascent updates for a given label $y _ { t }$ : ", + "bbox": [ + 169, + 152, + 825, + 181 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/ccb07e84729fec7c6eaa5d729a9476a83013e63f6780525686cdb2d6324e965c.jpg", + "text": "$$\n\\begin{array} { r } { e _ { t } \\gets e _ { t } + \\alpha \\cdot \\nabla _ { e _ { t } } \\left( \\underbrace { \\ell ( f _ { \\theta _ { T } } ( e _ { t } , t ) , y _ { t } ) - \\ell ( f _ { \\theta _ { t } } ( e _ { t } , t ) , y _ { t } ) } _ { \\mathrm { F o r g e t i n g l o s s } } - \\gamma \\underbrace { ( \\phi ( e _ { t } ) - \\phi _ { t } ) ^ { 2 } } _ { \\mathrm { M e a n e m b e d d i n g l o s s } } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 241, + 188, + 754, + 246 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where we recall that $\\phi$ denotes the feature extractor of the predictor, and $\\phi _ { t }$ is the neural mean embedding (Smola et al., 2007) of all observed examples from task $t$ which regularizes the constructed $e _ { t }$ against the outliers. Since the feature extractor is updated after experiencing each data point, the mean embeddings $\\phi _ { t }$ are computed as running averages. That is, after observing a minibatch $\\boldsymbol { B } = \\{ ( x , y , t ) \\}$ of task $t$ , we update: ", + "bbox": [ + 173, + 252, + 826, + 323 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/9dc3ae156a3a12b7b77d3577f826480fb6e574f1e3c38947613c4dd8fa61a1dd.jpg", + "text": "$$\n\\phi _ { t } \\beta \\cdot \\phi _ { t } + ( 1 - \\beta ) \\frac { 1 } { | \\mathcal { B } | } \\sum _ { x \\in \\mathcal { B } } \\phi ( x ) ,\n$$", + "text_format": "latex", + "bbox": [ + 377, + 330, + 619, + 367 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\phi _ { t }$ is initialized to zero at the beginning of the learning experience. In our experiments, we learn one $e _ { t }$ per class for each task. We fix the $y _ { t }$ to the corresponding class label, and discard $\\phi _ { t }$ after training on task $t$ . Learning $e _ { t }$ in this manner circumvents the requirement of storing the entire distribution $P _ { t }$ for a task. ", + "bbox": [ + 173, + 375, + 825, + 431 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Still, Eq. 7 requires the parameter vector $\\theta _ { T }$ , to be obtained in the distant future after all learning tasks have been experienced. To waive this impossible requirement, we will approximate the future by simulating the past. That is, instead of measuring the forgetting that would happen after the model is trained at future tasks, we will measure the forgetting that happens when the model is fine-tuned at past tasks. In this way, we say that forgetting is estimated in hindsight, using past experiences. More concretely, after training on task $t$ and obtaining the parameter vector $\\theta _ { t }$ , we minimize the loss during one epoch on the episodic memory $\\mathcal { M }$ to obtain the temporary parameter vector $\\theta _ { \\mathcal { M } }$ , and recast Eq. 7 as: ", + "bbox": [ + 173, + 436, + 825, + 549 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c74cd28e96eee403d915a48dd79920e5ac0be4f76b421f717a9566ae1da865dc.jpg", + "text": "$$\ne _ { t } \\gets e _ { t } + \\alpha \\cdot \\nabla _ { e _ { t } } \\left( \\ell ( f _ { \\theta _ { M } } ( e _ { t } , t ) , y _ { t } ) - \\ell ( f _ { \\theta _ { t } } ( e _ { t } , t ) , y _ { t } ) - \\gamma ( \\phi ( e _ { t } ) - \\phi _ { t } ) ^ { 2 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 550, + 746, + 570 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Our experiments show that the proposed approximation of future by replaying past data in hindsight reduces forgetting by $20 \\%$ when compared to a standard experience replay baseline. ", + "bbox": [ + 173, + 580, + 823, + 609 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "This completes the description of our proposed algorithm for continual learning, which combines experience replay with anchors learned in hindsight. We call our approach Hindsight Anchor Learning (HAL) and summarize the entire learning process as follows: ", + "bbox": [ + 174, + 614, + 825, + 659 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Hindsight Anchor Learning (HAL) ", + "text_level": 1, + "bbox": [ + 191, + 672, + 439, + 689 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Initialize $\\theta \\sim P ( \\theta )$ and $\\{ e _ { t } \\sim P ( e ) \\} _ { t = 1 } ^ { T }$ from normal distributions $P ( \\theta )$ and $P ( e )$ ", + "bbox": [ + 235, + 699, + 792, + 715 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Initialize $\\mathcal { M } = \\{ \\}$ • For each task $t = 1 , \\dots , T$ : ", + "bbox": [ + 236, + 719, + 372, + 733 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 236, + 738, + 431, + 752 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Initialize $\\phi _ { t } = 0$ \n– For each minibatch $\\boldsymbol { B }$ from task $t$ : $^ *$ Sample $B _ { { \\mathcal { M } } }$ from $\\mathcal { M }$ $^ *$ Update $\\theta$ using Eq. 5 $^ *$ Update $\\phi _ { t }$ using Eq. 8 $^ *$ Update $\\mathcal { M }$ by adding $\\boldsymbol { B }$ in a first-in first-out (FIFO) ring buffer \nFine-tune on $\\mathcal { M }$ to obtain $\\theta _ { \\mathcal { M } }$ \nBuild $e _ { t }$ using Eq. $9 k$ times \nDiscard $\\phi _ { t }$ ", + "bbox": [ + 261, + 757, + 720, + 898 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "• Return θ. ", + "bbox": [ + 235, + 904, + 312, + 916 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 326, + 117 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now evaluate the performance of HAL against a variety of state-of-the-art methods, across four popular continual learning benchmarks. ", + "bbox": [ + 174, + 133, + 823, + 161 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.1 DATESETS AND TASKS ", + "text_level": 1, + "bbox": [ + 176, + 179, + 370, + 193 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We perform experiments on four popular benchmarks for continual learning. ", + "bbox": [ + 174, + 204, + 673, + 219 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• Permuted MNIST is a variant of the MNIST dataset of handwritten digits (LeCun, 1998) where each task applies a fixed random pixel permutation to the original dataset. This benchmark contains 23 tasks, each with 1000 samples from 10 different classes. \n• Rotated MNIST is another variant of MNIST, where each task applies a fixed random image rotation (between 0 and 180 degrees) to the original dataset. This benchmark contains 23 tasks, each with 1000 samples from 10 different classes. \n• Split CIFAR is a variant of the CIFAR-100 dataset (Krizhevsky & Hinton, 2009; Zenke et al., 2017), where each task contains the data pertaining 5 random classes (without replacement) out of the total 100 classes. This benchmark contains 20 tasks, each with 250 samples per each of the 5 classes. \n• Split miniImageNet is a variant of the ImageNet dataset (Russakovsky et al., 2015; Vinyals et al., 2016), containing a subset of images and classes from the original dataset. This benchmark contains 20 tasks, each with 250 samples per each of the 5 classes. ", + "bbox": [ + 217, + 232, + 825, + 428 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For all datasets, the first 3 tasks are used for hyper-parameter optimization (grids available in Appendix B). The learners can perform multiple epochs on these three initial tasks, that are later discarded for evaluation. ", + "bbox": [ + 176, + 440, + 823, + 482 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4.2 BASELINES", + "text_level": 1, + "bbox": [ + 174, + 500, + 294, + 513 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We compare our proposed model HAL to the following baselines. ", + "bbox": [ + 173, + 525, + 606, + 540 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "• Finetune is a single model trained continually on the stream of data, without any regularization or episodic memory. \n• EWC (Kirkpatrick et al., 2016) is a continual learning method that limits changes to parameters critical to past tasks, as measured by the Fisher information matrix. \n• AGEM (Chaudhry et al., 2019a) is a continual learning method improving on (Lopez-Paz & Ranzato, 2017), which uses an episodic memory of parameter gradients to limit forgetting. \n• MER (Riemer et al., 2019) is a continual learning method that combines episodic memories with meta-learning to limit forgetting. \n• ER-Ring (Chaudhry et al., 2019b) is a continual learning method that uses a ring buffer as episodic memory. \n• Multitask is an oracle baseline that has access to all data to optimize Eq. 1, useful to estimate an upper bound on the obtainable Accuracy (Eq. 2). \n• Clone-and-finetune is an oracle baseline training one independent model per task, where the model for task $t ^ { \\prime }$ is initialized by cloning the parameters of the model for task $t ^ { \\prime } - 1$ . ", + "bbox": [ + 217, + 553, + 825, + 776 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "All baselines use the same neural network architectures: a perceptron with two hidden layers of 256 ReLU neurons in the MNIST experiments, and a ResNet18, with three times less feature maps across all layers, similar to Lopez-Paz & Ranzato (2017), their CIFAR and ImageNet experiments. The task identifiers are used to select the output head in the CIFAR and ImageNet experiments, while ignored in the MNIST experiments. Batch size is set to 10 for both the stream of data and episodic memories, across experiments and models. The size of episodic memories is set between 1 and 5 examples per class per task. The same unified code base runs all experiments, and is available at https://bit.ly/2mw8bsE.2 ", + "bbox": [ + 174, + 790, + 825, + 901 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/c60c7ce517d0161988c6df9386d96ab0cfcb3f5de0d2a6a4bee5e5e607e5e01f.jpg", + "table_caption": [ + "Table 1: Accuracy (Eq. 2) and Forgetting (Eq. 3) results of continual learning experiments. Averages and standard deviations are computed over five random seeds. When used, episodic memories contain up to one example per class per task. Last two rows are oracle baselines. " + ], + "table_footnote": [], + "table_body": "
MethodPermuted MNISTRotated MNIST
AccuracyForgettingAccuracyForgetting
Finetune53.5 (±1.46)0.29 (±0.01)41.9 (±1.37)0.50 (±0.01)
EWC63.1 (±1.40)0.18 (±0.01)44.1 (±0.99)0.47 (±0.01)
AGEM62.1 (±1.39)0.21 (±0.01)50.9 (±0.92)0.40 (±0.01)
MER69.9 (±0.40)0.14 (±0.01)66.0 (±2.04)0.23 (±0.01)
ER-Ring70.2 (±0.56)0.12 (±0.01)65.9 (±0.41)0.24 (±0.01)
HAL (ours)73.6 (±0.31)0.09 (±0.01)68.4 (±0.72)0.21 (±0.01)
Clone-and-finetune Multitask81.4 (±0.35)0.087.5 (±0.11)0.0
83.00.083.30.0
MethodSplit CIFARSplit miniImageNet
AccuracyForgettingAccuracyForgetting
Finetune42.9 (±2.07)0.25 (±0.03)34.7 (±2.69)0.26 (±0.03)
EWC42.4 (±3.02)0.26 (±0.02)37.7 (±3.29)0.21 (±0.03)
AGEM54.9 (±2.92)0.14 (±0.03)48.2 (±2.49)0.13 (±0.02)
MER49.7 (±2.97)0.19 (±0.03)45.5 (±1.49)0.15 (±0.01)
ER-Ring56.2 (±1.93)0.13 (±0.01)49.0 (±2.61)0.12 (±0.02)
HAL (ours)60.4 (±0.54)0.10 (±0.01)51.6 (±2.02)0.10 (±0.01)
Clone-and-finetune60.3 (±0.55)0.050.3 (±1.00)0.0
Multitask68.30.063.50.0
", + "bbox": [ + 220, + 155, + 774, + 506 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.3 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 553, + 279, + 568 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Table 1 summarizes the main results of our experiments. First, our proposed HAL is the method achieving maximum Accuracy (Eq. 2) and minimal Forgetting (Eq. 3) at all benchmarks. This does not include oracle baselines Multitask (which has access to all data simultaneously) and Clone-andfinetune (which trains a separate model per task). Second, the relative gains from the second-best method ER-Ring to HAL are significant, confirming that the anchoring objective (Eq. 5) allows experience-replay methods to generalize better from the same amount of episodic memory. ", + "bbox": [ + 173, + 588, + 825, + 672 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 1 shows a more fine grained analysis of average accuracy as new tasks are learned on Permuted MNIST and Split CIFAR. HAL preserves the performance of a predictor more effectively than other baselines. ", + "bbox": [ + 174, + 679, + 821, + 720 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Table 2 shows the Accuracy of methods that employ an episodic memory, when we allow the size of this memory to increase; 3 or 5 examples per class per task, resulting in a total memory size of 600 or 1000 for MNIST experiments, and 255 or 425 for CIFAR and ImageNet experiments. The corresponding numbers for Forgetting are given in Appendix Table 4. HAL outperforms all competitors at all benchmarks. ", + "bbox": [ + 173, + 728, + 825, + 797 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Figure 2 provides the training time of all the continual learning baselines on MNIST benchmarks. Although HAL adds a overhead on top of experience replay baseline, it is significantly faster than, MER, another meta-learning approach to reduce forgetting. ", + "bbox": [ + 176, + 804, + 823, + 848 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Finally, Table 3 summarizes the results of an ablation study to better understand the impact of anchor selection. We compare selecting anchors as random noise (Random-Anchors), as a random example for the associated class and task (Data-Anchors), or our proposed optimization in hindsight procedure (HAL). Our results confirm that optimizing anchors in hindsight is the most effective strategy in terms of both accuracy and forgetting metrics. ", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/f14896312758d6b8490d35a8d890359edf983ee590185c2c8239e80306f2822c.jpg", + "image_caption": [ + "Figure 1: Evolution of Accuracy (Eq. 2) as new tasks are learned. When used, episodic memories contain up to one example per class per task. " + ], + "image_footnote": [], + "bbox": [ + 179, + 102, + 820, + 220 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f824761106fa40d87c5c7d50c718626740a8c911cec0e322d8d22d60ccf5f6c6.jpg", + "table_caption": [ + "Table 2: Accuracy (Eq. 2) results for large (3 to 5 examples per class per task) episodic memory sizes. Here we only compare methods that use an episodic memory. Averages and standard deviations are computed over five random seeds. " + ], + "table_footnote": [], + "table_body": "
MethodPermuted MNISTRotated MNIST
|M|= 600|M|= 1000|M|= 600|M|= 1000
AGEM63.2 (±1.47)64.1 (±0.74)49.9 (±1.49)53.0 (±1.52)
MER74.9 (±0.49)78.3 (±0.19)76.5 (±0.30)77.3 (±1.13)
ER-Ring73.5 (±0.43)75.8 (±0.24)74.7 (±0.56)76.5 (±0.48)
HAL (ours)76.2 (±0.52)78.4 (±0.27)77.0 (±0.66)78.7 (±0.97)
MethodSplit CIFARSplit miniImageNet
|M| = 255|M| = 425M|= 255[M|= 425
AGEM56.9 (±3.45)59.9 (±2.64)51.6 (±2.69)54.3 (±1.56)
MER57.7 (±2.59)60.6 (±2.09)49.4 (±3.43)54.8 (±1.79)
ER-Ring60.9 (±1.44)62.6 (±1.77)53.5 (±1.42)54.2 (±3.23)
HAL (ours)62.9 (±1.49)64.4 (±2.15)56.5 (±0.87)57.2 (±1.54)
", + "bbox": [ + 240, + 333, + 756, + 563 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 RELATED WORK ", + "text_level": 1, + "bbox": [ + 174, + 590, + 341, + 606 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In continual learning (Ring, 1997), also called lifelong learning (Thrun, 1998), a learner addresses a sequence of changing tasks without storing the complete datasets of these tasks. This is in contrast to multitask learning (Caruana, 1997), where the learner assumes simultaneous access to data from all tasks. The main challenge in continual learning is to avoid catastrophic interference or forgetting (McCloskey & Cohen, 1989; McClelland et al., 1995; Goodfellow et al., 2013), that is, the learner forgetting previously acquired knowledge when learning new tasks. The state-of-the art methods in continual learning can be categorized into three classes. ", + "bbox": [ + 174, + 625, + 485, + 833 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "First, regularization approaches discourage updating parameters important for past tasks (Kirkpatrick et al., 2016; Aljundi et al., 2018; Nguyen et al., 2018; Zenke et al., 2017). While efficient in terms of memory and computation, these ap", + "bbox": [ + 174, + 840, + 483, + 910 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/42ba534765d99259f9f9127f5c62fef49a83981dae81575a5750156689ee6117.jpg", + "image_caption": [ + "Figure 2: Training time (s) of MNIST experiments for the entire continual learning experience. MER and HAL both use meta-learning objectives to reduce forgetting. " + ], + "image_footnote": [], + "bbox": [ + 495, + 650, + 823, + 818 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "proaches suffer from brittleness due to feature drift as the number of tasks increases (Titsias et al., ", + "bbox": [ + 173, + 910, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/5770cf222d7f0d8c5790191437c12505141e65165d5a7c0fc4a6c7603e3bec06.jpg", + "table_caption": [ + "Table 3: Impact of anchor selection, where we compare random-noise anchors (Random-), one random example-per-class anchors (Data-), and our optimized anchor selection (HAL). " + ], + "table_footnote": [], + "table_body": "
Anchor typePermuted MNISTSplit CIFAR
AccuracyForgettingAccuracyForgetting
ER-Ring70.20.1256.20.13
Random-Anchors72.70.1057.80.13
Data-Anchors73.20.1059.00.12
HAL (ours)73.60.0960.40.10
", + "bbox": [ + 259, + 140, + 738, + 247 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "2019). Additionally, these approaches are only effective when we can perform multiple passes over each dataset (Chaudhry et al., 2019a), a case deemed unrealistic in this work. ", + "bbox": [ + 173, + 277, + 823, + 305 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Second, modular approaches use different parts of the prediction function for each new task (Fernando et al., 2017; Aljundi et al., 2017; Rosenbaum et al., 2018; Chang et al., 2018; Xu & Zhu, 2018; Ferran Alet, 2018). Modular approaches do not scale to a large number of tasks, as they require searching over combinatorial space of module architectures. Another modular approach (Rusu et al., 2016; Lee et al., 2017) adds new parts to the prediction function as new tasks are learned. By construction, modular approaches have zero forgetting, but their memory requirements increase with the number of tasks. ", + "bbox": [ + 174, + 313, + 825, + 410 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Third, episodic memory approaches maintain and revisit a small episodic memory of datum from past tasks. In some of these methods (Li & Hoiem, 2016; Rebuffi et al., 2017), examples in the episodic memory are replayed and predictions are kept invariant by means of distillation (Hinton et al., 2014). In other approaches (Lopez-Paz & Ranzato, 2017; Chaudhry et al., 2019a; Aljundi et al., 2019b) the episodic memory is used as an optimization constraint that discourages increases in loss at past tasks. More recently, several works (Hayes et al., 2018; Riemer et al., 2019; Rolnick et al., 2018; Chaudhry et al., 2019b) have shown that directly optimizing the loss on the episodic memory, also known as experience replay, is cheaper than constraint-based approaches and improves prediction performance. Our contribution in this paper has been to improve experience replay methods with task anchors learned in hindsight. ", + "bbox": [ + 173, + 417, + 825, + 556 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "There are other definitions of continual learning, such as the one of task-free continual learning. The task-free formulation does not consider the notion of tasks, and instead works on undivided data streams (Aljundi et al., 2019a,b). We have focused on the task-based definition of continual learning and, similar to many recent works (Lopez-Paz & Ranzato, 2017; Hayes et al., 2018; Riemer et al., 2019; Chaudhry et al., 2019a), assumed that only a single pass through the data was possible. ", + "bbox": [ + 174, + 563, + 825, + 633 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finally, our gradient-based learning of anchors bears a similarity to (Simonyan et al., 2014) and (Wang et al., 2018). In Simonyan et al. (2014), the authors use gradient ascent on class scores to find saliency maps of a classification model. Contrary to them, our proposed hindsight learning objective optimizes for the forgetting metric, as reducing it is necessary while learning continually. Dataset distillation (Wang et al., 2018) proposes to encode the entire dataset in a few synthetic points at a given parameter vector by a gradient-based optimization process. Their method requires access to the entire dataset of a task for optimization purposes. We, instead, learn anchors in hindsight from the replay buffer of past tasks after training for current task. While Wang et al. (2018) aim to replicate the performance of the entire dataset from the synthetic points, we focus on reducing forgetting of an already learned task. ", + "bbox": [ + 173, + 640, + 825, + 779 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 805, + 318, + 821 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We introduced HAL, a meta-learning objective for continual learning. In our approach, we learn one “anchor point” per class per task, where predictions are requested to remain invariant. These anchors are learned using gradient-based optimization, and represent points that would maximize the forgetting of the current task throughout the entire learning experience. We simulate the forgetting that would happen during the learning of future tasks in hindsight, that is, by taking temporary gradient steps across a small episodic memory of past tasks. As shown in our experiments, anchoring in hindsight complements and improves the performance of continual learning methods based on experience replay, achieving a new state of the art on four standard continual learning benchmarks. ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 103, + 287, + 118 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Rahaf Aljundi, Punarjay Chakravarty, and Tinne Tuytelaars. Expert gate: Lifelong learning with a network of experts. In CVPR, pp. 7120–7129, 2017. ", + "bbox": [ + 176, + 126, + 825, + 155 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Rahaf Aljundi, Francesca Babiloni, Mohamed Elhoseiny, Marcus Rohrbach, and Tinne Tuytelaars. Memory aware synapses: Learning what (not) to forget. In ECCV, 2018. 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", + "bbox": [ + 173, + 900, + 825, + 916 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 103, + 264, + 117 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In Section A, we are report the forgetting 3 metric when large episodic memories are used. Section B provides the grid considered for hyper-parameters. Section C provides pseudo-code for HAL. ", + "bbox": [ + 173, + 133, + 823, + 162 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A MORE RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 183, + 344, + 199 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Table 4: Forgetting (Eq. 3) results for large (3 to 5 examples per class per task) episodic memory sizes. Here we only compare methods that use an episodic memory. Averages and standard deviations are computed over five random seeds. ", + "bbox": [ + 173, + 218, + 826, + 262 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/d9bc3ca889e786dbddf80f2270d283b3ee2c883fc7728fc2079f53786d55e30c.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MethodPermuted MNISTRotated MNIST
|M| = 600|M| = 1000|M|= 600|M|= 1000
AGEM0.20 (±0.01)0.19(±0.01)0.41 1 (±0.01)0.38 (±0.01)
MER0.14 (±0.01)0.09(±0.01)0.12(±0.01)0.11 (±0.01)
ER-Ring0.09 (±0.01)0.07 (±0.01)0.15 (±0.01)0.13 (±0.01)
HAL (ours)0.07 (±0.01)0.05 (±0.01)0.12 (±0.01)0.11 (±0.01)
", + "bbox": [ + 241, + 282, + 754, + 387 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/b4a340f4d00b93a2e6dee3332a863cf4bb6dd56a8022305e95ebfc7ba693e62d.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
MethodSplit CIFARSplit miniImageNet
M|= 255|M|= 425|M| = 255|M| = 425
AGEM0.13 (±0.03)0.10 (±0.02)0.10 (±0.02)0.08 (±0.01)
MER0.11 1(±0.01)0.09 (±0.02)0.12 (±0.02)0.07 (±0.01)
ER-Ring0.09 (±0.01)0.06 (±0.01)0.07 (±0.02)0.08 (±0.02)
HAL (ours)0.08 (±0.01)0.06 (±0.01)0.06 (±0.01)0.06 (±0.01)
", + "bbox": [ + 241, + 405, + 751, + 511 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B HYPER-PARAMETER SELECTION ", + "text_level": 1, + "bbox": [ + 174, + 541, + 478, + 558 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "In this section, we report the hyper-parameters grid considered for experiments. The best values for different benchmarks are given in parenthesis. ", + "bbox": [ + 174, + 573, + 825, + 602 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "– learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] ", + "bbox": [ + 232, + 635, + 821, + 664 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "– learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] ", + "bbox": [ + 228, + 688, + 821, + 717 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "• Finetune – learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] ", + "bbox": [ + 222, + 723, + 823, + 770 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "• EWC – learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] – regularization: [0.1, 1, 10 (MNIST perm, rot, CIFAR, miniImageNet), 100, 1000] ", + "bbox": [ + 217, + 773, + 823, + 852 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "• AGEM – learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] \n• MER \n– learning rate: [0.003, 0.01, 0.03 (MNIST, CIFAR, miniImageNet), 0.1, 0.3, 1.0] \n– within batch meta-learning rate: [0.01, 0.03, 0.1 (MNIST, CIFAR, miniImageNet), 0.3, 1.0] \n– current batch learning rate multiplier: [1, 2, 5 (CIFAR, miniImageNet), 10 (MNIST)] ", + "bbox": [ + 217, + 857, + 821, + 924 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 246, + 103, + 823, + 196 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• ER-Ring ", + "text_level": 1, + "bbox": [ + 218, + 204, + 294, + 219 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "– learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] ", + "bbox": [ + 240, + 227, + 820, + 257 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• HAL ", + "bbox": [ + 218, + 263, + 269, + 279 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "– learning rate: [0.003, 0.01, 0.03 (CIFAR, miniImageNet), 0.1 (MNIST perm, rot), 0.3, 1.0] \n– regularization (λ): [0.01, 0.03, 0.1 (MNIST perm, rot), 0.3 (miniImageNet), 1 (CIFAR), 3, 10] \n– mean embedding strength (γ): [0.01, 0.03, 0.1 (MNIST perm, rot, CIFAR, miniImageNet), 0.3, 1, 3, 10] ", + "bbox": [ + 246, + 286, + 821, + 381 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "C HAL ALGORITHM ", + "text_level": 1, + "bbox": [ + 174, + 404, + 362, + 420 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Algorithm 1 provides pseudo-code for HAL. ", + "bbox": [ + 176, + 439, + 467, + 453 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Algorithm 1 Training of HAL on sequential data $\\mathcal { D } = \\{ \\mathcal { D } _ { 1 } , \\cdot \\cdot \\cdot , \\mathcal { D } _ { T } \\}$ , with total replay buffer size ‘mem sz’, learning rate $\\cdot _ { \\alpha } ,$ , regularization strength $^ { \\bullet } \\lambda ^ { \\bullet }$ , mean embedding decay $\\cdot _ { \\beta } ,$ , mean embedding strength $\\cdot _ { \\eta } ,$ . ", + "bbox": [ + 174, + 472, + 826, + 516 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1: procedure $\\mathrm { H A L } ( { \\mathcal { D } } , \\mathrm { m e m . s z } , \\alpha , \\lambda , \\beta )$ \n2: M ← {} ∗ mem sz \n3: {e1, · · · , eT } ← {} \n4: for t ∈ {1, · · · , T } do \n5: φt ← \\~0 \n6: for B ∼ Dt do $\\triangleright$ Sample a batch from current task \n7: BM ∼ M $\\triangleright$ Sample a batch from episodic memory \n8: ˜θ ← θ − α · ∇θ \\`(B ∪ BM) . Temporary parameter update \n9: θ ← θ − α · ∇θ \u0010\\`(B ∪ BM) + λ · Pt0 t", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 420, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 268, + 117 + ], + "score": 1.0, + "content": "as to compute the final parameter vector", + "type": "text" + }, + { + "bbox": [ + 269, + 105, + 281, + 115 + ], + "score": 0.89, + "content": "\\theta _ { T }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 105, + 420, + 117 + ], + "score": 1.0, + "content": ". 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The same unified code base runs all experiments, and is available at", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 702, + 248, + 714 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 248, + 714 + ], + "score": 1.0, + "content": "https://bit.ly/2mw8bsE.2", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 118, + 721, + 417, + 732 + ], + "lines": [ + { + "bbox": [ + 118, + 719, + 418, + 734 + ], + "spans": [ + { + "bbox": [ + 118, + 719, + 418, + 734 + ], + "score": 1.0, + "content": "2The code base is anonymized and implements the Permuted MNIST experiments.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 201, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 201, + 95 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 106, + 504, + 128 + ], + "lines": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "We now evaluate the performance of HAL against a variety of state-of-the-art methods, across four", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 267, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 267, + 129 + ], + "score": 1.0, + "content": "popular continual learning benchmarks.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5, + "bbox_fs": [ + 105, + 106, + 506, + 129 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 142, + 227, + 153 + ], + "lines": [ + { + "bbox": [ + 106, + 142, + 227, + 154 + ], + "spans": [ + { + "bbox": [ + 106, + 142, + 227, + 154 + ], + "score": 1.0, + "content": "4.1 DATESETS AND TASKS", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 162, + 412, + 174 + ], + "lines": [ + { + "bbox": [ + 105, + 160, + 414, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 414, + 177 + ], + "score": 1.0, + "content": "We perform experiments on four popular benchmarks for continual learning.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 160, + 414, + 177 + ] + }, + { + "type": "list", + "bbox": [ + 133, + 184, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 133, + 184, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 133, + 184, + 505, + 196 + ], + "score": 1.0, + "content": "• Permuted MNIST is a variant of the MNIST dataset of handwritten digits (LeCun, 1998)", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 195, + 505, + 207 + ], + "spans": [ + { + "bbox": [ + 141, + 195, + 505, + 207 + ], + "score": 1.0, + "content": "where each task applies a fixed random pixel permutation to the original dataset. This", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 207, + 463, + 217 + ], + "spans": [ + { + "bbox": [ + 142, + 207, + 463, + 217 + ], + "score": 1.0, + "content": "benchmark contains 23 tasks, each with 1000 samples from 10 different classes.", + "type": "text" + } + ], + "index": 7, + "is_list_end_line": true + }, + { + "bbox": [ + 133, + 220, + 505, + 234 + ], + "spans": [ + { + "bbox": [ + 133, + 220, + 505, + 234 + ], + "score": 1.0, + "content": "• Rotated MNIST is another variant of MNIST, where each task applies a fixed random image", + "type": "text" + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 232, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 141, + 232, + 505, + 243 + ], + "score": 1.0, + "content": "rotation (between 0 and 180 degrees) to the original dataset. This benchmark contains 23", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 142, + 244, + 368, + 254 + ], + "spans": [ + { + "bbox": [ + 142, + 244, + 368, + 254 + ], + "score": 1.0, + "content": "tasks, each with 1000 samples from 10 different classes.", + "type": "text" + } + ], + "index": 10, + "is_list_end_line": true + }, + { + "bbox": [ + 133, + 257, + 505, + 269 + ], + "spans": [ + { + "bbox": [ + 133, + 257, + 505, + 269 + ], + "score": 1.0, + "content": "• Split CIFAR is a variant of the CIFAR-100 dataset (Krizhevsky & Hinton, 2009; Zenke et al.,", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 268, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 268, + 505, + 280 + ], + "score": 1.0, + "content": "2017), where each task contains the data pertaining 5 random classes (without replacement)", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 278, + 506, + 293 + ], + "spans": [ + { + "bbox": [ + 141, + 278, + 506, + 293 + ], + "score": 1.0, + "content": "out of the total 100 classes. This benchmark contains 20 tasks, each with 250 samples per", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 290, + 228, + 302 + ], + "spans": [ + { + "bbox": [ + 141, + 290, + 228, + 302 + ], + "score": 1.0, + "content": "each of the 5 classes.", + "type": "text" + } + ], + "index": 14, + "is_list_end_line": true + }, + { + "bbox": [ + 133, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 133, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "• Split miniImageNet is a variant of the ImageNet dataset (Russakovsky et al., 2015; Vinyals", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 141, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "et al., 2016), containing a subset of images and classes from the original dataset. This", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 327, + 457, + 339 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 457, + 339 + ], + "score": 1.0, + "content": "benchmark contains 20 tasks, each with 250 samples per each of the 5 classes.", + "type": "text" + } + ], + "index": 17, + "is_list_end_line": true + } + ], + "index": 11, + "bbox_fs": [ + 133, + 184, + 506, + 339 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 349, + 504, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "spans": [ + { + "bbox": [ + 105, + 348, + 505, + 361 + ], + "score": 1.0, + "content": "For all datasets, the first 3 tasks are used for hyper-parameter optimization (grids available in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "Appendix B). The learners can perform multiple epochs on these three initial tasks, that are later", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 370, + 207, + 382 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 207, + 382 + ], + "score": 1.0, + "content": "discarded for evaluation.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 348, + 506, + 382 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 396, + 180, + 407 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 182, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 182, + 408 + ], + "score": 1.0, + "content": "4.2 BASELINES", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 106, + 416, + 371, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 416, + 372, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 372, + 430 + ], + "score": 1.0, + "content": "We compare our proposed model HAL to the following baselines.", 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} + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 475, + 430, + 487 + ], + "spans": [ + { + "bbox": [ + 141, + 475, + 430, + 487 + ], + "score": 1.0, + "content": "ters critical to past tasks, as measured by the Fisher information matrix.", + "type": "text" + } + ], + "index": 26, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 489, + 506, + 502 + ], + "spans": [ + { + "bbox": [ + 132, + 489, + 506, + 502 + ], + "score": 1.0, + "content": "• AGEM (Chaudhry et al., 2019a) is a continual learning method improving on (Lopez-Paz &", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 141, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "Ranzato, 2017), which uses an episodic memory of parameter gradients to limit forgetting.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 133, + 515, + 505, + 528 + ], + "spans": [ + { + "bbox": 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Averages", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "spans": [ + { + "bbox": [ + 106, + 91, + 505, + 104 + ], + "score": 1.0, + "content": "and standard deviations are computed over five random seeds. When used, episodic memories contain", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 103, + 397, + 114 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 397, + 114 + ], + "score": 1.0, + "content": "up to one example per class per task. Last two rows are oracle baselines.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "table_body", + "bbox": [ + 135, + 123, + 474, + 401 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 135, + 123, + 474, + 401 + ], + "spans": [ + { + "bbox": [ + 135, + 123, + 474, + 401 + ], + "score": 0.631, + "html": "
MethodPermuted MNISTRotated MNIST
AccuracyForgettingAccuracyForgetting
Finetune53.5 (±1.46)0.29 (±0.01)41.9 (±1.37)0.50 (±0.01)
EWC63.1 (±1.40)0.18 (±0.01)44.1 (±0.99)0.47 (±0.01)
AGEM62.1 (±1.39)0.21 (±0.01)50.9 (±0.92)0.40 (±0.01)
MER69.9 (±0.40)0.14 (±0.01)66.0 (±2.04)0.23 (±0.01)
ER-Ring70.2 (±0.56)0.12 (±0.01)65.9 (±0.41)0.24 (±0.01)
HAL (ours)73.6 (±0.31)0.09 (±0.01)68.4 (±0.72)0.21 (±0.01)
Clone-and-finetune Multitask81.4 (±0.35)0.087.5 (±0.11)0.0
83.00.083.30.0
MethodSplit CIFARSplit miniImageNet
AccuracyForgettingAccuracyForgetting
Finetune42.9 (±2.07)0.25 (±0.03)34.7 (±2.69)0.26 (±0.03)
EWC42.4 (±3.02)0.26 (±0.02)37.7 (±3.29)0.21 (±0.03)
AGEM54.9 (±2.92)0.14 (±0.03)48.2 (±2.49)0.13 (±0.02)
MER49.7 (±2.97)0.19 (±0.03)45.5 (±1.49)0.15 (±0.01)
ER-Ring56.2 (±1.93)0.13 (±0.01)49.0 (±2.61)0.12 (±0.02)
HAL (ours)60.4 (±0.54)0.10 (±0.01)51.6 (±2.02)0.10 (±0.01)
Clone-and-finetune60.3 (±0.55)0.050.3 (±1.00)0.0
Multitask68.30.063.50.0
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MethodPermuted MNISTRotated MNIST
AccuracyForgettingAccuracyForgetting
Finetune53.5 (±1.46)0.29 (±0.01)41.9 (±1.37)0.50 (±0.01)
EWC63.1 (±1.40)0.18 (±0.01)44.1 (±0.99)0.47 (±0.01)
AGEM62.1 (±1.39)0.21 (±0.01)50.9 (±0.92)0.40 (±0.01)
MER69.9 (±0.40)0.14 (±0.01)66.0 (±2.04)0.23 (±0.01)
ER-Ring70.2 (±0.56)0.12 (±0.01)65.9 (±0.41)0.24 (±0.01)
HAL (ours)73.6 (±0.31)0.09 (±0.01)68.4 (±0.72)0.21 (±0.01)
Clone-and-finetune Multitask81.4 (±0.35)0.087.5 (±0.11)0.0
83.00.083.30.0
MethodSplit CIFARSplit miniImageNet
AccuracyForgettingAccuracyForgetting
Finetune42.9 (±2.07)0.25 (±0.03)34.7 (±2.69)0.26 (±0.03)
EWC42.4 (±3.02)0.26 (±0.02)37.7 (±3.29)0.21 (±0.03)
AGEM54.9 (±2.92)0.14 (±0.03)48.2 (±2.49)0.13 (±0.02)
MER49.7 (±2.97)0.19 (±0.03)45.5 (±1.49)0.15 (±0.01)
ER-Ring56.2 (±1.93)0.13 (±0.01)49.0 (±2.61)0.12 (±0.02)
HAL (ours)60.4 (±0.54)0.10 (±0.01)51.6 (±2.02)0.10 (±0.01)
Clone-and-finetune60.3 (±0.55)0.050.3 (±1.00)0.0
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Averages and standard deviations are", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 244, + 245, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 244, + 245, + 255 + ], + "score": 1.0, + "content": "computed over five random seeds.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "table_body", + "bbox": [ + 147, + 264, + 463, + 446 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 147, + 264, + 463, + 446 + ], + "spans": [ + { + "bbox": [ + 147, + 264, + 463, + 446 + ], + "score": 0.584, + "html": "
MethodPermuted MNISTRotated MNIST
|M|= 600|M|= 1000|M|= 600|M|= 1000
AGEM63.2 (±1.47)64.1 (±0.74)49.9 (±1.49)53.0 (±1.52)
MER74.9 (±0.49)78.3 (±0.19)76.5 (±0.30)77.3 (±1.13)
ER-Ring73.5 (±0.43)75.8 (±0.24)74.7 (±0.56)76.5 (±0.48)
HAL (ours)76.2 (±0.52)78.4 (±0.27)77.0 (±0.66)78.7 (±0.97)
MethodSplit CIFARSplit miniImageNet
|M| = 255|M| = 425M|= 255[M|= 425
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MER57.7 (±2.59)60.6 (±2.09)49.4 (±3.43)54.8 (±1.79)
ER-Ring60.9 (±1.44)62.6 (±1.77)53.5 (±1.42)54.2 (±3.23)
HAL (ours)62.9 (±1.49)64.4 (±2.15)56.5 (±0.87)57.2 (±1.54)
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MethodPermuted MNISTRotated MNIST
|M|= 600|M|= 1000|M|= 600|M|= 1000
AGEM63.2 (±1.47)64.1 (±0.74)49.9 (±1.49)53.0 (±1.52)
MER74.9 (±0.49)78.3 (±0.19)76.5 (±0.30)77.3 (±1.13)
ER-Ring73.5 (±0.43)75.8 (±0.24)74.7 (±0.56)76.5 (±0.48)
HAL (ours)76.2 (±0.52)78.4 (±0.27)77.0 (±0.66)78.7 (±0.97)
MethodSplit CIFARSplit miniImageNet
|M| = 255|M| = 425M|= 255[M|= 425
AGEM56.9 (±3.45)59.9 (±2.64)51.6 (±2.69)54.3 (±1.56)
MER57.7 (±2.59)60.6 (±2.09)49.4 (±3.43)54.8 (±1.79)
ER-Ring60.9 (±1.44)62.6 (±1.77)53.5 (±1.42)54.2 (±3.23)
HAL (ours)62.9 (±1.49)64.4 (±2.15)56.5 (±0.87)57.2 (±1.54)
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Anchor typePermuted MNISTSplit CIFAR
AccuracyForgettingAccuracyForgetting
ER-Ring70.20.1256.20.13
Random-Anchors72.70.1057.80.13
Data-Anchors73.20.1059.00.12
HAL (ours)73.60.0960.40.10
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Additionally, these approaches are only effective when we can perform multiple passes over", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 231, + 415, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 415, + 243 + ], + "score": 1.0, + "content": "each dataset (Chaudhry et al., 2019a), a case deemed unrealistic in this work.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "Second, modular approaches use different parts of the prediction function for each new task (Fernando", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "et al., 2017; Aljundi et al., 2017; Rosenbaum et al., 2018; Chang et al., 2018; Xu & Zhu, 2018;", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "Ferran Alet, 2018). Modular approaches do not scale to a large number of tasks, as they require", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "score": 1.0, + "content": "searching over combinatorial space of module architectures. Another modular approach (Rusu et al.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "2016; Lee et al., 2017) adds new parts to the prediction function as new tasks are learned. By", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "construction, modular approaches have zero forgetting, but their memory requirements increase with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 314, + 190, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 190, + 325 + ], + "score": 1.0, + "content": "the number of tasks.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "Third, episodic memory approaches maintain and revisit a small episodic memory of datum from past", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "tasks. In some of these methods (Li & Hoiem, 2016; Rebuffi et al., 2017), examples in the episodic", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "memory are replayed and predictions are kept invariant by means of distillation (Hinton et al., 2014).", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "In other approaches (Lopez-Paz & Ranzato, 2017; Chaudhry et al., 2019a; Aljundi et al., 2019b)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "the episodic memory is used as an optimization constraint that discourages increases in loss at past", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "tasks. More recently, several works (Hayes et al., 2018; Riemer et al., 2019; Rolnick et al., 2018;", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "Chaudhry et al., 2019b) have shown that directly optimizing the loss on the episodic memory, also", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 408, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 504, + 420 + ], + "score": 1.0, + "content": "known as experience replay, is cheaper than constraint-based approaches and improves prediction", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "performance. Our contribution in this paper has been to improve experience replay methods with task", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 430, + 223, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 223, + 442 + ], + "score": 1.0, + "content": "anchors learned in hindsight.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "There are other definitions of continual learning, such as the one of task-free continual learning. The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "task-free formulation does not consider the notion of tasks, and instead works on undivided data", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 466, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 506, + 482 + ], + "score": 1.0, + "content": "streams (Aljundi et al., 2019a,b). We have focused on the task-based definition of continual learning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "and, similar to many recent works (Lopez-Paz & Ranzato, 2017; Hayes et al., 2018; Riemer et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 489, + 482, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 482, + 503 + ], + "score": 1.0, + "content": "2019; Chaudhry et al., 2019a), assumed that only a single pass through the data was possible.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "Finally, our gradient-based learning of anchors bears a similarity to (Simonyan et al., 2014) and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "(Wang et al., 2018). In Simonyan et al. (2014), the authors use gradient ascent on class scores to find", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "saliency maps of a classification model. Contrary to them, our proposed hindsight learning objective", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "optimizes for the forgetting metric, as reducing it is necessary while learning continually. Dataset", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "distillation (Wang et al., 2018) proposes to encode the entire dataset in a few synthetic points at a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "given parameter vector by a gradient-based optimization process. Their method requires access to the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "entire dataset of a task for optimization purposes. We, instead, learn anchors in hindsight from the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 505, + 597 + ], + "score": 1.0, + "content": "replay buffer of past tasks after training for current task. While Wang et al. (2018) aim to replicate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "score": 1.0, + "content": "the performance of the entire dataset from the synthetic points, we focus on reducing forgetting of an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 606, + 191, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 191, + 618 + ], + "score": 1.0, + "content": "already learned task.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 638, + 195, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 197, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 197, + 654 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We introduced HAL, a meta-learning objective for continual learning. In our approach, we learn", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "one “anchor point” per class per task, where predictions are requested to remain invariant. These", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "anchors are learned using gradient-based optimization, and represent points that would maximize the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 713 + ], + "score": 1.0, + "content": "forgetting of the current task throughout the entire learning experience. 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Anchor typePermuted MNISTSplit CIFAR
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ER-Ring70.20.1256.20.13
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Additionally, these approaches are only effective when we can perform multiple passes over", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 231, + 415, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 415, + 243 + ], + "score": 1.0, + "content": "each dataset (Chaudhry et al., 2019a), a case deemed unrealistic in this work.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 219, + 505, + 243 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 248, + 505, + 325 + ], + "lines": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 505, + 260 + ], + "score": 1.0, + "content": "Second, modular approaches use different parts of the prediction function for each new task (Fernando", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 271 + ], + "score": 1.0, + "content": "et al., 2017; Aljundi et al., 2017; Rosenbaum et al., 2018; Chang et al., 2018; Xu & Zhu, 2018;", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 105, + 269, + 505, + 282 + ], + "score": 1.0, + "content": "Ferran Alet, 2018). Modular approaches do not scale to a large number of tasks, as they require", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 293 + ], + "score": 1.0, + "content": "searching over combinatorial space of module architectures. Another modular approach (Rusu et al.,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "2016; Lee et al., 2017) adds new parts to the prediction function as new tasks are learned. By", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 315 + ], + "score": 1.0, + "content": "construction, modular approaches have zero forgetting, but their memory requirements increase with", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 314, + 190, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 314, + 190, + 325 + ], + "score": 1.0, + "content": "the number of tasks.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 248, + 505, + 325 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 331, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "Third, episodic memory approaches maintain and revisit a small episodic memory of datum from past", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 342, + 505, + 353 + ], + "score": 1.0, + "content": "tasks. In some of these methods (Li & Hoiem, 2016; Rebuffi et al., 2017), examples in the episodic", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "memory are replayed and predictions are kept invariant by means of distillation (Hinton et al., 2014).", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 506, + 376 + ], + "score": 1.0, + "content": "In other approaches (Lopez-Paz & Ranzato, 2017; Chaudhry et al., 2019a; Aljundi et al., 2019b)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 374, + 506, + 387 + ], + "score": 1.0, + "content": "the episodic memory is used as an optimization constraint that discourages increases in loss at past", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 385, + 506, + 398 + ], + "score": 1.0, + "content": "tasks. More recently, several works (Hayes et al., 2018; Riemer et al., 2019; Rolnick et al., 2018;", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "Chaudhry et al., 2019b) have shown that directly optimizing the loss on the episodic memory, also", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 408, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 504, + 420 + ], + "score": 1.0, + "content": "known as experience replay, is cheaper than constraint-based approaches and improves prediction", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "performance. Our contribution in this paper has been to improve experience replay methods with task", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 430, + 223, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 430, + 223, + 442 + ], + "score": 1.0, + "content": "anchors learned in hindsight.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 330, + 506, + 442 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 505, + 502 + ], + "lines": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 505, + 458 + ], + "score": 1.0, + "content": "There are other definitions of continual learning, such as the one of task-free continual learning. The", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 106, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "task-free formulation does not consider the notion of tasks, and instead works on undivided data", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 466, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 104, + 466, + 506, + 482 + ], + "score": 1.0, + "content": "streams (Aljundi et al., 2019a,b). We have focused on the task-based definition of continual learning", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "and, similar to many recent works (Lopez-Paz & Ranzato, 2017; Hayes et al., 2018; Riemer et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 489, + 482, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 482, + 503 + ], + "score": 1.0, + "content": "2019; Chaudhry et al., 2019a), assumed that only a single pass through the data was possible.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 104, + 446, + 506, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 507, + 505, + 617 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "Finally, our gradient-based learning of anchors bears a similarity to (Simonyan et al., 2014) and", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "(Wang et al., 2018). In Simonyan et al. (2014), the authors use gradient ascent on class scores to find", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "saliency maps of a classification model. Contrary to them, our proposed hindsight learning objective", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "optimizes for the forgetting metric, as reducing it is necessary while learning continually. Dataset", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 506, + 564 + ], + "score": 1.0, + "content": "distillation (Wang et al., 2018) proposes to encode the entire dataset in a few synthetic points at a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 574 + ], + "score": 1.0, + "content": "given parameter vector by a gradient-based optimization process. Their method requires access to the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 106, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "entire dataset of a task for optimization purposes. 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(2018) aim to replicate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 608 + ], + "score": 1.0, + "content": "the performance of the entire dataset from the synthetic points, we focus on reducing forgetting of an", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 606, + 191, + 618 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 191, + 618 + ], + "score": 1.0, + "content": "already learned task.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 507, + 506, + 618 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 638, + 195, + 651 + ], + "lines": [ + { + "bbox": [ + 105, + 635, + 197, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 635, + 197, + 654 + ], + "score": 1.0, + "content": "6 CONCLUSION", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 666, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "We introduced HAL, a meta-learning objective for continual learning. 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MethodPermuted MNISTRotated MNIST
|M| = 600|M| = 1000|M|= 600|M|= 1000
AGEM0.20 (±0.01)0.19(±0.01)0.41 1 (±0.01)0.38 (±0.01)
MER0.14 (±0.01)0.09(±0.01)0.12(±0.01)0.11 (±0.01)
ER-Ring0.09 (±0.01)0.07 (±0.01)0.15 (±0.01)0.13 (±0.01)
HAL (ours)0.07 (±0.01)0.05 (±0.01)0.12 (±0.01)0.11 (±0.01)
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MethodSplit CIFARSplit miniImageNet
M|= 255|M|= 425|M| = 255|M| = 425
AGEM0.13 (±0.03)0.10 (±0.02)0.10 (±0.02)0.08 (±0.01)
MER0.11 1(±0.01)0.09 (±0.02)0.12 (±0.02)0.07 (±0.01)
ER-Ring0.09 (±0.01)0.06 (±0.01)0.07 (±0.02)0.08 (±0.02)
HAL (ours)0.08 (±0.01)0.06 (±0.01)0.06 (±0.01)0.06 (±0.01)
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MethodPermuted MNISTRotated MNIST
|M| = 600|M| = 1000|M|= 600|M|= 1000
AGEM0.20 (±0.01)0.19(±0.01)0.41 1 (±0.01)0.38 (±0.01)
MER0.14 (±0.01)0.09(±0.01)0.12(±0.01)0.11 (±0.01)
ER-Ring0.09 (±0.01)0.07 (±0.01)0.15 (±0.01)0.13 (±0.01)
HAL (ours)0.07 (±0.01)0.05 (±0.01)0.12 (±0.01)0.11 (±0.01)
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MethodSplit CIFARSplit miniImageNet
M|= 255|M|= 425|M| = 255|M| = 425
AGEM0.13 (±0.03)0.10 (±0.02)0.10 (±0.02)0.08 (±0.01)
MER0.11 1(±0.01)0.09 (±0.02)0.12 (±0.02)0.07 (±0.01)
ER-Ring0.09 (±0.01)0.06 (±0.01)0.07 (±0.02)0.08 (±0.02)
HAL (ours)0.08 (±0.01)0.06 (±0.01)0.06 (±0.01)0.06 (±0.01)
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MethodPermuted MNISTRotated MNIST
AccuracyForgettingAccuracyForgetting
Finetune53.5 (±1.46)0.29 (±0.01)41.9 (±1.37)0.50 (±0.01)
EWC63.1 (±1.40)0.18 (±0.01)44.1 (±0.99)0.47 (±0.01)
AGEM62.1 (±1.39)0.21 (±0.01)50.9 (±0.92)0.40 (±0.01)
MER69.9 (±0.40)0.14 (±0.01)66.0 (±2.04)0.23 (±0.01)
ER-Ring70.2 (±0.56)0.12 (±0.01)65.9 (±0.41)0.24 (±0.01)
HAL (ours)73.6 (±0.31)0.09 (±0.01)68.4 (±0.72)0.21 (±0.01)
Clone-and-finetune Multitask81.4 (±0.35)0.087.5 (±0.11)0.0
83.00.083.30.0
MethodSplit CIFARSplit miniImageNet
AccuracyForgettingAccuracyForgetting
Finetune42.9 (±2.07)0.25 (±0.03)34.7 (±2.69)0.26 (±0.03)
EWC42.4 (±3.02)0.26 (±0.02)37.7 (±3.29)0.21 (±0.03)
AGEM54.9 (±2.92)0.14 (±0.03)48.2 (±2.49)0.13 (±0.02)
MER49.7 (±2.97)0.19 (±0.03)45.5 (±1.49)0.15 (±0.01)
ER-Ring56.2 (±1.93)0.13 (±0.01)49.0 (±2.61)0.12 (±0.02)
HAL (ours)60.4 (±0.54)0.10 (±0.01)51.6 (±2.02)0.10 (±0.01)
Clone-and-finetune60.3 (±0.55)0.050.3 (±1.00)0.0
Multitask68.30.063.50.0
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MethodPermuted MNISTRotated MNIST
|M|= 600|M|= 1000|M|= 600|M|= 1000
AGEM63.2 (±1.47)64.1 (±0.74)49.9 (±1.49)53.0 (±1.52)
MER74.9 (±0.49)78.3 (±0.19)76.5 (±0.30)77.3 (±1.13)
ER-Ring73.5 (±0.43)75.8 (±0.24)74.7 (±0.56)76.5 (±0.48)
HAL (ours)76.2 (±0.52)78.4 (±0.27)77.0 (±0.66)78.7 (±0.97)
MethodSplit CIFARSplit miniImageNet
|M| = 255|M| = 425M|= 255[M|= 425
AGEM56.9 (±3.45)59.9 (±2.64)51.6 (±2.69)54.3 (±1.56)
MER57.7 (±2.59)60.6 (±2.09)49.4 (±3.43)54.8 (±1.79)
ER-Ring60.9 (±1.44)62.6 (±1.77)53.5 (±1.42)54.2 (±3.23)
HAL (ours)62.9 (±1.49)64.4 (±2.15)56.5 (±0.87)57.2 (±1.54)
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MethodSplit CIFARSplit miniImageNet
M|= 255|M|= 425|M| = 255|M| = 425
AGEM0.13 (±0.03)0.10 (±0.02)0.10 (±0.02)0.08 (±0.01)
MER0.11 1(±0.01)0.09 (±0.02)0.12 (±0.02)0.07 (±0.01)
ER-Ring0.09 (±0.01)0.06 (±0.01)0.07 (±0.02)0.08 (±0.02)
HAL (ours)0.08 (±0.01)0.06 (±0.01)0.06 (±0.01)0.06 (±0.01)
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MethodPermuted MNISTRotated MNIST
|M| = 600|M| = 1000|M|= 600|M|= 1000
AGEM0.20 (±0.01)0.19(±0.01)0.41 1 (±0.01)0.38 (±0.01)
MER0.14 (±0.01)0.09(±0.01)0.12(±0.01)0.11 (±0.01)
ER-Ring0.09 (±0.01)0.07 (±0.01)0.15 (±0.01)0.13 (±0.01)
HAL (ours)0.07 (±0.01)0.05 (±0.01)0.12 (±0.01)0.11 (±0.01)
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0000000000000000000000000000000000000000..408411bb90d8db9c4b4aa94f6d4f33718a1c768c --- /dev/null +++ b/parse/train/HkxStoC5F7/HkxStoC5F7.md @@ -0,0 +1,443 @@ +# META-LEARN ING PROBAB IL IST IC INFERENCE FOR PRED ICT ION + +Jonathan Gordon∗, John Bronskill∗ University of Cambridge {jg801,jfb54}@cam.ac.uk + +Matthias Bauer +University of Cambridge +Max Planck Institute for Intelligent Systems +bauer@tue.mpg.de + +Sebastian Nowozin† Google AI Berlin nowozin@google.com + +Richard E. Turner University of Cambridge Microsoft Research ret26@cam.ac.uk + +# ABSTRACT + +This paper introduces a new framework for data efficient and versatile learning. Specifically: 1) We develop ML-PIP, a general framework for Meta-Learning approximate Probabilistic Inference for Prediction. ML-PIP extends existing probabilistic interpretations of meta-learning to cover a broad class of methods. 2) We introduce VERSA, an instance of the framework employing a flexible and versatile amortization network that takes few-shot learning datasets as inputs, with arbitrary numbers of shots, and outputs a distribution over task-specific parameters in a single forward pass. VERSA substitutes optimization at test time with forward passes through inference networks, amortizing the cost of inference and relieving the need for second derivatives during training. 3) We evaluate VERSA on benchmark datasets where the method sets new state-of-the-art results, handles arbitrary numbers of shots, and for classification, arbitrary numbers of classes at train and test time. The power of the approach is then demonstrated through a challenging few-shot ShapeNet view reconstruction task. + +# 1 INTRODUCT ION + +Many applications require predictions to be made on myriad small, but related datasets. In such cases, it is natural to desire learners that can rapidly adapt to new datasets at test time. These applications have given rise to vast interest in few-shot learning (Fei-Fei et al., 2006; Lake et al., 2011), which emphasizes data efficiency via information sharing across related tasks. Despite recent advances, notably in meta-learning based approaches (Ravi and Larochelle, 2017; Vinyals et al., 2016; Edwards and Storkey, 2017; Finn et al., 2017; Lacoste et al., 2018), there remains a lack of general purpose methods for flexible, data-efficient learning. + +Due to the ubiquity of recent work, a unifying view is needed to understand and improve these methods. Existing frameworks (Grant et al., 2018; Finn et al., 2018) are limited to specific families of approaches. In this paper we develop a framework for meta-learning approximate probabilistic inference for prediction (ML-PIP), providing this view in terms of amortizing posterior predictive distributions. In Section 4, we show that ML-PIP re-frames and extends existing point-estimate probabilistic interpretations of meta-learning (Grant et al., 2018; Finn et al., 2018) to cover a broader class of methods, including gradient based meta-learning (Finn et al., 2017; Ravi and Larochelle, 2017), metric based meta-learning (Snell et al., 2017), amortized MAP inference (Qiao et al., 2017) and conditional probability modelling (Garnelo et al., 2018a;b). + +The framework incorporates three key elements. First, we leverage shared statistical structure between tasks via hierarchical probabilistic models developed for multi-task and transfer learning (Heskes, + +2000; Bakker and Heskes, 2003). Second, we share information between tasks about how to learn and perform inference using meta-learning (Naik and Mammone, 1992; Thrun and Pratt, 2012; Schmidhuber, 1987). Since uncertainty is rife in small datasets, we provide a procedure for metalearning probabilistic inference. Third, we enable fast learning that can flexibly handle a wide range of tasks and learning settings via amortization (Kingma and Welling, 2014; Rezende et al., 2014). + +Building on the framework, we propose a new method – VERSA – which substitutes optimization procedures at test time with forward passes through inference networks. This amortizes the cost of inference, resulting in faster test-time performance, and relieves the need for second derivatives during training. VERSA employs a flexible amortization network that takes few-shot learning datasets, and outputs a distribution over task-specific parameters in a single forward pass. The network can handle arbitrary numbers of shots, and for classification, arbitrary numbers of classes at train and test time (see Section 3). In Section 5, we evaluate VERSA on (i) standard benchmarks where the method sets new state-of-the-art results, (ii) settings where test conditions (shot and way) differ from training, and (iii) a challenging one-shot view reconstruction task. + +# 2 META-LEARN ING PROBAB IL IST IC INFERENCE FOR PRED ICT ION + +We now present the framework that consists of (i) a multi-task probabilistic model, and (ii) a method for meta-learning probabilistic inference. + +# 2.1 PROBAB IL IST IC MODEL + +Two principles guide the choice of model. First, the use of discriminative models to maximize predictive performance on supervised learning tasks $\mathrm { N g }$ and Jordan, 2002). Second, the need to leverage shared statistical structure between tasks (i.e. multi-task learning). These criteria are met by the standard multi-task directed graphical model shown in Fig. 1 that employs shared parameters $\theta$ , which are common to all tasks, and task specific parameters $\{ \psi ^ { ( t ) } \} _ { t = 1 } ^ { T }$ . Inputs are denoted $x$ and outputs $y$ . Training data $D ^ { ( t ) } = \{ ( x _ { n } ^ { ( t ) } , y _ { n } ^ { ( t ) } ) \} _ { n = 1 } ^ { N _ { t } }$ , and test data $\{ ( \tilde { x } _ { m } ^ { ( t ) } , \tilde { y } _ { m } ^ { ( t ) } ) \} _ { m = 1 } ^ { M _ { t } }$ are explicitly distinguished for each task , as this is key for few-shot learning. + +![](images/68029ff8f47704e3ae4f511823dad2c320911504c271f201f15d84dbc23b17c9.jpg) +Figure 1: Directed graphical model for multi-task learning. + +Let $X ^ { ( t ) }$ and $Y ^ { ( t ) }$ denote all the inputs and outputs (both test and train) for task $t$ . The joint probability of the outputs and task specific parameters for $T$ tasks, given the inputs and global parameters is: + +$$ +\small \rho \left( \{ Y ^ { ( t ) } , \psi ^ { ( t ) } \} _ { t = 1 } ^ { T } | \{ X ^ { ( t ) } \} _ { t = 1 } ^ { T } , \boldsymbol \theta \right) = \prod _ { t = 1 } ^ { T } p \left( \psi ^ { ( t ) } | \theta \right) \prod _ { n = 1 } ^ { N _ { t } } p \left( y _ { n } ^ { ( t ) } | x _ { n } ^ { ( t ) } , \psi ^ { ( t ) } , \theta \right) \prod _ { m = 1 } ^ { M _ { t } } p \left( \tilde { y } _ { m } ^ { ( t ) } | \tilde { x } _ { m } ^ { ( t ) } , \psi ^ { ( t ) } , \theta \right) +$$ + +In the next section, the goal is to meta-learn fast and accurate approximations to the posterior predictive distribution $\begin{array} { r } { p ( \tilde { y } ^ { ( t ) } | \tilde { x } ^ { ( t ) } , \theta ) = \int p ( \tilde { y } ^ { ( t ) } | \tilde { x } ^ { ( t ) } , \psi ^ { ( t ) } , \theta ) p ( \psi ^ { ( t ) } | \tilde { x } ^ { ( t ) } , D ^ { ( t ) } , \theta ) \mathrm { d } \psi ^ { ( t ) } } \end{array}$ for unseen tasks $t$ . + +# 2.2 PROBAB IL IST IC INFERENCE + +This section provides a framework for meta-learning approximate inference that is a simple reframing and extension of existing approaches (Finn et al., 2017; Grant et al., 2018). We will employ point estimates for the shared parameters $\theta$ since data across all tasks will pin down their value. Distributional estimates will be used for the task-specific parameters since only a few shots constrain them. + +Once the shared parameters are learned, the probabilistic solution to few-shot learning in the model above comprises two steps. First, form the posterior distribution over the task-specific parameters $p ( \psi ^ { ( t ) } | \tilde { x } ^ { ( t ) } , D ^ { ( t ) } , \theta )$ . Second, compute the posterior predictive $p ( \tilde { y } ^ { ( t ) } | \tilde { x } ^ { ( t ) } , \theta )$ . These steps will require approximation and the emphasis here is on performing this quickly at test time. We will describe the form of the approximation, the optimization problem used to learn it, and how to implement this efficiently below. In what follows we initially suppress dependencies on the inputs $\tilde { x }$ and shared parameters $\theta$ to reduce notational clutter, but will reintroduce these at the end of the section. + +Specification of the approximate posterior predictive distribution. Our framework approximates the posterior predictive distribution by an amortized distribution $q _ { \phi } ( \tilde { y } | D )$ . That is, we learn a feed-forward inference network with parameters $\phi$ that takes any training dataset $D ^ { ( t ) }$ and test input $\tilde { x }$ as inputs and returns the predictive distribution over the test output $\tilde { y } ^ { ( t ) }$ . We construct this by amortizing the approximate posterior $q _ { \phi } ( \psi | D )$ and then form the approximate posterior predictive distribution using: + +$$ +q _ { \phi } ( \tilde { y } | D ) = \int p ( \tilde { y } | \psi ) q _ { \phi } ( \psi | D ) \mathrm { d } \psi . +$$ + +This step may require additional approximation e.g. Monte Carlo sampling. The amortization will enable fast predictions at test time. The form of these distributions is identical to those used in amortized variational inference (Edwards and Storkey, 2017; Kingma and Welling, 2014). In this work, we use a factorized Gaussian distribution for $q _ { \phi } \big ( \psi | D ^ { ( t ) } \big )$ with means and variances set by the amortization network. However, the training method described next is different. + +Meta-learning the approximate posterior predictive distribution. The quality of the approximate posterior predictive for a single task will be measured by the KL-divergence between the true and approximate posterior predictive distribution KL $[ p ( \tilde { y } | D ) \rvert | \dot { q } _ { \phi } ( \tilde { y } | D ) ]$ . The goal of learning will be to minimize the expected value of this KL averaged over tasks, + +$$ +\phi ^ { * } = \underset { \phi } { \operatorname { a r g m i n } } \ \underset { p ( D ) } { \mathbb { E } } \left[ \mathrm { K L } \left[ p ( \tilde { y } | D ) \lVert q _ { \phi } ( \tilde { y } | D ) \right] \right] = \underset { \phi } { \operatorname { a r g m a x } } \ \underset { p ( \tilde { y } , D ) } { \mathbb { E } } \left[ \log \int p ( \tilde { y } | \psi ) q _ { \phi } ( \psi | D ) \mathrm { d } \psi \right] . +$$ + +Training will therefore return parameters $\phi$ that best approximate the posterior predictive distribution in an average KL sense. So, if the approximate posterior $q _ { \phi } ( \psi | D )$ is rich enough, global optimization will recover the true posterior $p ( \psi | D )$ (assuming $p ( \psi | D )$ obeys identifiability conditions (Casella and Berger, 2002)).1 Thus, the amortized procedure meta-learns approximate inference that supports accurate prediction. Appendix A provides a generalized derivation of the framework, grounded in Bayesian decision theory (Jaynes, 2003). + +The right hand side of Eq. (2) indicates how training could proceed: (i) select a task $t$ at random, (ii) sample some training data $D ^ { ( t ) }$ , (iii) form the posterior predictive $q _ { \phi } \big ( \cdot | D ^ { ( t ) } \big )$ and, (iv) compute the log-density $\log q _ { \phi } ( \tilde { y } ^ { ( t ) } | D ^ { ( t ) } )$ at test data $\tilde { y } ^ { ( t ) }$ not included in $D ^ { ( t ) }$ . Repeating this process many times and averaging the results would provide an unbiased estimate of the objective which can then be optimized. This perspective also makes it clear that the procedure is scoring the approximate inference procedure by simulating approximate Bayesian held-out log-likelihood evaluation. Importantly, while an inference network is used to approximate posterior distributions, the training procedure differs significantly from standard variational inference. In particular, rather than minimizing $\mathrm { K L } ( q _ { \phi } ( \psi | D \bar { ) } | | p ( \psi | D \bar { ) } )$ , our objective function directly focuses on the posterior predictive distribution and minimizes $\mathrm { K L } ( p ( \tilde { y } | D ) \| q _ { \phi } ( \tilde { y } | D ) )$ . + +End-to-end stochastic training. Armed by the insights above we now layout the full training procedure. We reintroduce inputs and shared parameters $\theta$ and the objective becomes: + +$$ +\mathcal { L } \left( \phi \right) = - \underset { p ( D , \tilde { y } , \tilde { x } ) } { \mathbb { E } } \left[ \log q _ { \phi } ( \tilde { y } | \tilde { x } , \theta ) \right] = - \underset { p ( D , \tilde { y } , \tilde { x } ) } { \mathbb { E } } \left[ \log \int p ( \tilde { y } | \tilde { x } , \psi , \theta ) q _ { \phi } ( \psi | D , \theta ) \mathrm { d } \psi \right] . +$$ + +We optimize the objective over the shared parameters $\theta$ as this will maximize predictive performance (i.e., Bayesian held out likelihood). An end-to-end stochastic training objective for $\theta$ and $\phi$ is: + +$$ +\hat { \mathcal { L } } \left( \boldsymbol { \theta } , \boldsymbol { \phi } \right) = \frac { 1 } { M T } \sum _ { M , T } \log \frac { 1 } { L } \sum _ { l = 1 } ^ { L } p \left( \tilde { y } _ { m } ^ { ( t ) } | \tilde { x } _ { m } ^ { ( t ) } , \psi _ { l } ^ { ( t ) } , \boldsymbol { \theta } \right) , \quad \mathrm { w i t h ~ } \psi _ { l } ^ { ( t ) } \sim q _ { \boldsymbol { \phi } } ( \boldsymbol { \psi } | D ^ { ( t ) } , \boldsymbol { \theta } ) +$$ + +![](images/1947f1f187d900ab04d56c92f56cfc49c1300157d804236f663049757cd354da.jpg) +Figure 2: Computational flow of VERSA for few-shot classification with the context-independent approximation. Left: A test point $\tilde { x }$ is mapped to its softmax output through a feature extractor neural network and a linear classifier (fully connected layer). The global parameters $\theta$ of the feature extractor are shared between tasks whereas the weight vectors $\dot { w } _ { t } ^ { ( c ) }$ of the linear classifier are task specific and inferred through an amortization network with parameters $\phi$ . Right: Amortization network that maps the extracted features of the $k$ training examples of a particular class to the corresponding weight vector of the linear classifier. + +and $\{ \tilde { y } _ { m } ^ { ( t ) } , \tilde { x } _ { m } ^ { ( t ) } , D ^ { ( t ) } \} \sim p ( \tilde { y } , \tilde { x } , D )$ , where $p$ represents the data distribution (e.g., sampling tasks and splitting them into disjoint training data $D$ and test data $\{ ( \tilde { x } _ { m } ^ { ( t ) } , \tilde { y } _ { m } ^ { ( t ) } ) \} _ { m = 1 } ^ { M _ { t } } )$ . This type of training therefore uses episodic train / test splits at meta-train time. We have also approximated the integral over $\psi$ using $L$ Monte Carlo samples. The local reparametrization (Kingma et al., 2015) trick enables optimization. Interestingly, the learning objective does not require an explicit specification of the prior distribution over parameters, $p ( \psi ^ { ( t ) } | \theta )$ , learning it implicitly through $q _ { \phi } ( \psi | D , \theta )$ instead. + +In summary, we have developed an approach for Meta-Learning Probabilistic Inference for Prediction (ML-PIP). A simple investigation of the inference method with synthetic data is provided in Section 5.1. In Section 4 we will show that this formulation unifies a number of existing approaches, but first we discuss a particular instance of the ML-PIP framework that supports versatile learning. + +# 3 VERSAT ILE AMORT IZED INFERENCE + +A versatile system is one that makes inferences both rapidly and flexibly. By rapidly we mean that test-time inference involves only simple computation such as a feed-forward pass through a neural network. By flexibly we mean that the system supports a variety of tasks – including variable numbers of shots or numbers of classes in classification problems – without retraining. Rapid inference comes automatically with the use of a deep neural network to amortize the approximate posterior distribution $q$ . However, it typically comes at the cost of flexibility: amortized inference is usually limited to a single specific task. Below, we discuss design choices that enable us to retain flexibility. + +Inference with sets as inputs. The amortization network takes data sets of variable size as inputs whose ordering we should be invariant to. We use permutation-invariant instance-pooling operations to process these sets similarly to Qi et al. (2017) and as formalized in Zaheer et al. (2017). The instance-pooling operation ensures that the network can process any number of training observations. + +VERSA for Few-Shot Image Classification. For few-shot image classification, our parameterization of the probabilistic model is inspired by early work from Heskes (2000); Bakker and Heskes (2003) and recent extensions to deep learning (Bauer et al., 2017; Qiao et al., 2017). A feature extractor neural network $h _ { \theta } ( x ) \in \mathbb { R } ^ { d _ { \theta } }$ , shared across all tasks, feeds into a set of task-specific linear classifiers with softmax outputs and weights and biases $\psi ^ { ( t ) } = \{ W ^ { ( t ) } , b ^ { ( t ) } \}$ (see Fig. 2). + +A naive amortization requires the approximate posterior $q _ { \phi } ( \psi | D , \theta )$ to model the distribution over full weight matrices in $\mathbb { R } ^ { d _ { \theta } \times C }$ (and biases). This requires the specification of the number of few-shot classes $C$ ahead of time and limits inference to this chosen number. Moreover, it is difficult to metalearn systems that directly output large matrices as the output dimensionality is high. We therefore propose specifying $q _ { \phi } ( \psi | D , \theta )$ in a context independent manner such that each weight vector $\psi _ { c }$ depends only on examples from class $c$ , by amortizing individual weight vectors associated with a single softmax output instead of the entire weight matrix directly. To reduce the number of learned parameters, the amortization network operates directly on the extracted features $h _ { \theta } ( x )$ : + +![](images/29e22e0c9c513f6c2befeab130b574ebc7b494ef8f9a5f6a7797d92132a91970.jpg) +Figure 3: Computational flow of VERSA for few-shot view reconstruction. Left: A set of training images and angles $\{ ( y _ { n } ^ { ( t ) } , x _ { n } ^ { ( t ) } ) \} _ { n = 1 } ^ { k }$ are mapped to a stochastic input $\psi ^ { ( t ) }$ through the amortization network $q _ { \phi }$ . $\psi ^ { ( t ) }$ is then concatenated with a test angle $\tilde { x }$ and mapped onto a new image through the generator $\theta$ . Right: Amortization network that maps $k$ image/angle examples of a particular object-instance to the corresponding stochastic input. + +$$ +q _ { \phi } ( \psi | D , \theta ) = \prod _ { c = 1 } ^ { C } q _ { \phi } \left( \psi _ { c } | \{ h _ { \theta } \left( x _ { n } ^ { c } \right) \} _ { n = 1 } ^ { k _ { c } } , \theta \right) . +$$ + +Note that in our implementation, end-to-end training is employed, i.e., we backpropagate to $\theta$ through the inference network. Here $k _ { c }$ is the number of observed examples in class $c$ and $\psi _ { c } = \{ w _ { c } , b _ { c } \}$ denotes the weight vector and bias of the linear classifier associated with that class. Thus, we construct the classification matrix $\psi ^ { ( t ) }$ by performing $C$ feed-forward passes through the inference network $q _ { \phi } ( \psi | D , \theta )$ (see Fig. 2). + +The assumption of context independent inference is an approximation. In Appendix B, we provide theoretical and empirical justification for its validity. Our theoretical arguments use insights from Density Ratio Estimation (Mohamed, 2018; Sugiyama et al., 2012), and we empirically demonstrate that full approximate posterior distributions are close to their context independent counterparts. Critically, the context independent approximation addresses all the limitations of a naive amortization mentioned above: (i) the inference network needs to amortize far fewer parameters whose number does not scale with number of classes $C$ (a single weight vector instead of the entire matrix); (ii) the amortization network can be meta-trained with different numbers of classes per task, and (iii) the number of classes $C$ can vary at test-time. + +VERSA for Few-Shot Image Reconstruction (Regression). We consider a challenging few-shot learning task with a complex (high dimensional and continuous) output space. We define view reconstruction as the ability to infer how an object looks from any desired angle based on a small set of observed views. We frame this as a multi-output regression task from a set of training images with known orientations to output images with specified orientations. + +Our generative model is similar to the generator of a GAN or the decoder of a VAE: A latent vector $\psi ^ { ( t ) } \in \mathbb { R } ^ { d _ { \psi } }$ , which acts as an object-instance level input to the generator, is concatenated with an angle representation and mapped through the generator to produce an image at the specified orientation. In this setting, we treat all parameters $\theta$ of the generator network as global parameters (see Appendix E.1 for full details of the architecture), whereas the latent inputs $\psi ^ { ( t ) }$ are the task-specific parameters. We use a Gaussian likelihood in pixel space for the outputs of the generator. To ensure that the output means are between zero and one, we use a sigmoid activation after the final layer. $\phi$ parameterizes an amortization network that first processes the image representations of an object, concatenates them with their associated view orientations, and processes them further before instance-pooling. From the pooled representations, $q _ { \phi } ( \psi | D , \theta )$ produces a distribution over vectors $\psi ^ { ( t ) }$ . This process is illustrated in Fig. 3. + +# 4 ML-PIP UN IFIES DISPARATE RELATED WORK + +In this section, we continue in the spirit of Grant et al. (2018), and recast a broader class of metalearning approaches as approximate inference in hierarchical models. We show that ML-PIP unifies a number of important approaches to meta-learning, including both gradient and metric based variants, as well as amortized MAP inference and conditional modelling approaches (Garnelo et al., 2018a). We lay out these connections, most of which rely on point estimates for the task-specific parameters corresponding to $q ( \psi ^ { ( t ) } | D ^ { ( t ) } , \theta ) = \delta \left( \psi ^ { ( t ) } \bar { - } \psi ^ { * } ( D ^ { ( t ) } , \theta ) \right)$ . In addition, we compare previous approaches to VERSA. + +Gradient-Based Meta-Learning. Let the task-specific parameters $\psi ^ { ( t ) }$ be all the parameters in a neural network. Consider a point estimate formed by taking a step of gradient ascent of the training loss, initialized at $\psi _ { 0 }$ and with learning rate $\eta$ . + +$$ +\psi ^ { * } ( D ^ { ( t ) } , \theta ) = \psi _ { 0 } + \eta \frac { \partial } { \partial \psi } \log \sum _ { n = 1 } ^ { N _ { t } } p ( y _ { n } ^ { ( t ) } | x _ { n } ^ { ( t ) } , \psi , \theta ) \bigg | _ { \psi _ { 0 } } . +$$ + +This is an example of semi-amortized inference (Kim et al., 2018b), as the only shared inference parameters are the initialization and learning rate, and optimization is required for each task (albeit only for one step). Importantly, Eq. (6) recovers Model-agnostic meta-learning (Finn et al., 2017), providing a perspective as semi-amortized ML-PIP. This perspective is complementary to that of Grant et al. (2018) who justify the one-step gradient parameter update employed by MAML through MAP inference and the form of the prior $p ( \psi | \theta )$ . Note that the episodic meta-train / meta-test splits do not fall out of this perspective. Instead we view the update choice as one of amortization which is trained using the predictive KL and naturally recovers the test-train splits. More generally, multiple gradient steps could be fed into an RNN to compute $\psi ^ { * }$ which recovers Ravi and Larochelle (2017). In comparison to these methods, besides being distributional over $\psi$ , VERSA relieves the need to back-propagate through gradient based updates during training and compute gradients at test time, as well as enables the treatment of both local and global parameters which simplifies inference. + +Metric-Based Few-Shot Learning. Let the task-specific parameters be the top layer softmax weights and biases of a neural network $\boldsymbol { \psi } ^ { ( t ) } = \{ w _ { c } ^ { ( t ) } , b _ { c } ^ { ( t ) } \} _ { c = 1 } ^ { C }$ . The shared parameters are the lower layer weights. Consider amortized point estimates for these parameters constructed by averaging the top-layer activations for each class, + +$$ +\psi ^ { * } ( D ^ { ( t ) } , \theta ) = \big \{ w _ { c } ^ { * } , b _ { c } ^ { * } \big \} _ { c = 1 } ^ { C } = \Big \{ \mu _ { c } ^ { ( t ) } , - \| \mu _ { c } ^ { ( t ) } \| ^ { 2 } / 2 \Big \} _ { c = 1 } ^ { C } \mathrm { ~ w h e r e ~ } \mu _ { c } ^ { ( t ) } = \frac { 1 } { k _ { c } } \sum _ { n = 1 } ^ { k _ { c } } h _ { \theta } ( x _ { n } ^ { ( c ) } ) +$$ + +These choices lead to the following predictive distribution: + +$$ +p ( \tilde { y } ^ { ( t ) } = c | \tilde { x } ^ { ( t ) } , \theta ) \propto \exp \left( - d ( h _ { \theta } ( \tilde { x } ^ { ( t ) } ) , \mu _ { c } ^ { ( t ) } ) \right) = \exp \left( h _ { \theta } ( \tilde { x } ^ { ( t ) } ) ^ { T } \mu _ { c } ^ { ( t ) } - \frac { 1 } { 2 } \| \mu _ { c } ^ { ( t ) } \| ^ { 2 } \right) , +$$ + +which recovers prototypical networks (Snell et al., 2017) using a Euclidean distance function $d$ with the final hidden layer being the embedding space. In comparison, VERSA is distributional and it uses a more flexible amortization function that goes beyond averaging of activations. + +Amortized MAP inference. Qiao et al. (2017) proposed a method for predicting weights of classes from activations of a pre-trained network to support i) online learning on a single task to which new few-shot classes are incrementally added, ii) transfer from a high-shot classification task to a separate low-shot classification task. This is an example usage of hyper-networks (Ha et al., 2016) to amortize learning about weights, and can be recovered by the ML-PIP framework by pre-training $\theta$ and performing MAP inference for $\psi$ . VERSA goes beyond point estimates and although its amortization network is similar in spirit, it is more general, employing end-to-end training and supporting full multi-task learning by sharing information between many tasks. + +Conditional models trained via maximum likelihood. In cases where a point estimate of the task-specific parameters are used the predictive becomes + +$$ +q _ { \phi } ( \tilde { y } | D , \theta ) = \int p ( \tilde { y } | \psi , \theta ) q _ { \phi } ( \psi | D , \theta ) \mathrm { d } \psi = p ( \tilde { y } | \psi ^ { * } ( D , \theta ) , \theta ) . +$$ + +In such cases the amortization network that computes $\psi ^ { * } ( D , \theta )$ can be equivalently viewed as part of the model specification rather than the inference scheme. From this perspective, the ML-PIP training procedure for $\phi$ and $\theta$ is equivalent to training a conditional model $p ( \tilde { y } | \psi _ { \phi } ^ { * } ( D , \theta ) , \theta )$ via maximum likelihood estimation, establishing a strong connection to neural processes (Garnelo et al., 2018a;b). + +Comparison to Variational Inference (VI). Standard application of amortized VI (Kingma and Welling, 2014; Rezende et al., 2014; Kingma et al., 2015; Blundell et al., 2015) for $\psi$ in the multi-task discriminative model optimizes the Monte Carlo approximated free-energy w.r.t. $\phi$ and $\theta$ : + +$$ +\hat { \mathcal { L } } ( \boldsymbol { \theta } , \boldsymbol { \phi } ) = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \left( \sum _ { ( \boldsymbol { x } , \boldsymbol { y } ) \in D ^ { ( t ) } } \left( \frac { 1 } { L } \sum _ { l = 1 } ^ { L } \log p ( \boldsymbol { y } ^ { ( t ) } | \boldsymbol { x } ^ { ( t ) } , \boldsymbol { \psi } _ { l } ^ { ( t ) } , \boldsymbol { \theta } ) \right) - \mathrm { K L } \left[ q _ { \boldsymbol { \phi } } ( \boldsymbol { \psi } | D ^ { ( t ) } , \boldsymbol { \theta } ) \| p ( \boldsymbol { \psi } | \boldsymbol { \theta } ) \right] \right) , +$$ + +where $\psi _ { l } ^ { ( t ) } \sim q _ { \phi } ( \psi | D ^ { ( t ) } , \theta )$ . In addition to the conceptual difference from ML-PIP (discussed in Section 2.1), this differs from the ML-PIP objective by i) not employing meta train / test splits, and ii) including the KL for regularization instead. In Section 5, we show that VERSA significantly improves over standard VI in the few-shot classification case and compare to recent VI/meta-learning hybrids. + +# 5 EXPER IMENTS AND RESULTS + +We evaluate VERSA on several few-shot learning tasks. We begin with toy experiments to investigate the properties of the amortized posterior inference achieved by VERSA. We then report few-shot classification results using the Omniglot and miniImageNet datasets in Section 5.2, and demonstrate VERSA’s ability to retain high accuracy as the shot and way are varied at test time. In Section 5.3, we examine VERSA’s performance on a one-shot view reconstruction task with ShapeNet objects.2 + +# 5.1 POSTER IOR INFERENCE W ITH TOY DATA + +To investigate the approximate inference performed by our training procedure, we run the following experiment. We first generate data from a Gaussian distribution with a mean that varies across tasks: + +$$ +p ( \theta ) = \delta ( \theta - 0 ) ; \quad p \left( \psi ^ { ( t ) } | \theta \right) = \mathcal { N } \left( \psi ^ { ( t ) } ; \theta , \sigma _ { \psi } ^ { 2 } \right) ; \quad p \left( y _ { n } ^ { ( t ) } | \psi ^ { ( t ) } \right) = \mathcal { N } \left( y _ { n } ^ { ( t ) } ; \psi ^ { ( t ) } , \sigma _ { y } ^ { 2 } \right) . +$$ + +We generate $T = 2 5 0$ tasks in two separate experiments, having $N \in \{ 5 , 1 0 \}$ train observations and $M = 1 5$ test observations. We introduce the inference network $q _ { \phi } ( \psi | D ^ { ( t ) } ) = \mathcal { N } ( \psi ; \mu _ { q } ^ { ( t ) } , \sigma _ { q } ^ { ( t ) 2 } )$ amortizing inference as: + +$$ +\mu _ { q } ^ { ( t ) } = w _ { \mu } \sum _ { n = 1 } ^ { N } y _ { n } ^ { ( t ) } + b _ { \mu } , \quad \sigma _ { q } ^ { ( t ) 2 } = \exp \left( w _ { \sigma } \sum _ { n = 1 } ^ { N } y _ { n } ^ { ( t ) } + b _ { \sigma } \right) . +$$ + +The learnable parameters $\phi = \{ w _ { \boldsymbol { \mu } } , b _ { \boldsymbol { \mu } } , w _ { \sigma } , b _ { \sigma } \}$ are trained with the objective function in Eq. (4). The model is trained to convergence with Adam (Kingma and Ba, 2015) using mini-batches of tasks from the generated dataset. Then, a separate set of tasks is generated from the same generative process, and the posterior $q _ { \phi } ( \psi | D )$ is inferred with the learned amortization parameters. The true posterior over $\psi$ is Gaussian with a mean that depends on the task, and may be computed analytically. Fig. 4 shows the approximate posterior distributions inferred for unseen test sets by the trained amortization networks. The evaluation shows that the inference procedure is able to recover accurate posterior distributions over $\psi$ , despite minimizing a predictive KL divergence in data space. + +# 5.2 FEW-SHOT CLASS IFICAT ION + +We evaluate VERSA on standard few-shot classification tasks in comparison to previous work. Specifically, we consider the Omniglot (Lake et al., 2011) and miniImageNet (Ravi and Larochelle, 2017) datasets which are $C$ -way classification tasks with $k _ { c }$ examples per class. VERSA follows the implementation in Sections 2 and 3, and the approximate inference scheme in Eq. (5). We follow the experimental protocol established by Vinyals et al. (2016) for Omniglot and Ravi and Larochelle (2017) for miniImagenet, using equivalent architectures for $h _ { \theta }$ . Training is carried out in an episodic manner: for each task, $k _ { c }$ examples are used as training inputs to infer $q _ { \phi } ( \psi ^ { ( c ) } | D , \theta )$ for each class, and an additional set of examples is used to evaluate the objective function. Full details of data preparation and network architectures are provided in Appendix D. + +![](images/18148ee79ba818f9ab1287c364b34b0e0ff936907e4fbaa3b299579ffbed0ca9.jpg) +Figure 4: True posteriors $p ( \psi | D )$ ( ) and approximate posteriors $q _ { \phi } ( \psi | D )$ ( ) for unseen test sets $( { \star } )$ in the experiment. In both cases (five and ten shot), the approximate posterior closely resembles the true posterior given the observed data. + +Table 3 details few-shot classification performance for VERSA as well as competitive approaches. The tables include results for only those approaches with comparable training procedures and convolutional feature extraction architectures. Approaches that employ pre-training and/or residual networks (Bauer et al., 2017; Qiao et al., 2017; Rusu et al., 2018; Gidaris and Komodakis, 2018; Oreshkin et al., 2018; Garcia and Bruna, 2017; Lacoste et al., 2018) have been excluded so that the quality of the learning algorithm can be assessed separately from the power of the underlying discriminative model. + +For Omniglot, the training, validation, and test splits have not been specified for previous methods, affecting the comparison. VERSA achieves a new state-of-the-art results $( 6 7 . 3 7 \%$ - up $1 . 3 8 \%$ over the previous best) on 5-way - 5-shot classification on the miniImageNet benchmark and $9 7 . 6 6 \%$ - up $0 . 0 2 \% )$ on the 20-way - 1 shot Omniglot benchmark for systems using a convolution-based network architecture and an end-to-end training procedure. VERSA is within error bars of state-of-the-art on three other benchmarks including 5-way - 1-shot miniImageNet, 5-way - 5-shot Omniglot, and 5-way - 1-shot Omniglot. Results on the Omniglot 20 way - 5-shot benchmark are very competitive with, but lower than other approaches. While most of the methods evaluated in Table 3 adapt all of the learned parameters for new tasks, VERSA is able to achieve state-of-the-art performance despite adapting only the weights of the top-level classifier. + +Comparison to standard and amortized VI. To investigate the performance of our inference procedure, we compare it in terms of log-likelihood (Table 1) and accuracy (Table 3) to training the same model using both amortized and non-amortized VI (i.e., Eq. (10)). Derivations and further experimental details are provided in Appendix C. VERSA improves substantially over amortized VI even though the same amortization network is used for both. This is due to VI’s tendency to under-fit, especially for small numbers of data points (Trippe and Turner, 2018; Turner and Sahani, 2011) which is compounded when using inference networks (Cremer et al., 2018). Using non-amortized VI improves performance substantially, but does not reach the level of VERSA and forming the posterior is significantly slower as it requires many forward / backward passes through the network. This is similar in spirit to MAML (Finn et al., 2017), though MAML dramatically reduces the number of required iterations by finding good global initializations e.g., five gradient steps for miniImageNet. This is in contrast to the single forward pass required by VERSA. + +Table 1: Negative Log-likelihood (NLL) results for different few-shot settings on Omniglot and miniImageNet. The $\pm$ sign indicates the $9 5 \%$ confidence interval over tasks using a Student’s t-distribution approximation. + +
OmniglotminilmageNet
5-way NLL20-way NLL5-way NLL
Method1-shot5-shot1-shot5-shot1-shot5-shot
Amortized VI0.179 ± 0.0090.137 ± 0.0040.456± 0.0100.253 ± 0.0041.328 ± 0.0241.165 ± 0.010
Non-Amortized VI0.144 ± 0.0050.025 ± 0.0010.393 ± 0.0050.078 ± 0.002
VERSA0.010± 0.0050.007 ± 0.0030.079 ± 0.0090.031 ± 0.0041.183 ± 0.0230.859 ± 0.015
+ +Versatility. VERSA allows us to vary the number of classes $C$ and shots $k _ { c }$ between training and testing (Eq. (5)). Fig. 5a shows that a model trained for a particular $C$ -way retains very high accuracy as $C$ is varied. For example, when VERSA is trained for the 20-Way, 5-Shot condition, at test-time it can handle $C = 1 0 0$ way conditions and retain an accuracy of approximately $94 \%$ . Fig. 5b shows similar robustness as the number of shots $k _ { c }$ is varied. VERSA therefore demonstrates considerable flexibility and robustness to the test-time conditions, but at the same time it is efficient as it only requires forward passes through the network. The time taken to evaluate 1000 test tasks with a 5-way, 5-shot miniImageNet trained model using MAML (https://github.com/cbfinn/maml) is 302.9 seconds whereas VERSA took 53.5 seconds on a NVIDIA Tesla P100-PCIE-16GB GPU. This is more than $5 \times$ speed advantage in favor of VERSA while bettering MAML in accuracy by $4 . 2 6 \%$ . + +![](images/69a7c60c0513a773fe18756537a4337d39a44e0da27404102321c16b9171f824.jpg) +Figure 5: Test accuracy on Omniglot when varying (a) way (fixing shot to be that used for training) and (b) shot. In Fig. 5b, all models are evaluated on 5-way classification. Colors indicate models trained with different way-shot episodic combinations. +Table 2: View reconstruction test results. Mean squared error (MSE – lower is better) and the structural similarity index (SSIM - higher is better) (Wang et al., 2004) are measured between the generated and ground truth images. Error bars not shown as they are insignificant. + +# 5.3 SHAPENET VIEW RECONSTRUCT ION + +ShapeNetCore v2 (Chang et al., 2015) is a database of 3D objects covering 55 common object categories with ${ \sim } 5 1 { , } 3 0 0$ unique objects. For our experiments, we use 12 of the largest object categories. We concatenate all instances from all 12 of the object categories together to obtain a dataset of 37,108 objects. This dataset is then randomly shuffled and we use $70 \%$ of the objects for training, $10 \%$ for validation, and $20 \%$ for testing. For each object, we generate 36 views of size $3 2 \times 3 2$ pixels spaced evenly every 10 degrees in azimuth around the object. + +We evaluate VERSA by comparing it to a conditional variational autoencoder (C-VAE) with view angles as labels (Kingma et al., 2014; Narayanaswamy et al., 2017) and identical architectures. We train VERSA in an episodic manner and the C-VAE in batch-mode on all 12 object classes at once. We train on a single view selected at random and use the remaining views to evaluate the objective function. For full experimentation details see Appendix E. Fig. 6 shows views of unseen objects from the test set generated from a single shot with VERSA as well as a C-VAE and compares both to ground truth views. Both VERSA and the C-VAE capture the correct orientation of the object in the generated images. However, VERSA produces images that contain much more detail and are visually sharper than the C-VAE images. Although important information is missing due to occlusion in the single shot, VERSA is often able to accurately impute this information presumably due to learning the statistics of these objects. Table 2 provides quantitative comparison results between VERSA with varying shot and the C-VAE. The quantitative metrics all show the superiority of VERSA over a C-VAE. As the number of shots increase to 5, the measurements show a corresponding improvement. + +
ModelMSESSIM
C-VAE 1-shot0.02690.5705
VERSA 1-shot0.01080.7893
VERSA 5-shot0.00690.8483
+ +# 6 CONCLUS IONS + +We have introduced ML-PIP, a probabilistic framework for meta-learning. ML-PIP unifies a broad class of recently proposed meta-learning methods, and suggests alternative approaches. Building on ML-PIP, we developed VERSA, a few-shot learning algorithm that avoids the use of gradient based optimization at test time by amortizing posterior inference of task-specific parameters. We evaluated VERSA on several few-shot learning tasks and demonstrated state-of-the-art performance and compelling visual results on a challenging 1-shot view reconstruction task. + +Table 3: Accuracy results for different few-shot settings on Omniglot and miniImageNet. The $\pm$ sign indicates the $9 5 \%$ confidence interval over tasks using a Student’s t-distribution approximation. Bold text indicates the highest scores that overlap in their confidence intervals. + +
OmniglotminilmageNet
Method5-way accuracy (%)5-shot20-way accuracy(%) 1-shot5-shot5-way accuracy (%) 1-shot 5-shot
1-shot98.488.197.0
Siamese Nets (Koch et al., 2015)97.398.946.6
Matching Nets (Vinyals et al.,2016)98.193.898.560.0
Neural Statistician (Edwards and Storkey, 2017)98.199.593.298.1
Memory Mod (Kaiser et al.,2017)98.499.695.098.6
Meta LSTM (Ravi and Larochelle, 2017)43.44 ± 0.77 60.60± 0.71
MAML (Finn et al., 2017)98.7 ± 0.499.9 ± 0.195.8 ± 0.398.9± 0.248.7 ± 1.84 63.11 ± 0.92
Prototypical Netsa (Snell et al., 2017)97.499.395.498.7 46.61 ± 0.7865.77 ± 0.70
mAP-SSVM (Triantafillou et al., 2017)98.699.695.298.650.32 ± 0.80 63.94 ± 0.72
mAP-DLM (Triantafillou et al., 2017)98.899.695.498.6 50.28 ± 0.8063.70± 0.70
LLAMA (Grant et al., 2018)49.40 ± 1.83
PLATIPUS (Finn et al., 2018)50.13 ± 1.86
Meta-SGD (Li et al., 2017)99.53± 0.2699.93 ± 0.0995.93 ± 0.3898.97 ± 0.1950.47 ± 1.87 64.03 ± 0.94
SNAIL (Mishra et al., 2018)99.07 ± 0.1699.78 ± 0.0997.64 ± 0.3099.36 ± 0.18 45.155.2
Relation Net (Yang et al.,2018)99.6 ± 0.299.8 ± 0.197.6 ± 0.299.1 ± 0.1 50.44 ± 0.8265.32 ± 0.70
Reptile (Nichol and Schulman,2018)97.68 ± 0.0499.48 ± 0.0689.43 ± 0.1497.12 ± 0.32 49.97 ± 0.3265.99 ± 0.58
BMAML (Kim et al., 2018a)53.8 ± 1.46
Amortized VI98.71 ± 0.2290.56 ± 0.5444.13 ± 1.78
Non-Amortized VI97.77 ± 0.55 98.77 ± 0.1899.74 ± 0.0695.28 ± 0.1996.12 ± 0.23 98.84± 0.0955.68 ± 0.91
VERSA (Ours)99.70 ± 0.2099.75 ± 0.1397.66 ± 0.2998.77 ± 0.1853.40 ± 1.82 67.37 ± 0.86
+ +![](images/4148f14ceb3482059d23c8517735680aca8d7c7eeb406656865087c074ca44fb.jpg) +Figure 6: Results for ShapeNet view reconstruction for unseen objects from the test set (shown left). The model was trained to reconstruct views from a single orientation. Top row: images/views generated by a C-VAE model; middle row images/views generated by VERSA; bottom row: ground truth images. Views are spaced evenly every 30 degrees in azimuth. + +# ACKNOWLEDGEMENTS + +We thank Ravi and Larochelle for providing the miniImageNet dataset, and Yingzhen Li, Niki Kilbertus, Will Tebbutt, Maria Lomelli, and Robert Pinsler for their useful feedback. J.G. acknowledges funding from a Samsung Doctoral Scholarship. M.B. acknowledges funding by the EPSRC and a Qualcomm European Scholarship in Technology. R.E.T. acknowledges support from EPSRC grants EP/M0269571 and EP/L000776/1. + +# REFERENCES + +B. Bakker and T. Heskes. Task clustering and gating for Bayesian multitask learning. 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Few-shot learning through an information retrieval lens. In Advances in Neural Information Processing Systems, pages 2255–2265, 2017. +B. Trippe and R. Turner. Overpruning in variational bayesian neural networks. arXiv preprint arXiv:1801.06230, 2018. +R. E. Turner and M. Sahani. Two problems with variational expectation maximisation for time-series models. Bayesian Time series models, 1(3.1):3–1, 2011. +O. Vinyals, C. Blundell, T. Lillicrap, D. Wierstra, et al. Matching networks for one shot learning. In Advances in Neural Information Processing Systems, pages 3630–3638, 2016. +M. J. Wainwright and M. I. Jordan. Graphical models, exponential families, and variational inference. Foundations and Trends $\textsuperscript { \textregistered }$ in Machine Learning, 1(1-2):1–305, 2008. +Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE transactions on image processing, 13(4):600–612, 2004. +F. S. Y. Yang, L. Zhang, T. Xiang, P. H. Torr, and T. M. Hospedales. Learning to compare: Relation network for few-shot learning. 2018. +M. Zaheer, S. Kottur, S. Ravanbakhsh, B. Poczos, R. R. Salakhutdinov, and A. J. Smola. Deep sets. In Advances in Neural Information Processing Systems, pages 3394–3404, 2017. + +# A BAYES IAN DEC IS ION THEORET IC GENERAL IZAT ION OF ML-PIP + +A generalization of the new inference framework presented in Section 2 is based upon Bayesian decision theory (BDT). BDT provides a recipe for making predictions $\hat { y }$ for an unknown test variable $\tilde { y }$ by combining information from observed training data $\bar { D } ^ { ( t ) }$ (here from a single task $t$ ) and a loss function $L ( \tilde { y } , \hat { y } )$ that encodes the cost of predicting $\hat { y }$ when the true value is $\tilde { y }$ (Berger, 2013; Jaynes, 2003). In BDT an optimal prediction minimizes the expected loss (suppressing dependencies on the inputs and $\theta$ to reduce notational clutter):3 + +$$ +\hat { y } ^ { * } = \underset { \hat { y } } { \mathrm { a r g m i n } } \int p ( \tilde { y } | D ^ { ( t ) } ) L ( \tilde { y } , \hat { y } ) \mathrm { d } \tilde { y } , \quad \mathrm { w h e r e } \quad p ( \tilde { y } | D ^ { ( t ) } ) = \int p ( \tilde { y } | \psi ^ { ( t ) } ) p ( \psi ^ { ( t ) } | D ^ { ( t ) } ) \mathrm { d } \psi ^ { ( t ) } +$$ + +is the Bayesian predictive distribution and $p ( \psi ^ { ( t ) } | D ^ { ( t ) } )$ the posterior distribution of $\psi ^ { ( t ) }$ given the training data from task $t$ . + +BDT separates test and training data and so is a natural lens through which to view recent episodic approaches to training that utilize many internal training/test splits (Vinyals et al., 2016). Based on this insight, what follows is a fairly dense derivation of an ultimately simple stochastic variational objective for meta-learning probabilistic inference that is rigorously grounded in Bayesian inference and decision theory. + +Distributional BDT. We generalize BDT to cases where the goal is to return a full predictive distribution $q ( \cdot )$ over the unknown test variable $\tilde { y }$ rather than a point prediction. The quality of $q$ is quantified through a distributional loss function $L ( \tilde { y } , q ( \cdot ) )$ . Typically, if $\tilde { y }$ (the true value of the underlying variable) falls in a low probability region of $q ( \cdot )$ the loss will be high, and vice versa. The optimal predictive $q ^ { * }$ is found by optimizing the expected distributional loss with $q$ constrained to a distributional family $\mathcal { Q }$ : + +$$ +q ^ { * } = \operatorname * { a r g m i n } _ { \boldsymbol { q } \in \mathcal { Q } } \int p ( \boldsymbol { \tilde { y } } | D ^ { ( t ) } ) L ( \boldsymbol { \tilde { y } } , \boldsymbol { q } ( \cdot ) ) \mathrm { d } \boldsymbol { \tilde { y } } . +$$ + +Amortized variational training. Here, we amortize $q$ to form quick predictions at test time and learn parameters by minimizing average expected loss over tasks. Let $\phi$ be a set of shared variational parameters such that $q ( \tilde { y } ) = q _ { \phi } \overline { { ( \tilde { y } | D ) } }$ (or $q _ { \phi }$ for shorthand). Now the approximate predictive distribution can take any training dataset $D ^ { ( t ) }$ as an argument and directly perform prediction of $\tilde { y } ^ { ( t ) }$ . The optimal variational parameters are found by minimizing the expected distributional loss across tasks + +$$ +\phi ^ { * } = \underset { \phi } { \mathrm { a r g m i n } } \mathcal { L } \left[ q _ { \phi } \right] , \mathcal { L } \left[ q _ { \phi } \right] = \int p ( D ) p ( \tilde { y } | D ) L ( \tilde { y } , q _ { \phi } ( \cdot | D ) ) \mathrm { d } \tilde { y } \mathrm { d } D = \mathbb { E } _ { p ( D , \tilde { y } ) } [ L ( \tilde { y } , q _ { \phi } ( \cdot | D ) ) ] . +$$ + +Here the variables $D , { \tilde { x } }$ and $\tilde { y }$ are placeholders for integration over all possible datasets, test inputs and outputs. Note that Eq. (A.3) can be stochastically approximated by sampling a task $t$ and randomly partitioning into training data $D$ and test data $\{ \tilde { x } _ { m } , \bar { \tilde { y } } _ { m } \} _ { m = 1 } ^ { M }$ , which naturally recovers episodic minibatch training over tasks and data (Vinyals et al., 2016; Ravi and Larochelle, 2017). Critically, this does not require computation of the true predictive distribution. It also emphasizes the meta-learning aspect of the procedure, as the model is learning how to infer predictive distributions from training tasks. + +Loss functions. We employ the log-loss: the negative log density of $q _ { \phi }$ at $\tilde { y }$ . In this case, + +$$ +\mathcal { L } \left[ q _ { \phi } \right] = \mathbb { E } _ { p \left( D , \tilde { y } \right) } \left[ - \log q _ { \phi } ( \tilde { y } | D ) \right] = \mathbb { E } _ { p \left( D \right) } \left[ \mathrm { K L } \left[ p ( \tilde { y } | D ) \| q _ { \phi } ( \tilde { y } | D ) \right] + \mathrm { H } \left[ p ( \tilde { y } | D ) \right] \right] , +$$ + +where $\mathrm { K L } [ p ( y ) | | q ( y ) ]$ is the KL-divergence, and $\mathrm { ~ H ~ } [ p ( y ) ]$ is the entropy of $p$ . Eq. (A.4) has the elegant property that the optimal $q _ { \phi }$ is the closest member of $\mathcal { Q }$ (in a KL sense) to the true predictive $p ( \tilde { y } | D )$ , which is unsurprising as the log-loss is a proper scoring rule (Huszar, 2013). This is reminiscent of the sleep phase in the wake-sleep algorithm (Hinton et al., 1995). Exploration of alternative proper scoring rules (Dawid, 2007) and more task-specific losses (Lacoste-Julien et al., 2011) is left for future work. + +Specification of the approximate predictive distribution. Next, we consider the form of $q _ { \phi }$ . Motivated by the optimal predictive distribution, we replace the true posterior by an approximation: + +$$ +q _ { \phi } ( \tilde { y } | D ) = \int p ( \tilde { y } | \psi ) q _ { \phi } ( \psi | D ) \mathrm { d } \psi . +$$ + +# B JUST IFICAT ION FOR CONTEXT-INDEPENDENT APPROX IMAT ION + +In this section we lay out both theoretical and empirical justifications for the context-independent approximation detailed in Section 3. + +# B.1 THEORET ICAL ARGUMENT – DENS ITY RAT IO EST IMAT ION + +A principled justification for the approximation is best understood through the lens of density ratio estimation (Mohamed, 2018; Sugiyama et al., 2012). We denote the conditional density of each class as $p ( x | y = c )$ and assume equal a priori class probability $p ( y = c ) = 1 / C$ . Density ratio theory then uses Bayes’ theorem to show that the optimal softmax classifier can be expressed in terms of the conditional densities (Mohamed, 2018; Sugiyama et al., 2012): + +$$ +\mathrm { S o f t m a x } ( y = c | x ) = \frac { \exp ( h ( x ) ^ { \top } w _ { c } ) } { \sum _ { c ^ { \prime } } \exp ( h ( x ) ^ { \top } w _ { c ^ { \prime } } ) } = p ( y = c | x ) = \frac { p ( x | y = c ) } { \sum _ { c ^ { \prime } } p ( x | y = c ^ { \prime } ) } , +$$ + +This implies that the optimal classifier will construct estimators for the conditional density for each class, that is $\mathrm { e x p } ( h ( x ) ^ { \top } w _ { c } ) \propto p ( x | y = c )$ . Importantly for our approximation, notice that these estimates are constructed independently for each class, similarly to training a naive Bayes classifier. VERSA mirrors this optimal form using: + +$$ +\log p ( x | y = c ) \propto h _ { \theta } ( \tilde { x } ) ^ { \top } w _ { c } , +$$ + +where $w _ { c } \sim q _ { \phi } \left( w | \{ x _ { n } | y _ { n } = c \} \right)$ ) for each class in a given task. Under ideal conditions (i.e., if one could perfectly estimate $p ( \tilde { x } | y = c )$ ), the context-independent assumption holds, further motivating our design. + +# B.2 EMP IR ICAL JUST IFICAT ION + +Here we detail a simple experiment to evaluate the validity of the context-independent inference assumption. The goal of the experiment is to examine if weights may be context-independent without imposing the assumption on the amortization network. To see this, we randomly generate fifty tasks from a dataset, where classes may appear a number of times in different tasks. We then perform free-form (non-amortized) variational inference on the weights for each of the tasks, with a Gaussian variational distribution: + +$$ +\begin{array} { r } { q _ { \phi } \left( W ^ { ( t ) } | \mathcal { D } ^ { ( t ) } , \theta \right) = \mathcal { N } \left( W ^ { ( t ) } ; \mu _ { \phi } ^ { ( t ) } , \sigma _ { \phi } ^ { ( t ) 2 } \right) . } \end{array} +$$ + +If the assumption is reasonable, we may expect the distribution of the weights of a specific class to be similar regardless of the additional classes in the task. + +We examine 5-way classification in the MNIST dataset. We randomly sample and fix fifty such tasks. We train the model twice using the same feature extraction network used in the few-shot classification experiments, and fix the $d _ { \theta }$ to be 16 and 2. We then train the model in an episodic manner by minibatching tasks at each iteration. The model is trained to convergence, and achieves $9 9 \%$ accuracy on held out test examples for the tasks. After training is complete we examine the optimized $\mu _ { \phi } ^ { ( t ) }$ for each class in each task. Fig. B.1a shows a t-SNE (Maaten and Hinton, 2008) plot for the 16-dimensional weights. We see that when reduced to 2-dimensions, the weights cluster according to class. Fig. B.1b visualizes the weights in their original space. In this plot, weights from the same class are grouped together, and clear similarity patterns are evident across the image, showing that weights from the same class have similar means across tasks. Fig. B.2 details the task weights in 2-dimensional space. Here, each pentagon represents the weight means learned for one training task, where the nodes of the pentagon are colored according to the class the weights represent. In Fig. B.2a we see that overall, the classes cluster in 2-dimensional space as well. However, there is some overlap (e.g., classes ‘1’ and $\cdot _ { 2 } \cdot ,$ ), and that for some tasks a class-weight may appear away from the cluster. Fig. B.2b shows the same plot, but only for tasks that contain both class ‘1’ and ‘2’. Here we can see that for these tasks, class $\bullet _ { 2 } \cdot$ weights are all located away from their cluster. + +![](images/07839a600a28d6118a99a997c66579d785371a45435e63c2ca530bf156ab5c22.jpg) +Figure B.1: Visualizing the learned weights for $d _ { \theta } = 1 6$ . (a) Weight dimensionality is reduced using T-SNE (Maaten and Hinton, 2008). Weights are colored according to class. (b) Each weight represents one column of the image. Weights are grouped by class. + +![](images/6a6bb38f30c8cc0c065163afb7af7683b5fc6f6bb5a982c629606c9ac13733bb.jpg) +Figure B.2: Visualizing the task weights for $d _ { \theta } = 2$ . (a) All training tasks. (b) Only training tasks containing both ‘1’s and $\mathrm { { } ^ { \circ } 2 \mathrm { { } s } }$ . + +This implies that each class-weights are typically well-approximated as being independent of the task. However, if the model lacks capacity to properly assign each set of class weights to different regions of space, for tasks where classes from similar regions of space appear, the inference procedure will ‘move’ one of the class weights to an ‘empty’ region of the space. + +# C VAR IAT IONAL INFERENCE DER IVAT IONS FOR THE MODEL + +We derive a VI-based objective for our probabilistic model. By “amortized” VI we mean that $q _ { \phi } ( \psi | D ^ { ( t ) } , \theta )$ is parameterized by a neural network with a fixed-sized $\phi$ . Conversely, “non-amortized” VI refers to local parameters $\phi ^ { ( t ) }$ that are optimized independently (at test time) for each new task $t$ , such that $\begin{array} { r } { q ( \psi | D ^ { ( t ) } , \theta ) = \mathcal { N } ( \psi | \mu _ { \phi ^ { ( t ) } } , \Sigma _ { \phi ^ { ( t ) } } ) } \end{array}$ . However, the derivation of the objective function does not change between these options. For a single task $t$ , an evidence lower bound (ELBO; (Wainwright and Jordan, 2008)) may be expressed as: + +$$ +\mathcal { L } _ { t } = \mathbb { E } _ { q _ { \phi } ( \psi | D ^ { ( t ) } , \theta ) } \left[ \sum _ { ( \boldsymbol { x } , \boldsymbol { y } ) \in D ^ { ( t ) } } \log p ( \boldsymbol { y } | \boldsymbol { x } , \psi , \theta ) \right] - \mathrm { K L } \left[ q _ { \phi } ( \psi | D ^ { ( t ) } , \theta ) \| p ( \psi | \theta ) \right] . +$$ + +We can then derive a stochastic estimator to optimize Eq. (C.1) by sampling $D ^ { ( t ) } \sim p ( D )$ (approximated with a training set of tasks) and simple Monte Carlo integration over $\psi$ such that $\mathsf { \bar { \psi } } ^ { ( l ) } \sim q _ { \phi } ( \psi | D ^ { ( t ) } , \theta )$ : + +$$ +\hat { \mathcal { L } } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \left( \sum _ { \left( x , y \right) \in D ^ { \left( t \right) } } \left( \frac { 1 } { L } \sum _ { l = 1 } ^ { L } \log p ( y | x , \psi ^ { \left( l \right) } , \theta ) \right) - \mathrm { K L } \left[ q _ { \phi } ( \psi | D ^ { \left( t \right) } , \theta ) \| p ( \psi | \theta ) \right] \right) , +$$ + +Eq. (C.2) differs from our objective function in Eq. (4) in two important ways: (i) Eq. (4) does not contain a KL term for ${ q _ { \phi } } ( \psi | D ^ { ( t ) } , \theta )$ (nor any other form of prior distribution over $\psi$ , and (ii) Eq. (C.1) does not distinguish between training and test data within a task, and therefore does not explicitly encourage the model to generalize in any way. + +# D EXPER IMENTAT ION DETA ILS + +In this section we provide comprehensive details on the few-shot classification experiments. + +# D.1 OMN IGLOT FEW-SHOT CLASS IFICAT ION TRA IN ING PROCEDURE + +Omniglot (Lake et al., 2011) is a few-shot learning dataset consisting of 1623 handwritten characters (each with 20 instances) derived from 50 alphabets. We follow a pre-processing and training procedure akin to that defined in (Vinyals et al., 2016). First the images are resized to $2 8 \times 2 8$ pixels and then character classes are augmented with rotations of 90 degrees. The training, validation and test sets consist of a random split of 1100, 100, and 423 characters, respectively. When augmented this results in 4400 training, 400 validation, and 1292 test classes, each having 20 character instances. For $C$ -way, $k _ { c }$ -shot classification, training proceeds in an episodic manner. Each training iteration consists of a batch of one or more tasks. For each task $C$ classes are selected at random from the training set. During training, $k _ { c }$ character instances are used as training inputs and 15 character instances are used as test inputs. The validation set is used to monitor the progress of learning and to select the best model to test, but does not affect the training process. Final evaluation of the trained model is done on 600 randomly selected tasks from the test set. During evaluation, $k _ { c }$ character instances are used as training inputs and $k _ { c }$ character instances are used as test inputs. We use the Adam (Kingma and Ba, 2015) optimizer with a constant learning rate of 0.0001 with 16 tasks per batch to train all models. The 5-way - 5-shot and 5-way - 1-shot models are trained for 80,000 iterations while the 20-way - 5-shot model is trained for 60,000 iterations, and the 20-way - 1-shot model is trained for 100,000 iterations. In addition, we use a Gaussian form for $q$ and set the number of $\psi$ samples to $L = 1 0$ . + +# D.2 miniIMAGENET FEW-SHOT CLASS IFICAT ION TRA IN ING PROCEDURE + +miniImageNet (Vinyals et al., 2016) is a dataset of 60,000 color images that is sub-divided into 100 classes, each with 600 instances. The images have dimensions of $8 4 \times 8 4$ pixels. For our experiments, we use the 64 training, 16 validation, and 20 test class splits defined by (Ravi and Larochelle, 2017). Training proceeds in the same episodic manner as with Omniglot. We use the Adam (Kingma and Ba, 2015) optimizer and a Gaussian form for $q$ and set the number of $\psi$ samples to $L = 1 0$ . For the 5-way - 5-shot model, we train using 4 tasks per batch for 100,000 iterations and use a constant learning rate of 0.0001. For the 5-way - 1-shot model, we train with 8 tasks per batch for 50,000 iterations and use a constant learning rate of 0.00025. + +# D.3 FEW-SHOT CLASS IFICAT ION NETWORK ARCH ITECTURES + +Tables D.1 to D.4 detail the neural network architectures for the feature extractor $\theta$ , amortization network $\phi$ , and linear classifier $\psi$ , respectively. The feature extraction network is very similar to that used in (Vinyals et al., 2016). The output of the amortization network yields mean-field Gaussian parameters for the weight distributions of the linear classifier $\psi$ . When sampling from the weight distributions, we employ the local-reparameterization trick (Kingma et al., 2015), that is we sample from the implied distribution over the logits rather than directly from the variational distribution. To reduce the number of learned parameters, we share the feature extraction network $\theta$ with the pre-processing phase of the amortizaion network $\psi$ . + +Table D.1: Feature extraction network used for Omniglot few-shot learning. Batch Normalization and dropout with a keep probability of 0.9 used throughout. +Omniglot Shared Feature Extraction Network $\mathbf { \eta } ^ { ( \theta ) }$ : $\tilde { x } h _ { \theta } ( \tilde { x } )$ + +
Output sizeLayers
28×28×1Input image
14 ×14 × 64conv2d (3 × 3, stride 1, SAME,RELU),dropout, pool (2 × 2, stride 2, SAME)
7×7×64conv2d (3 × 3, stride 1, SAME,RELU),dropout, pool (2 × 2, stride 2, SAME)
4×4×64conv2d (3 × 3, stride 1, SAME,RELU),dropout, pool (2 × 2, stride 2, SAME)
2×2×64conv2d (3 × 3,stride 1, SAME,RELU),dropout,pool (2 × 2, stride 2, SAME)
256flatten
+ +Table D.2: Feature extraction network used for miniImageNet few-shot learning. Batch Normalization and dropout with a keep probability of 0.5 used throughout. + +miniImageNet Shared Feature Extraction Network (θ): $\tilde { x } h _ { \theta } ( \tilde { x } )$ + +
Output sizeLayers
84×84×1Input image
42 × 42× 64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
21×21×64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
10 ×10×64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
5×5×64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
2×2×64 256conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID) flatten
+ +# E SHAPENET EXPER IMENTAT ION DETA ILS + +# E.1 VIEW RECONSTRUCT ION TRA IN ING PROCEDURE AND NETWORK ARCH ITECTURES + +ShapeNetCore v2 (Chang et al., 2015) is an annotated database of 3D objects covering 55 common object categories with ${ \sim } 5 1 { , } 3 0 0$ unique objects. For our experiments, we use 12 of the largest object categories. Refer to Table E.1 for a complete list. We concatenate all instances from all 12 of the object categories together to obtain a dataset of 37,108 objects. This concatenated dataset is then randomly shuffled and we use $70 \%$ of the objects (25,975 in total) for training, $10 \%$ for validation (3,710 in total) , and $20 \%$ (7423 in total) for testing. For each object, we generate $V = 3 6$ , $1 2 8 \times 1 2 8$ pixel image views spaced evenly every 10 degrees in azimuth around the object. We then convert the rendered images to gray-scale and reduce their size to be $3 2 \times 3 2$ pixels. Again, we train our model in an episodic manner. Each training iteration consists a batch of one or more tasks. For each task an object is selected at random from the training set. We train on a single view selected at random from the $V = 3 6$ views associated with each object and use the remaining 35 views to evaluate the objective function. We then generate 36 views of the object with a modified version of our amortization network which is shown diagrammatically in Fig. 3. To evaluate the system, we generate views and compute quantitative metrics over the entire test set. Tables E.2 to E.4 describe the network architectures for the encoder, amortization, and generator networks, respectively. To train, we use the Adam (Kingma and Ba, 2015) optimizer with a constant learning rate of 0.0001 with 24 tasks per batch for 500,000 training iterations. In addition, we set $d _ { \phi } = 2 5 6$ , $d _ { \psi } = 2 5 6$ and number of $\psi$ samples to 1. + +Table D.3: Amortization network used for Omniglot and miniImageNet few-shot learning. + +
PhaseOutput size Layers
feature extraction instance pooling weight distributionk×256shared feature network (0)
256mean
2562 × fully connected, ELU+
linear fully connected to μw(c), ² .2 w(c)
+ +Table D.4: Linear classifier used for Omniglot and miniImageNet few-shot learning. + +
Linear Classifier (ψ): hθ(x) →p(y|𝑥,0,ψt)
Output sizeLayers
256Input features
Cfully connected, softmax
+ +Table E.1: List of ShapeNet categories used in the VERSA view reconstruction experiments. + +
Object Categorysysnet IDInstances
airplane026911564045
bench028288841813
cabinet029331121571
car029583433533
phone02992529831
chair030016276778
display032111171093
lamp036366492318
speaker036914591597
sofa042565203173
table043792438436
boat045305661939
+ +Table E.2: Encoder network used for ShapeNet few-shot learning. No dropout or batch normalization is used. +ShapeNet Encoder Network (φ): $y h$ + +
Output size Layers
32 ×32×1Input image
16 ×16×64conv2d (3 × 3, stride 1, SAME,RELU), pool (2 × 2, stride2,VALID)
8×8×64conv2d (3 × 3, stride 1, SAME,RELU),pool (2 × 2, stride 2, VALID)
4×4×64conv2d (3 × 3, stride 1, SAME,RELU), pool (2 × 2, stride 2,VALID)
2×2×64conv2d (3 × 3, stride 1, SAME,RELU),pool (2 × 2, stride 2, VALID)
dfully connected, RELU
+ +Table E.3: Amortization network used for ShapeNet few-shot learning. +ShapeNet Amortization Network $( \phi )$ $: x _ { 1 } ^ { ( t ) } , . . . , x _ { k } ^ { ( t ) } , y _ { 1 } ^ { ( t ) } , . . . , y _ { k } ^ { ( t ) } \mu _ { \psi } , \sigma _ { \psi } ^ { 2 }$ + +
PhaseOutput sizeLayers
Φprek×dencoder network ()
concatenate h and Xk ×(dp+dx)concat(h, X)
Φmidkxd2 × 2 fully connected,ELU
instance pooling1×daverage
post1×d2× fully connected, ELU
distributiondfully connected linear layers to μψ,
+ +Table E.4: Generator network used for ShapeNet few-shot learning. No dropout or batch normalization is used. +ShapeNet Generator Network (θ): $\tilde { x } p ( \tilde { y } | \tilde { x } , \theta , \psi ^ { ( t ) } )$ + +
Output sizeLayers
d+dxconcat(y,x)
512fully connected, RELU
1024fully connected, RELU
2×2×256reshape
4×4×128deconv2d (3 × 3, stride 2, SAME, RELU)
8×8×64deconv2d (3 × 3, stride 2, SAME,RELU)
16 ×16 × 32deconv2d (3 × 3, stride 2, SAME,RELU)
32×32×1deconv2d (3 × 3, stride 2, SAME,sigmoid)
\ No newline at end of file diff --git a/parse/train/HkxStoC5F7/HkxStoC5F7_content_list.json b/parse/train/HkxStoC5F7/HkxStoC5F7_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..14182221009c98b458f35c3cc7dd5f9646e64b3a --- /dev/null +++ b/parse/train/HkxStoC5F7/HkxStoC5F7_content_list.json @@ -0,0 +1,1913 @@ +[ + { + "type": "text", + "text": "META-LEARN ING PROBAB IL IST IC INFERENCE FOR PRED ICT ION ", + "text_level": 1, + "bbox": [ + 174, + 98, + 823, + 145 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jonathan Gordon∗, John Bronskill∗ University of Cambridge {jg801,jfb54}@cam.ac.uk ", + "bbox": [ + 184, + 170, + 424, + 213 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Matthias Bauer \nUniversity of Cambridge \nMax Planck Institute for Intelligent Systems \nbauer@tue.mpg.de ", + "bbox": [ + 522, + 170, + 813, + 226 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Sebastian Nowozin† Google AI Berlin nowozin@google.com ", + "bbox": [ + 183, + 247, + 362, + 289 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Richard E. Turner University of Cambridge Microsoft Research ret26@cam.ac.uk ", + "bbox": [ + 519, + 247, + 683, + 303 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 452, + 339, + 545, + 354 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This paper introduces a new framework for data efficient and versatile learning. Specifically: 1) We develop ML-PIP, a general framework for Meta-Learning approximate Probabilistic Inference for Prediction. ML-PIP extends existing probabilistic interpretations of meta-learning to cover a broad class of methods. 2) We introduce VERSA, an instance of the framework employing a flexible and versatile amortization network that takes few-shot learning datasets as inputs, with arbitrary numbers of shots, and outputs a distribution over task-specific parameters in a single forward pass. VERSA substitutes optimization at test time with forward passes through inference networks, amortizing the cost of inference and relieving the need for second derivatives during training. 3) We evaluate VERSA on benchmark datasets where the method sets new state-of-the-art results, handles arbitrary numbers of shots, and for classification, arbitrary numbers of classes at train and test time. The power of the approach is then demonstrated through a challenging few-shot ShapeNet view reconstruction task. ", + "bbox": [ + 233, + 371, + 766, + 565 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCT ION ", + "text_level": 1, + "bbox": [ + 176, + 592, + 341, + 607 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many applications require predictions to be made on myriad small, but related datasets. In such cases, it is natural to desire learners that can rapidly adapt to new datasets at test time. These applications have given rise to vast interest in few-shot learning (Fei-Fei et al., 2006; Lake et al., 2011), which emphasizes data efficiency via information sharing across related tasks. Despite recent advances, notably in meta-learning based approaches (Ravi and Larochelle, 2017; Vinyals et al., 2016; Edwards and Storkey, 2017; Finn et al., 2017; Lacoste et al., 2018), there remains a lack of general purpose methods for flexible, data-efficient learning. ", + "bbox": [ + 174, + 622, + 825, + 720 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Due to the ubiquity of recent work, a unifying view is needed to understand and improve these methods. Existing frameworks (Grant et al., 2018; Finn et al., 2018) are limited to specific families of approaches. In this paper we develop a framework for meta-learning approximate probabilistic inference for prediction (ML-PIP), providing this view in terms of amortizing posterior predictive distributions. In Section 4, we show that ML-PIP re-frames and extends existing point-estimate probabilistic interpretations of meta-learning (Grant et al., 2018; Finn et al., 2018) to cover a broader class of methods, including gradient based meta-learning (Finn et al., 2017; Ravi and Larochelle, 2017), metric based meta-learning (Snell et al., 2017), amortized MAP inference (Qiao et al., 2017) and conditional probability modelling (Garnelo et al., 2018a;b). ", + "bbox": [ + 174, + 727, + 825, + 853 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The framework incorporates three key elements. First, we leverage shared statistical structure between tasks via hierarchical probabilistic models developed for multi-task and transfer learning (Heskes, ", + "bbox": [ + 176, + 859, + 823, + 887 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "2000; Bakker and Heskes, 2003). Second, we share information between tasks about how to learn and perform inference using meta-learning (Naik and Mammone, 1992; Thrun and Pratt, 2012; Schmidhuber, 1987). Since uncertainty is rife in small datasets, we provide a procedure for metalearning probabilistic inference. Third, we enable fast learning that can flexibly handle a wide range of tasks and learning settings via amortization (Kingma and Welling, 2014; Rezende et al., 2014). ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Building on the framework, we propose a new method – VERSA – which substitutes optimization procedures at test time with forward passes through inference networks. This amortizes the cost of inference, resulting in faster test-time performance, and relieves the need for second derivatives during training. VERSA employs a flexible amortization network that takes few-shot learning datasets, and outputs a distribution over task-specific parameters in a single forward pass. The network can handle arbitrary numbers of shots, and for classification, arbitrary numbers of classes at train and test time (see Section 3). In Section 5, we evaluate VERSA on (i) standard benchmarks where the method sets new state-of-the-art results, (ii) settings where test conditions (shot and way) differ from training, and (iii) a challenging one-shot view reconstruction task. ", + "bbox": [ + 173, + 180, + 825, + 305 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 META-LEARN ING PROBAB IL IST IC INFERENCE FOR PRED ICT ION ", + "text_level": 1, + "bbox": [ + 171, + 325, + 759, + 342 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We now present the framework that consists of (i) a multi-task probabilistic model, and (ii) a method for meta-learning probabilistic inference. ", + "bbox": [ + 173, + 356, + 823, + 385 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 PROBAB IL IST IC MODEL ", + "text_level": 1, + "bbox": [ + 174, + 401, + 387, + 416 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Two principles guide the choice of model. First, the use of discriminative models to maximize predictive performance on supervised learning tasks $\\mathrm { N g }$ and Jordan, 2002). Second, the need to leverage shared statistical structure between tasks (i.e. multi-task learning). These criteria are met by the standard multi-task directed graphical model shown in Fig. 1 that employs shared parameters $\\theta$ , which are common to all tasks, and task specific parameters $\\{ \\psi ^ { ( t ) } \\} _ { t = 1 } ^ { T }$ . Inputs are denoted $x$ and outputs $y$ . Training data $D ^ { ( t ) } = \\{ ( x _ { n } ^ { ( t ) } , y _ { n } ^ { ( t ) } ) \\} _ { n = 1 } ^ { N _ { t } }$ , and test data $\\{ ( \\tilde { x } _ { m } ^ { ( t ) } , \\tilde { y } _ { m } ^ { ( t ) } ) \\} _ { m = 1 } ^ { M _ { t } }$ are explicitly distinguished for each task , as this is key for few-shot learning. ", + "bbox": [ + 173, + 426, + 826, + 530 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/68029ff8f47704e3ae4f511823dad2c320911504c271f201f15d84dbc23b17c9.jpg", + "image_caption": [ + "Figure 1: Directed graphical model for multi-task learning. " + ], + "image_footnote": [], + "bbox": [ + 326, + 541, + 671, + 661 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Let $X ^ { ( t ) }$ and $Y ^ { ( t ) }$ denote all the inputs and outputs (both test and train) for task $t$ . The joint probability of the outputs and task specific parameters for $T$ tasks, given the inputs and global parameters is: ", + "bbox": [ + 173, + 708, + 825, + 737 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/035e1ab8e7de4fa6f78a83d361a72f279275f343d4d3707139eb0bded10cd1ad.jpg", + "text": "$$\n\\small \\rho \\left( \\{ Y ^ { ( t ) } , \\psi ^ { ( t ) } \\} _ { t = 1 } ^ { T } | \\{ X ^ { ( t ) } \\} _ { t = 1 } ^ { T } , \\boldsymbol \\theta \\right) = \\prod _ { t = 1 } ^ { T } p \\left( \\psi ^ { ( t ) } | \\theta \\right) \\prod _ { n = 1 } ^ { N _ { t } } p \\left( y _ { n } ^ { ( t ) } | x _ { n } ^ { ( t ) } , \\psi ^ { ( t ) } , \\theta \\right) \\prod _ { m = 1 } ^ { M _ { t } } p \\left( \\tilde { y } _ { m } ^ { ( t ) } | \\tilde { x } _ { m } ^ { ( t ) } , \\psi ^ { ( t ) } , \\theta \\right)\n$$", + "text_format": "latex", + "bbox": [ + 181, + 741, + 839, + 785 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the next section, the goal is to meta-learn fast and accurate approximations to the posterior predictive distribution $\\begin{array} { r } { p ( \\tilde { y } ^ { ( t ) } | \\tilde { x } ^ { ( t ) } , \\theta ) = \\int p ( \\tilde { y } ^ { ( t ) } | \\tilde { x } ^ { ( t ) } , \\psi ^ { ( t ) } , \\theta ) p ( \\psi ^ { ( t ) } | \\tilde { x } ^ { ( t ) } , D ^ { ( t ) } , \\theta ) \\mathrm { d } \\psi ^ { ( t ) } } \\end{array}$ for unseen tasks $t$ . ", + "bbox": [ + 173, + 795, + 828, + 827 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 PROBAB IL IST IC INFERENCE ", + "text_level": 1, + "bbox": [ + 176, + 842, + 415, + 856 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "This section provides a framework for meta-learning approximate inference that is a simple reframing and extension of existing approaches (Finn et al., 2017; Grant et al., 2018). We will employ point estimates for the shared parameters $\\theta$ since data across all tasks will pin down their value. Distributional estimates will be used for the task-specific parameters since only a few shots constrain them. ", + "bbox": [ + 174, + 867, + 826, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Once the shared parameters are learned, the probabilistic solution to few-shot learning in the model above comprises two steps. First, form the posterior distribution over the task-specific parameters $p ( \\psi ^ { ( t ) } | \\tilde { x } ^ { ( t ) } , D ^ { ( t ) } , \\theta )$ . Second, compute the posterior predictive $p ( \\tilde { y } ^ { ( t ) } | \\tilde { x } ^ { ( t ) } , \\theta )$ . These steps will require approximation and the emphasis here is on performing this quickly at test time. We will describe the form of the approximation, the optimization problem used to learn it, and how to implement this efficiently below. In what follows we initially suppress dependencies on the inputs $\\tilde { x }$ and shared parameters $\\theta$ to reduce notational clutter, but will reintroduce these at the end of the section. ", + "bbox": [ + 173, + 103, + 825, + 203 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Specification of the approximate posterior predictive distribution. Our framework approximates the posterior predictive distribution by an amortized distribution $q _ { \\phi } ( \\tilde { y } | D )$ . That is, we learn a feed-forward inference network with parameters $\\phi$ that takes any training dataset $D ^ { ( t ) }$ and test input $\\tilde { x }$ as inputs and returns the predictive distribution over the test output $\\tilde { y } ^ { ( t ) }$ . We construct this by amortizing the approximate posterior $q _ { \\phi } ( \\psi | D )$ and then form the approximate posterior predictive distribution using: ", + "bbox": [ + 173, + 217, + 825, + 304 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/5fcbd72fe66f96372df83943188d1cb5d896503448e0fbbb27d455be38843bbb.jpg", + "text": "$$\nq _ { \\phi } ( \\tilde { y } | D ) = \\int p ( \\tilde { y } | \\psi ) q _ { \\phi } ( \\psi | D ) \\mathrm { d } \\psi .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 303, + 614, + 334 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "This step may require additional approximation e.g. Monte Carlo sampling. The amortization will enable fast predictions at test time. The form of these distributions is identical to those used in amortized variational inference (Edwards and Storkey, 2017; Kingma and Welling, 2014). In this work, we use a factorized Gaussian distribution for $q _ { \\phi } \\big ( \\psi | D ^ { ( t ) } \\big )$ with means and variances set by the amortization network. However, the training method described next is different. ", + "bbox": [ + 173, + 333, + 825, + 405 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Meta-learning the approximate posterior predictive distribution. The quality of the approximate posterior predictive for a single task will be measured by the KL-divergence between the true and approximate posterior predictive distribution KL $[ p ( \\tilde { y } | D ) \\rvert | \\dot { q } _ { \\phi } ( \\tilde { y } | D ) ]$ . The goal of learning will be to minimize the expected value of this KL averaged over tasks, ", + "bbox": [ + 174, + 419, + 825, + 474 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/73402fc6a9affa5c4af8ffa0ef78bfb98365b92d1d80544d33b37e20aa546a9c.jpg", + "text": "$$\n\\phi ^ { * } = \\underset { \\phi } { \\operatorname { a r g m i n } } \\ \\underset { p ( D ) } { \\mathbb { E } } \\left[ \\mathrm { K L } \\left[ p ( \\tilde { y } | D ) \\lVert q _ { \\phi } ( \\tilde { y } | D ) \\right] \\right] = \\underset { \\phi } { \\operatorname { a r g m a x } } \\ \\underset { p ( \\tilde { y } , D ) } { \\mathbb { E } } \\left[ \\log \\int p ( \\tilde { y } | \\psi ) q _ { \\phi } ( \\psi | D ) \\mathrm { d } \\psi \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 191, + 477, + 785, + 511 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Training will therefore return parameters $\\phi$ that best approximate the posterior predictive distribution in an average KL sense. So, if the approximate posterior $q _ { \\phi } ( \\psi | D )$ is rich enough, global optimization will recover the true posterior $p ( \\psi | D )$ (assuming $p ( \\psi | D )$ obeys identifiability conditions (Casella and Berger, 2002)).1 Thus, the amortized procedure meta-learns approximate inference that supports accurate prediction. Appendix A provides a generalized derivation of the framework, grounded in Bayesian decision theory (Jaynes, 2003). ", + "bbox": [ + 173, + 513, + 825, + 598 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The right hand side of Eq. (2) indicates how training could proceed: (i) select a task $t$ at random, (ii) sample some training data $D ^ { ( t ) }$ , (iii) form the posterior predictive $q _ { \\phi } \\big ( \\cdot | D ^ { ( t ) } \\big )$ and, (iv) compute the log-density $\\log q _ { \\phi } ( \\tilde { y } ^ { ( t ) } | D ^ { ( t ) } )$ at test data $\\tilde { y } ^ { ( t ) }$ not included in $D ^ { ( t ) }$ . Repeating this process many times and averaging the results would provide an unbiased estimate of the objective which can then be optimized. This perspective also makes it clear that the procedure is scoring the approximate inference procedure by simulating approximate Bayesian held-out log-likelihood evaluation. Importantly, while an inference network is used to approximate posterior distributions, the training procedure differs significantly from standard variational inference. In particular, rather than minimizing $\\mathrm { K L } ( q _ { \\phi } ( \\psi | D \\bar { ) } | | p ( \\psi | D \\bar { ) } )$ , our objective function directly focuses on the posterior predictive distribution and minimizes $\\mathrm { K L } ( p ( \\tilde { y } | D ) \\| q _ { \\phi } ( \\tilde { y } | D ) )$ . ", + "bbox": [ + 173, + 603, + 826, + 748 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "End-to-end stochastic training. Armed by the insights above we now layout the full training procedure. We reintroduce inputs and shared parameters $\\theta$ and the objective becomes: ", + "bbox": [ + 173, + 761, + 823, + 790 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2b2598aea2f3de525143ec26a3f6c035748d11c4938a1aa4c554af4054d9f0db.jpg", + "text": "$$\n\\mathcal { L } \\left( \\phi \\right) = - \\underset { p ( D , \\tilde { y } , \\tilde { x } ) } { \\mathbb { E } } \\left[ \\log q _ { \\phi } ( \\tilde { y } | \\tilde { x } , \\theta ) \\right] = - \\underset { p ( D , \\tilde { y } , \\tilde { x } ) } { \\mathbb { E } } \\left[ \\log \\int p ( \\tilde { y } | \\tilde { x } , \\psi , \\theta ) q _ { \\phi } ( \\psi | D , \\theta ) \\mathrm { d } \\psi \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 220, + 791, + 777, + 825 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We optimize the objective over the shared parameters $\\theta$ as this will maximize predictive performance (i.e., Bayesian held out likelihood). An end-to-end stochastic training objective for $\\theta$ and $\\phi$ is: ", + "bbox": [ + 173, + 828, + 823, + 856 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/3b588e77f672e8637fd114392caa0319218d9384bc3c5775eee70d4cc290ddb3.jpg", + "text": "$$\n\\hat { \\mathcal { L } } \\left( \\boldsymbol { \\theta } , \\boldsymbol { \\phi } \\right) = \\frac { 1 } { M T } \\sum _ { M , T } \\log \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } p \\left( \\tilde { y } _ { m } ^ { ( t ) } | \\tilde { x } _ { m } ^ { ( t ) } , \\psi _ { l } ^ { ( t ) } , \\boldsymbol { \\theta } \\right) , \\quad \\mathrm { w i t h ~ } \\psi _ { l } ^ { ( t ) } \\sim q _ { \\boldsymbol { \\phi } } ( \\boldsymbol { \\psi } | D ^ { ( t ) } , \\boldsymbol { \\theta } )\n$$", + "text_format": "latex", + "bbox": [ + 227, + 858, + 769, + 902 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/1947f1f187d900ab04d56c92f56cfc49c1300157d804236f663049757cd354da.jpg", + "image_caption": [ + "Figure 2: Computational flow of VERSA for few-shot classification with the context-independent approximation. Left: A test point $\\tilde { x }$ is mapped to its softmax output through a feature extractor neural network and a linear classifier (fully connected layer). The global parameters $\\theta$ of the feature extractor are shared between tasks whereas the weight vectors $\\dot { w } _ { t } ^ { ( c ) }$ of the linear classifier are task specific and inferred through an amortization network with parameters $\\phi$ . Right: Amortization network that maps the extracted features of the $k$ training examples of a particular class to the corresponding weight vector of the linear classifier. " + ], + "image_footnote": [], + "bbox": [ + 189, + 102, + 823, + 247 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "and $\\{ \\tilde { y } _ { m } ^ { ( t ) } , \\tilde { x } _ { m } ^ { ( t ) } , D ^ { ( t ) } \\} \\sim p ( \\tilde { y } , \\tilde { x } , D )$ , where $p$ represents the data distribution (e.g., sampling tasks and splitting them into disjoint training data $D$ and test data $\\{ ( \\tilde { x } _ { m } ^ { ( t ) } , \\tilde { y } _ { m } ^ { ( t ) } ) \\} _ { m = 1 } ^ { M _ { t } } )$ . This type of training therefore uses episodic train / test splits at meta-train time. We have also approximated the integral over $\\psi$ using $L$ Monte Carlo samples. The local reparametrization (Kingma et al., 2015) trick enables optimization. Interestingly, the learning objective does not require an explicit specification of the prior distribution over parameters, $p ( \\psi ^ { ( t ) } | \\theta )$ , learning it implicitly through $q _ { \\phi } ( \\psi | D , \\theta )$ instead. ", + "bbox": [ + 174, + 366, + 825, + 458 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In summary, we have developed an approach for Meta-Learning Probabilistic Inference for Prediction (ML-PIP). A simple investigation of the inference method with synthetic data is provided in Section 5.1. In Section 4 we will show that this formulation unifies a number of existing approaches, but first we discuss a particular instance of the ML-PIP framework that supports versatile learning. ", + "bbox": [ + 174, + 463, + 826, + 520 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3 VERSAT ILE AMORT IZED INFERENCE ", + "text_level": 1, + "bbox": [ + 176, + 541, + 521, + 558 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A versatile system is one that makes inferences both rapidly and flexibly. By rapidly we mean that test-time inference involves only simple computation such as a feed-forward pass through a neural network. By flexibly we mean that the system supports a variety of tasks – including variable numbers of shots or numbers of classes in classification problems – without retraining. Rapid inference comes automatically with the use of a deep neural network to amortize the approximate posterior distribution $q$ . However, it typically comes at the cost of flexibility: amortized inference is usually limited to a single specific task. Below, we discuss design choices that enable us to retain flexibility. ", + "bbox": [ + 173, + 573, + 825, + 671 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Inference with sets as inputs. The amortization network takes data sets of variable size as inputs whose ordering we should be invariant to. We use permutation-invariant instance-pooling operations to process these sets similarly to Qi et al. (2017) and as formalized in Zaheer et al. (2017). The instance-pooling operation ensures that the network can process any number of training observations. ", + "bbox": [ + 174, + 688, + 825, + 743 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "VERSA for Few-Shot Image Classification. For few-shot image classification, our parameterization of the probabilistic model is inspired by early work from Heskes (2000); Bakker and Heskes (2003) and recent extensions to deep learning (Bauer et al., 2017; Qiao et al., 2017). A feature extractor neural network $h _ { \\theta } ( x ) \\in \\mathbb { R } ^ { d _ { \\theta } }$ , shared across all tasks, feeds into a set of task-specific linear classifiers with softmax outputs and weights and biases $\\psi ^ { ( t ) } = \\{ W ^ { ( t ) } , b ^ { ( t ) } \\}$ (see Fig. 2). ", + "bbox": [ + 174, + 760, + 825, + 832 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A naive amortization requires the approximate posterior $q _ { \\phi } ( \\psi | D , \\theta )$ to model the distribution over full weight matrices in $\\mathbb { R } ^ { d _ { \\theta } \\times C }$ (and biases). This requires the specification of the number of few-shot classes $C$ ahead of time and limits inference to this chosen number. Moreover, it is difficult to metalearn systems that directly output large matrices as the output dimensionality is high. We therefore propose specifying $q _ { \\phi } ( \\psi | D , \\theta )$ in a context independent manner such that each weight vector $\\psi _ { c }$ depends only on examples from class $c$ , by amortizing individual weight vectors associated with a single softmax output instead of the entire weight matrix directly. To reduce the number of learned parameters, the amortization network operates directly on the extracted features $h _ { \\theta } ( x )$ : ", + "bbox": [ + 174, + 838, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/29e22e0c9c513f6c2befeab130b574ebc7b494ef8f9a5f6a7797d92132a91970.jpg", + "image_caption": [ + "Figure 3: Computational flow of VERSA for few-shot view reconstruction. Left: A set of training images and angles $\\{ ( y _ { n } ^ { ( t ) } , x _ { n } ^ { ( t ) } ) \\} _ { n = 1 } ^ { k }$ are mapped to a stochastic input $\\psi ^ { ( t ) }$ through the amortization network $q _ { \\phi }$ . $\\psi ^ { ( t ) }$ is then concatenated with a test angle $\\tilde { x }$ and mapped onto a new image through the generator $\\theta$ . Right: Amortization network that maps $k$ image/angle examples of a particular object-instance to the corresponding stochastic input. " + ], + "image_footnote": [], + "bbox": [ + 181, + 103, + 821, + 231 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 340, + 825, + 369 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/90f49e542378c37cd9e8624a652d209073acc48e68bf1e81c2fc7fff87e3bac4.jpg", + "text": "$$\nq _ { \\phi } ( \\psi | D , \\theta ) = \\prod _ { c = 1 } ^ { C } q _ { \\phi } \\left( \\psi _ { c } | \\{ h _ { \\theta } \\left( x _ { n } ^ { c } \\right) \\} _ { n = 1 } ^ { k _ { c } } , \\theta \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 349, + 391, + 648, + 434 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Note that in our implementation, end-to-end training is employed, i.e., we backpropagate to $\\theta$ through the inference network. Here $k _ { c }$ is the number of observed examples in class $c$ and $\\psi _ { c } = \\{ w _ { c } , b _ { c } \\}$ denotes the weight vector and bias of the linear classifier associated with that class. Thus, we construct the classification matrix $\\psi ^ { ( t ) }$ by performing $C$ feed-forward passes through the inference network $q _ { \\phi } ( \\psi | D , \\theta )$ (see Fig. 2). ", + "bbox": [ + 174, + 454, + 825, + 527 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The assumption of context independent inference is an approximation. In Appendix B, we provide theoretical and empirical justification for its validity. Our theoretical arguments use insights from Density Ratio Estimation (Mohamed, 2018; Sugiyama et al., 2012), and we empirically demonstrate that full approximate posterior distributions are close to their context independent counterparts. Critically, the context independent approximation addresses all the limitations of a naive amortization mentioned above: (i) the inference network needs to amortize far fewer parameters whose number does not scale with number of classes $C$ (a single weight vector instead of the entire matrix); (ii) the amortization network can be meta-trained with different numbers of classes per task, and (iii) the number of classes $C$ can vary at test-time. ", + "bbox": [ + 174, + 532, + 825, + 659 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "VERSA for Few-Shot Image Reconstruction (Regression). We consider a challenging few-shot learning task with a complex (high dimensional and continuous) output space. We define view reconstruction as the ability to infer how an object looks from any desired angle based on a small set of observed views. We frame this as a multi-output regression task from a set of training images with known orientations to output images with specified orientations. ", + "bbox": [ + 174, + 690, + 825, + 760 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our generative model is similar to the generator of a GAN or the decoder of a VAE: A latent vector $\\psi ^ { ( t ) } \\in \\mathbb { R } ^ { d _ { \\psi } }$ , which acts as an object-instance level input to the generator, is concatenated with an angle representation and mapped through the generator to produce an image at the specified orientation. In this setting, we treat all parameters $\\theta$ of the generator network as global parameters (see Appendix E.1 for full details of the architecture), whereas the latent inputs $\\psi ^ { ( t ) }$ are the task-specific parameters. We use a Gaussian likelihood in pixel space for the outputs of the generator. To ensure that the output means are between zero and one, we use a sigmoid activation after the final layer. $\\phi$ parameterizes an amortization network that first processes the image representations of an object, concatenates them with their associated view orientations, and processes them further before instance-pooling. From the pooled representations, $q _ { \\phi } ( \\psi | D , \\theta )$ produces a distribution over vectors $\\psi ^ { ( t ) }$ . This process is illustrated in Fig. 3. ", + "bbox": [ + 174, + 766, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 ML-PIP UN IFIES DISPARATE RELATED WORK ", + "text_level": 1, + "bbox": [ + 173, + 102, + 606, + 118 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we continue in the spirit of Grant et al. (2018), and recast a broader class of metalearning approaches as approximate inference in hierarchical models. We show that ML-PIP unifies a number of important approaches to meta-learning, including both gradient and metric based variants, as well as amortized MAP inference and conditional modelling approaches (Garnelo et al., 2018a). We lay out these connections, most of which rely on point estimates for the task-specific parameters corresponding to $q ( \\psi ^ { ( t ) } | D ^ { ( t ) } , \\theta ) = \\delta \\left( \\psi ^ { ( t ) } \\bar { - } \\psi ^ { * } ( D ^ { ( t ) } , \\theta ) \\right)$ . In addition, we compare previous approaches to VERSA. ", + "bbox": [ + 173, + 131, + 826, + 233 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Gradient-Based Meta-Learning. Let the task-specific parameters $\\psi ^ { ( t ) }$ be all the parameters in a neural network. Consider a point estimate formed by taking a step of gradient ascent of the training loss, initialized at $\\psi _ { 0 }$ and with learning rate $\\eta$ . ", + "bbox": [ + 174, + 247, + 825, + 290 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/7d060f52c6d2a1ee859d2459d8dfab8fa2d3add7c90d8311a5cfc786c6ecf8de.jpg", + "text": "$$\n\\psi ^ { * } ( D ^ { ( t ) } , \\theta ) = \\psi _ { 0 } + \\eta \\frac { \\partial } { \\partial \\psi } \\log \\sum _ { n = 1 } ^ { N _ { t } } p ( y _ { n } ^ { ( t ) } | x _ { n } ^ { ( t ) } , \\psi , \\theta ) \\bigg | _ { \\psi _ { 0 } } .\n$$", + "text_format": "latex", + "bbox": [ + 318, + 294, + 679, + 337 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "This is an example of semi-amortized inference (Kim et al., 2018b), as the only shared inference parameters are the initialization and learning rate, and optimization is required for each task (albeit only for one step). Importantly, Eq. (6) recovers Model-agnostic meta-learning (Finn et al., 2017), providing a perspective as semi-amortized ML-PIP. This perspective is complementary to that of Grant et al. (2018) who justify the one-step gradient parameter update employed by MAML through MAP inference and the form of the prior $p ( \\psi | \\theta )$ . Note that the episodic meta-train / meta-test splits do not fall out of this perspective. Instead we view the update choice as one of amortization which is trained using the predictive KL and naturally recovers the test-train splits. More generally, multiple gradient steps could be fed into an RNN to compute $\\psi ^ { * }$ which recovers Ravi and Larochelle (2017). In comparison to these methods, besides being distributional over $\\psi$ , VERSA relieves the need to back-propagate through gradient based updates during training and compute gradients at test time, as well as enables the treatment of both local and global parameters which simplifies inference. ", + "bbox": [ + 173, + 337, + 826, + 506 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Metric-Based Few-Shot Learning. Let the task-specific parameters be the top layer softmax weights and biases of a neural network $\\boldsymbol { \\psi } ^ { ( t ) } = \\{ w _ { c } ^ { ( t ) } , b _ { c } ^ { ( t ) } \\} _ { c = 1 } ^ { C }$ . The shared parameters are the lower layer weights. Consider amortized point estimates for these parameters constructed by averaging the top-layer activations for each class, ", + "bbox": [ + 173, + 518, + 825, + 579 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/ee67c7c3e5a42acf9658aaa2eb7880f88fc72ba536326f5c3d8f5c7e566667f0.jpg", + "text": "$$\n\\psi ^ { * } ( D ^ { ( t ) } , \\theta ) = \\big \\{ w _ { c } ^ { * } , b _ { c } ^ { * } \\big \\} _ { c = 1 } ^ { C } = \\Big \\{ \\mu _ { c } ^ { ( t ) } , - \\| \\mu _ { c } ^ { ( t ) } \\| ^ { 2 } / 2 \\Big \\} _ { c = 1 } ^ { C } \\mathrm { ~ w h e r e ~ } \\mu _ { c } ^ { ( t ) } = \\frac { 1 } { k _ { c } } \\sum _ { n = 1 } ^ { k _ { c } } h _ { \\theta } ( x _ { n } ^ { ( c ) } )\n$$", + "text_format": "latex", + "bbox": [ + 210, + 582, + 785, + 626 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "These choices lead to the following predictive distribution: ", + "bbox": [ + 174, + 627, + 558, + 642 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/23010b57029acf028828c91fca0a3880fe24aa126dbbc6e728251582b8b4aed2.jpg", + "text": "$$\np ( \\tilde { y } ^ { ( t ) } = c | \\tilde { x } ^ { ( t ) } , \\theta ) \\propto \\exp \\left( - d ( h _ { \\theta } ( \\tilde { x } ^ { ( t ) } ) , \\mu _ { c } ^ { ( t ) } ) \\right) = \\exp \\left( h _ { \\theta } ( \\tilde { x } ^ { ( t ) } ) ^ { T } \\mu _ { c } ^ { ( t ) } - \\frac { 1 } { 2 } \\| \\mu _ { c } ^ { ( t ) } \\| ^ { 2 } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 215, + 645, + 781, + 679 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "which recovers prototypical networks (Snell et al., 2017) using a Euclidean distance function $d$ with the final hidden layer being the embedding space. In comparison, VERSA is distributional and it uses a more flexible amortization function that goes beyond averaging of activations. ", + "bbox": [ + 174, + 681, + 823, + 724 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Amortized MAP inference. Qiao et al. (2017) proposed a method for predicting weights of classes from activations of a pre-trained network to support i) online learning on a single task to which new few-shot classes are incrementally added, ii) transfer from a high-shot classification task to a separate low-shot classification task. This is an example usage of hyper-networks (Ha et al., 2016) to amortize learning about weights, and can be recovered by the ML-PIP framework by pre-training $\\theta$ and performing MAP inference for $\\psi$ . VERSA goes beyond point estimates and although its amortization network is similar in spirit, it is more general, employing end-to-end training and supporting full multi-task learning by sharing information between many tasks. ", + "bbox": [ + 173, + 737, + 825, + 851 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Conditional models trained via maximum likelihood. In cases where a point estimate of the task-specific parameters are used the predictive becomes ", + "bbox": [ + 174, + 864, + 823, + 893 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/fc06a6df173a21c00e71f5e7b83cece086ccd15fa74546f7f29eafe70805ec8a.jpg", + "text": "$$\nq _ { \\phi } ( \\tilde { y } | D , \\theta ) = \\int p ( \\tilde { y } | \\psi , \\theta ) q _ { \\phi } ( \\psi | D , \\theta ) \\mathrm { d } \\psi = p ( \\tilde { y } | \\psi ^ { * } ( D , \\theta ) , \\theta ) .\n$$", + "text_format": "latex", + "bbox": [ + 295, + 896, + 702, + 929 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In such cases the amortization network that computes $\\psi ^ { * } ( D , \\theta )$ can be equivalently viewed as part of the model specification rather than the inference scheme. From this perspective, the ML-PIP training procedure for $\\phi$ and $\\theta$ is equivalent to training a conditional model $p ( \\tilde { y } | \\psi _ { \\phi } ^ { * } ( D , \\theta ) , \\theta )$ via maximum likelihood estimation, establishing a strong connection to neural processes (Garnelo et al., 2018a;b). ", + "bbox": [ + 174, + 103, + 825, + 161 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison to Variational Inference (VI). Standard application of amortized VI (Kingma and Welling, 2014; Rezende et al., 2014; Kingma et al., 2015; Blundell et al., 2015) for $\\psi$ in the multi-task discriminative model optimizes the Monte Carlo approximated free-energy w.r.t. $\\phi$ and $\\theta$ : ", + "bbox": [ + 173, + 175, + 826, + 218 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/9a7a2798835e23ac425f38572b265a4f2e6ee3f798c6481b9b91502faf72a79d.jpg", + "text": "$$\n\\hat { \\mathcal { L } } ( \\boldsymbol { \\theta } , \\boldsymbol { \\phi } ) = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\left( \\sum _ { ( \\boldsymbol { x } , \\boldsymbol { y } ) \\in D ^ { ( t ) } } \\left( \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } \\log p ( \\boldsymbol { y } ^ { ( t ) } | \\boldsymbol { x } ^ { ( t ) } , \\boldsymbol { \\psi } _ { l } ^ { ( t ) } , \\boldsymbol { \\theta } ) \\right) - \\mathrm { K L } \\left[ q _ { \\boldsymbol { \\phi } } ( \\boldsymbol { \\psi } | D ^ { ( t ) } , \\boldsymbol { \\theta } ) \\| p ( \\boldsymbol { \\psi } | \\boldsymbol { \\theta } ) \\right] \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 224, + 818, + 273 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\psi _ { l } ^ { ( t ) } \\sim q _ { \\phi } ( \\psi | D ^ { ( t ) } , \\theta )$ . In addition to the conceptual difference from ML-PIP (discussed in Section 2.1), this differs from the ML-PIP objective by i) not employing meta train / test splits, and ii) including the KL for regularization instead. In Section 5, we show that VERSA significantly improves over standard VI in the few-shot classification case and compare to recent VI/meta-learning hybrids. ", + "bbox": [ + 174, + 285, + 825, + 343 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5 EXPER IMENTS AND RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 363, + 457, + 380 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate VERSA on several few-shot learning tasks. We begin with toy experiments to investigate the properties of the amortized posterior inference achieved by VERSA. We then report few-shot classification results using the Omniglot and miniImageNet datasets in Section 5.2, and demonstrate VERSA’s ability to retain high accuracy as the shot and way are varied at test time. In Section 5.3, we examine VERSA’s performance on a one-shot view reconstruction task with ShapeNet objects.2 ", + "bbox": [ + 173, + 395, + 825, + 465 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.1 POSTER IOR INFERENCE W ITH TOY DATA ", + "text_level": 1, + "bbox": [ + 173, + 482, + 508, + 496 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To investigate the approximate inference performed by our training procedure, we run the following experiment. We first generate data from a Gaussian distribution with a mean that varies across tasks: ", + "bbox": [ + 174, + 507, + 825, + 536 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/56c6083b9994f207e06d5066bb811accea5a7ed3113fc5f0d552d440c33b6bdc.jpg", + "text": "$$\np ( \\theta ) = \\delta ( \\theta - 0 ) ; \\quad p \\left( \\psi ^ { ( t ) } | \\theta \\right) = \\mathcal { N } \\left( \\psi ^ { ( t ) } ; \\theta , \\sigma _ { \\psi } ^ { 2 } \\right) ; \\quad p \\left( y _ { n } ^ { ( t ) } | \\psi ^ { ( t ) } \\right) = \\mathcal { N } \\left( y _ { n } ^ { ( t ) } ; \\psi ^ { ( t ) } , \\sigma _ { y } ^ { 2 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 184, + 541, + 785, + 569 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We generate $T = 2 5 0$ tasks in two separate experiments, having $N \\in \\{ 5 , 1 0 \\}$ train observations and $M = 1 5$ test observations. We introduce the inference network $q _ { \\phi } ( \\psi | D ^ { ( t ) } ) = \\mathcal { N } ( \\psi ; \\mu _ { q } ^ { ( t ) } , \\sigma _ { q } ^ { ( t ) 2 } )$ amortizing inference as: ", + "bbox": [ + 174, + 574, + 826, + 621 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/3c616af307bd84b679e0cbd405f9716ac21b732b7ad1b18487cb88fb01843057.jpg", + "text": "$$\n\\mu _ { q } ^ { ( t ) } = w _ { \\mu } \\sum _ { n = 1 } ^ { N } y _ { n } ^ { ( t ) } + b _ { \\mu } , \\quad \\sigma _ { q } ^ { ( t ) 2 } = \\exp \\left( w _ { \\sigma } \\sum _ { n = 1 } ^ { N } y _ { n } ^ { ( t ) } + b _ { \\sigma } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 627, + 704, + 670 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The learnable parameters $\\phi = \\{ w _ { \\boldsymbol { \\mu } } , b _ { \\boldsymbol { \\mu } } , w _ { \\sigma } , b _ { \\sigma } \\}$ are trained with the objective function in Eq. (4). The model is trained to convergence with Adam (Kingma and Ba, 2015) using mini-batches of tasks from the generated dataset. Then, a separate set of tasks is generated from the same generative process, and the posterior $q _ { \\phi } ( \\psi | D )$ is inferred with the learned amortization parameters. The true posterior over $\\psi$ is Gaussian with a mean that depends on the task, and may be computed analytically. Fig. 4 shows the approximate posterior distributions inferred for unseen test sets by the trained amortization networks. The evaluation shows that the inference procedure is able to recover accurate posterior distributions over $\\psi$ , despite minimizing a predictive KL divergence in data space. ", + "bbox": [ + 173, + 675, + 826, + 787 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "5.2 FEW-SHOT CLASS IFICAT ION ", + "text_level": 1, + "bbox": [ + 176, + 804, + 416, + 819 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We evaluate VERSA on standard few-shot classification tasks in comparison to previous work. Specifically, we consider the Omniglot (Lake et al., 2011) and miniImageNet (Ravi and Larochelle, 2017) datasets which are $C$ -way classification tasks with $k _ { c }$ examples per class. VERSA follows the implementation in Sections 2 and 3, and the approximate inference scheme in Eq. (5). We follow the experimental protocol established by Vinyals et al. (2016) for Omniglot and Ravi and Larochelle (2017) for miniImagenet, using equivalent architectures for $h _ { \\theta }$ . Training is carried out in an episodic manner: for each task, $k _ { c }$ examples are used as training inputs to infer $q _ { \\phi } ( \\psi ^ { ( c ) } | D , \\theta )$ for each class, and an additional set of examples is used to evaluate the objective function. Full details of data preparation and network architectures are provided in Appendix D. ", + "bbox": [ + 174, + 830, + 826, + 900 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/18148ee79ba818f9ab1287c364b34b0e0ff936907e4fbaa3b299579ffbed0ca9.jpg", + "image_caption": [ + "Figure 4: True posteriors $p ( \\psi | D )$ ( ) and approximate posteriors $q _ { \\phi } ( \\psi | D )$ ( ) for unseen test sets $( { \\star } )$ in the experiment. In both cases (five and ten shot), the approximate posterior closely resembles the true posterior given the observed data. " + ], + "image_footnote": [], + "bbox": [ + 210, + 106, + 785, + 207 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 282, + 825, + 339 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3 details few-shot classification performance for VERSA as well as competitive approaches. The tables include results for only those approaches with comparable training procedures and convolutional feature extraction architectures. Approaches that employ pre-training and/or residual networks (Bauer et al., 2017; Qiao et al., 2017; Rusu et al., 2018; Gidaris and Komodakis, 2018; Oreshkin et al., 2018; Garcia and Bruna, 2017; Lacoste et al., 2018) have been excluded so that the quality of the learning algorithm can be assessed separately from the power of the underlying discriminative model. ", + "bbox": [ + 174, + 347, + 825, + 430 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For Omniglot, the training, validation, and test splits have not been specified for previous methods, affecting the comparison. VERSA achieves a new state-of-the-art results $( 6 7 . 3 7 \\%$ - up $1 . 3 8 \\%$ over the previous best) on 5-way - 5-shot classification on the miniImageNet benchmark and $9 7 . 6 6 \\%$ - up $0 . 0 2 \\% )$ on the 20-way - 1 shot Omniglot benchmark for systems using a convolution-based network architecture and an end-to-end training procedure. VERSA is within error bars of state-of-the-art on three other benchmarks including 5-way - 1-shot miniImageNet, 5-way - 5-shot Omniglot, and 5-way - 1-shot Omniglot. Results on the Omniglot 20 way - 5-shot benchmark are very competitive with, but lower than other approaches. While most of the methods evaluated in Table 3 adapt all of the learned parameters for new tasks, VERSA is able to achieve state-of-the-art performance despite adapting only the weights of the top-level classifier. ", + "bbox": [ + 173, + 436, + 825, + 575 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Comparison to standard and amortized VI. To investigate the performance of our inference procedure, we compare it in terms of log-likelihood (Table 1) and accuracy (Table 3) to training the same model using both amortized and non-amortized VI (i.e., Eq. (10)). Derivations and further experimental details are provided in Appendix C. VERSA improves substantially over amortized VI even though the same amortization network is used for both. This is due to VI’s tendency to under-fit, especially for small numbers of data points (Trippe and Turner, 2018; Turner and Sahani, 2011) which is compounded when using inference networks (Cremer et al., 2018). Using non-amortized VI improves performance substantially, but does not reach the level of VERSA and forming the posterior is significantly slower as it requires many forward / backward passes through the network. This is similar in spirit to MAML (Finn et al., 2017), though MAML dramatically reduces the number of required iterations by finding good global initializations e.g., five gradient steps for miniImageNet. This is in contrast to the single forward pass required by VERSA. ", + "bbox": [ + 174, + 598, + 825, + 696 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/c4921189b20c7868e1c9678a5585ba64b6d48985866b71cfba1af1ea687c2968.jpg", + "table_caption": [ + "Table 1: Negative Log-likelihood (NLL) results for different few-shot settings on Omniglot and miniImageNet. The $\\pm$ sign indicates the $9 5 \\%$ confidence interval over tasks using a Student’s t-distribution approximation. " + ], + "table_footnote": [], + "table_body": "
OmniglotminilmageNet
5-way NLL20-way NLL5-way NLL
Method1-shot5-shot1-shot5-shot1-shot5-shot
Amortized VI0.179 ± 0.0090.137 ± 0.0040.456± 0.0100.253 ± 0.0041.328 ± 0.0241.165 ± 0.010
Non-Amortized VI0.144 ± 0.0050.025 ± 0.0010.393 ± 0.0050.078 ± 0.002
VERSA0.010± 0.0050.007 ± 0.0030.079 ± 0.0090.031 ± 0.0041.183 ± 0.0230.859 ± 0.015
", + "bbox": [ + 181, + 747, + 816, + 825 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 853, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Versatility. VERSA allows us to vary the number of classes $C$ and shots $k _ { c }$ between training and testing (Eq. (5)). Fig. 5a shows that a model trained for a particular $C$ -way retains very high accuracy as $C$ is varied. For example, when VERSA is trained for the 20-Way, 5-Shot condition, at test-time it can handle $C = 1 0 0$ way conditions and retain an accuracy of approximately $94 \\%$ . Fig. 5b shows similar robustness as the number of shots $k _ { c }$ is varied. VERSA therefore demonstrates considerable flexibility and robustness to the test-time conditions, but at the same time it is efficient as it only requires forward passes through the network. The time taken to evaluate 1000 test tasks with a 5-way, 5-shot miniImageNet trained model using MAML (https://github.com/cbfinn/maml) is 302.9 seconds whereas VERSA took 53.5 seconds on a NVIDIA Tesla P100-PCIE-16GB GPU. This is more than $5 \\times$ speed advantage in favor of VERSA while bettering MAML in accuracy by $4 . 2 6 \\%$ . ", + "bbox": [ + 173, + 103, + 825, + 243 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/69a7c60c0513a773fe18756537a4337d39a44e0da27404102321c16b9171f824.jpg", + "image_caption": [ + "Figure 5: Test accuracy on Omniglot when varying (a) way (fixing shot to be that used for training) and (b) shot. In Fig. 5b, all models are evaluated on 5-way classification. Colors indicate models trained with different way-shot episodic combinations. ", + "Table 2: View reconstruction test results. Mean squared error (MSE – lower is better) and the structural similarity index (SSIM - higher is better) (Wang et al., 2004) are measured between the generated and ground truth images. Error bars not shown as they are insignificant. " + ], + "image_footnote": [], + "bbox": [ + 217, + 262, + 792, + 410 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5.3 SHAPENET VIEW RECONSTRUCT ION ", + "text_level": 1, + "bbox": [ + 176, + 505, + 478, + 518 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ShapeNetCore v2 (Chang et al., 2015) is a database of 3D objects covering 55 common object categories with ${ \\sim } 5 1 { , } 3 0 0$ unique objects. For our experiments, we use 12 of the largest object categories. We concatenate all instances from all 12 of the object categories together to obtain a dataset of 37,108 objects. This dataset is then randomly shuffled and we use $70 \\%$ of the objects for training, $10 \\%$ for validation, and $20 \\%$ for testing. For each object, we generate 36 views of size $3 2 \\times 3 2$ pixels spaced evenly every 10 degrees in azimuth around the object. ", + "bbox": [ + 174, + 532, + 825, + 617 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We evaluate VERSA by comparing it to a conditional variational autoencoder (C-VAE) with view angles as labels (Kingma et al., 2014; Narayanaswamy et al., 2017) and identical architectures. We train VERSA in an episodic manner and the C-VAE in batch-mode on all 12 object classes at once. We train on a single view selected at random and use the remaining views to evaluate the objective function. For full experimentation details see Appendix E. Fig. 6 shows views of unseen objects from the test set generated from a single shot with VERSA as well as a C-VAE and compares both to ground truth views. Both VERSA and the C-VAE capture the correct orientation of the object in the generated images. However, VERSA produces images that contain much more detail and are visually sharper than the C-VAE images. Although important information is missing due to occlusion in the single shot, VERSA is often able to accurately impute this information presumably due to learning the statistics of these objects. Table 2 provides quantitative comparison results between VERSA with varying shot and the C-VAE. The quantitative metrics all show the superiority of VERSA over a C-VAE. As the number of shots increase to 5, the measurements show a corresponding improvement. ", + "bbox": [ + 173, + 625, + 826, + 804 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/37c95680e68a04f94a7cc3c5686bf03f41359f572c63d114eba6b1f1904c0e74.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
ModelMSESSIM
C-VAE 1-shot0.02690.5705
VERSA 1-shot0.01080.7893
VERSA 5-shot0.00690.8483
", + "bbox": [ + 218, + 823, + 449, + 890 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CONCLUS IONS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 333, + 118 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We have introduced ML-PIP, a probabilistic framework for meta-learning. ML-PIP unifies a broad class of recently proposed meta-learning methods, and suggests alternative approaches. Building on ML-PIP, we developed VERSA, a few-shot learning algorithm that avoids the use of gradient based optimization at test time by amortizing posterior inference of task-specific parameters. We evaluated VERSA on several few-shot learning tasks and demonstrated state-of-the-art performance and compelling visual results on a challenging 1-shot view reconstruction task. ", + "bbox": [ + 173, + 133, + 825, + 217 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/11999d8dbc1084f71e2073d44ed042be9a9e3b74934e195d253b124f4d0af769.jpg", + "table_caption": [ + "Table 3: Accuracy results for different few-shot settings on Omniglot and miniImageNet. The $\\pm$ sign indicates the $9 5 \\%$ confidence interval over tasks using a Student’s t-distribution approximation. Bold text indicates the highest scores that overlap in their confidence intervals. " + ], + "table_footnote": [], + "table_body": "
OmniglotminilmageNet
Method5-way accuracy (%)5-shot20-way accuracy(%) 1-shot5-shot5-way accuracy (%) 1-shot 5-shot
1-shot98.488.197.0
Siamese Nets (Koch et al., 2015)97.398.946.6
Matching Nets (Vinyals et al.,2016)98.193.898.560.0
Neural Statistician (Edwards and Storkey, 2017)98.199.593.298.1
Memory Mod (Kaiser et al.,2017)98.499.695.098.6
Meta LSTM (Ravi and Larochelle, 2017)43.44 ± 0.77 60.60± 0.71
MAML (Finn et al., 2017)98.7 ± 0.499.9 ± 0.195.8 ± 0.398.9± 0.248.7 ± 1.84 63.11 ± 0.92
Prototypical Netsa (Snell et al., 2017)97.499.395.498.7 46.61 ± 0.7865.77 ± 0.70
mAP-SSVM (Triantafillou et al., 2017)98.699.695.298.650.32 ± 0.80 63.94 ± 0.72
mAP-DLM (Triantafillou et al., 2017)98.899.695.498.6 50.28 ± 0.8063.70± 0.70
LLAMA (Grant et al., 2018)49.40 ± 1.83
PLATIPUS (Finn et al., 2018)50.13 ± 1.86
Meta-SGD (Li et al., 2017)99.53± 0.2699.93 ± 0.0995.93 ± 0.3898.97 ± 0.1950.47 ± 1.87 64.03 ± 0.94
SNAIL (Mishra et al., 2018)99.07 ± 0.1699.78 ± 0.0997.64 ± 0.3099.36 ± 0.18 45.155.2
Relation Net (Yang et al.,2018)99.6 ± 0.299.8 ± 0.197.6 ± 0.299.1 ± 0.1 50.44 ± 0.8265.32 ± 0.70
Reptile (Nichol and Schulman,2018)97.68 ± 0.0499.48 ± 0.0689.43 ± 0.1497.12 ± 0.32 49.97 ± 0.3265.99 ± 0.58
BMAML (Kim et al., 2018a)53.8 ± 1.46
Amortized VI98.71 ± 0.2290.56 ± 0.5444.13 ± 1.78
Non-Amortized VI97.77 ± 0.55 98.77 ± 0.1899.74 ± 0.0695.28 ± 0.1996.12 ± 0.23 98.84± 0.0955.68 ± 0.91
VERSA (Ours)99.70 ± 0.2099.75 ± 0.1397.66 ± 0.2998.77 ± 0.1853.40 ± 1.82 67.37 ± 0.86
", + "bbox": [ + 165, + 268, + 834, + 705 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/4148f14ceb3482059d23c8517735680aca8d7c7eeb406656865087c074ca44fb.jpg", + "image_caption": [ + "Figure 6: Results for ShapeNet view reconstruction for unseen objects from the test set (shown left). The model was trained to reconstruct views from a single orientation. Top row: images/views generated by a C-VAE model; middle row images/views generated by VERSA; bottom row: ground truth images. Views are spaced evenly every 30 degrees in azimuth. " + ], + "image_footnote": [], + "bbox": [ + 196, + 183, + 810, + 786 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "ACKNOWLEDGEMENTS ", + "text_level": 1, + "bbox": [ + 176, + 104, + 339, + 117 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We thank Ravi and Larochelle for providing the miniImageNet dataset, and Yingzhen Li, Niki Kilbertus, Will Tebbutt, Maria Lomelli, and Robert Pinsler for their useful feedback. J.G. acknowledges funding from a Samsung Doctoral Scholarship. M.B. acknowledges funding by the EPSRC and a Qualcomm European Scholarship in Technology. R.E.T. acknowledges support from EPSRC grants EP/M0269571 and EP/L000776/1. ", + "bbox": [ + 174, + 130, + 826, + 199 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 220, + 289, + 236 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "B. Bakker and T. Heskes. Task clustering and gating for Bayesian multitask learning. 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IEEE transactions on image processing, 13(4):600–612, 2004. \nF. S. Y. Yang, L. Zhang, T. Xiang, P. H. Torr, and T. M. Hospedales. Learning to compare: Relation network for few-shot learning. 2018. \nM. Zaheer, S. Kottur, S. Ravanbakhsh, B. Poczos, R. R. Salakhutdinov, and A. J. Smola. Deep sets. In Advances in Neural Information Processing Systems, pages 3394–3404, 2017. ", + "bbox": [ + 171, + 242, + 828, + 925 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 85, + 828, + 928 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 98, + 828, + 820 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A BAYES IAN DEC IS ION THEORET IC GENERAL IZAT ION OF ML-PIP ", + "text_level": 1, + "bbox": [ + 171, + 101, + 766, + 119 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A generalization of the new inference framework presented in Section 2 is based upon Bayesian decision theory (BDT). BDT provides a recipe for making predictions $\\hat { y }$ for an unknown test variable $\\tilde { y }$ by combining information from observed training data $\\bar { D } ^ { ( t ) }$ (here from a single task $t$ ) and a loss function $L ( \\tilde { y } , \\hat { y } )$ that encodes the cost of predicting $\\hat { y }$ when the true value is $\\tilde { y }$ (Berger, 2013; Jaynes, 2003). In BDT an optimal prediction minimizes the expected loss (suppressing dependencies on the inputs and $\\theta$ to reduce notational clutter):3 ", + "bbox": [ + 173, + 133, + 826, + 219 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/60928715b8dbb5475b7d91211cf150f200f00758dae8d2bcd21cf0891ecc53e2.jpg", + "text": "$$\n\\hat { y } ^ { * } = \\underset { \\hat { y } } { \\mathrm { a r g m i n } } \\int p ( \\tilde { y } | D ^ { ( t ) } ) L ( \\tilde { y } , \\hat { y } ) \\mathrm { d } \\tilde { y } , \\quad \\mathrm { w h e r e } \\quad p ( \\tilde { y } | D ^ { ( t ) } ) = \\int p ( \\tilde { y } | \\psi ^ { ( t ) } ) p ( \\psi ^ { ( t ) } | D ^ { ( t ) } ) \\mathrm { d } \\psi ^ { ( t ) }\n$$", + "text_format": "latex", + "bbox": [ + 196, + 226, + 800, + 261 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "is the Bayesian predictive distribution and $p ( \\psi ^ { ( t ) } | D ^ { ( t ) } )$ the posterior distribution of $\\psi ^ { ( t ) }$ given the training data from task $t$ . ", + "bbox": [ + 174, + 272, + 825, + 301 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "BDT separates test and training data and so is a natural lens through which to view recent episodic approaches to training that utilize many internal training/test splits (Vinyals et al., 2016). Based on this insight, what follows is a fairly dense derivation of an ultimately simple stochastic variational objective for meta-learning probabilistic inference that is rigorously grounded in Bayesian inference and decision theory. ", + "bbox": [ + 173, + 308, + 825, + 378 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Distributional BDT. We generalize BDT to cases where the goal is to return a full predictive distribution $q ( \\cdot )$ over the unknown test variable $\\tilde { y }$ rather than a point prediction. The quality of $q$ is quantified through a distributional loss function $L ( \\tilde { y } , q ( \\cdot ) )$ . Typically, if $\\tilde { y }$ (the true value of the underlying variable) falls in a low probability region of $q ( \\cdot )$ the loss will be high, and vice versa. The optimal predictive $q ^ { * }$ is found by optimizing the expected distributional loss with $q$ constrained to a distributional family $\\mathcal { Q }$ : ", + "bbox": [ + 173, + 393, + 825, + 478 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/49935d4751621a415bb24532383d3a4324fb48df6c7256c54ccca5ad6ce39a0f.jpg", + "text": "$$\nq ^ { * } = \\operatorname * { a r g m i n } _ { \\boldsymbol { q } \\in \\mathcal { Q } } \\int p ( \\boldsymbol { \\tilde { y } } | D ^ { ( t ) } ) L ( \\boldsymbol { \\tilde { y } } , \\boldsymbol { q } ( \\cdot ) ) \\mathrm { d } \\boldsymbol { \\tilde { y } } .\n$$", + "text_format": "latex", + "bbox": [ + 367, + 477, + 630, + 512 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Amortized variational training. Here, we amortize $q$ to form quick predictions at test time and learn parameters by minimizing average expected loss over tasks. Let $\\phi$ be a set of shared variational parameters such that $q ( \\tilde { y } ) = q _ { \\phi } \\overline { { ( \\tilde { y } | D ) } }$ (or $q _ { \\phi }$ for shorthand). Now the approximate predictive distribution can take any training dataset $D ^ { ( t ) }$ as an argument and directly perform prediction of $\\tilde { y } ^ { ( t ) }$ . The optimal variational parameters are found by minimizing the expected distributional loss across tasks ", + "bbox": [ + 173, + 525, + 825, + 599 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/d401d81cacc54cdf4fde52363316e227bd52d894fa8a72e0ee3db7e57924da5e.jpg", + "text": "$$\n\\phi ^ { * } = \\underset { \\phi } { \\mathrm { a r g m i n } } \\mathcal { L } \\left[ q _ { \\phi } \\right] , \\mathcal { L } \\left[ q _ { \\phi } \\right] = \\int p ( D ) p ( \\tilde { y } | D ) L ( \\tilde { y } , q _ { \\phi } ( \\cdot | D ) ) \\mathrm { d } \\tilde { y } \\mathrm { d } D = \\mathbb { E } _ { p ( D , \\tilde { y } ) } [ L ( \\tilde { y } , q _ { \\phi } ( \\cdot | D ) ) ] .\n$$", + "text_format": "latex", + "bbox": [ + 192, + 606, + 803, + 640 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Here the variables $D , { \\tilde { x } }$ and $\\tilde { y }$ are placeholders for integration over all possible datasets, test inputs and outputs. Note that Eq. (A.3) can be stochastically approximated by sampling a task $t$ and randomly partitioning into training data $D$ and test data $\\{ \\tilde { x } _ { m } , \\bar { \\tilde { y } } _ { m } \\} _ { m = 1 } ^ { M }$ , which naturally recovers episodic minibatch training over tasks and data (Vinyals et al., 2016; Ravi and Larochelle, 2017). Critically, this does not require computation of the true predictive distribution. It also emphasizes the meta-learning aspect of the procedure, as the model is learning how to infer predictive distributions from training tasks. ", + "bbox": [ + 173, + 651, + 826, + 750 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Loss functions. We employ the log-loss: the negative log density of $q _ { \\phi }$ at $\\tilde { y }$ . In this case, ", + "bbox": [ + 171, + 765, + 764, + 781 + ], + "page_idx": 14 + }, + { + "type": "equation", + "img_path": "images/f564fa62d5fd7411e3e03f1c83b1f7b787d2b3b7c754dd4b4587d4964800b2a3.jpg", + "text": "$$\n\\mathcal { L } \\left[ q _ { \\phi } \\right] = \\mathbb { E } _ { p \\left( D , \\tilde { y } \\right) } \\left[ - \\log q _ { \\phi } ( \\tilde { y } | D ) \\right] = \\mathbb { E } _ { p \\left( D \\right) } \\left[ \\mathrm { K L } \\left[ p ( \\tilde { y } | D ) \\| q _ { \\phi } ( \\tilde { y } | D ) \\right] + \\mathrm { H } \\left[ p ( \\tilde { y } | D ) \\right] \\right] ,\n$$", + "text_format": "latex", + "bbox": [ + 210, + 789, + 751, + 808 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "where $\\mathrm { K L } [ p ( y ) | | q ( y ) ]$ is the KL-divergence, and $\\mathrm { ~ H ~ } [ p ( y ) ]$ is the entropy of $p$ . Eq. (A.4) has the elegant property that the optimal $q _ { \\phi }$ is the closest member of $\\mathcal { Q }$ (in a KL sense) to the true predictive $p ( \\tilde { y } | D )$ , which is unsurprising as the log-loss is a proper scoring rule (Huszar, 2013). This is reminiscent of the sleep phase in the wake-sleep algorithm (Hinton et al., 1995). Exploration of alternative proper scoring rules (Dawid, 2007) and more task-specific losses (Lacoste-Julien et al., 2011) is left for future work. ", + "bbox": [ + 173, + 815, + 826, + 900 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Specification of the approximate predictive distribution. Next, we consider the form of $q _ { \\phi }$ . Motivated by the optimal predictive distribution, we replace the true posterior by an approximation: ", + "bbox": [ + 169, + 103, + 826, + 133 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/8e8f447d5b00fb9d4b78abf81dd402ee7405ed4a1ad19a7279713f0aad5632d1.jpg", + "text": "$$\nq _ { \\phi } ( \\tilde { y } | D ) = \\int p ( \\tilde { y } | \\psi ) q _ { \\phi } ( \\psi | D ) \\mathrm { d } \\psi .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 138, + 612, + 171 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B JUST IFICAT ION FOR CONTEXT-INDEPENDENT APPROX IMAT ION ", + "text_level": 1, + "bbox": [ + 171, + 185, + 753, + 203 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In this section we lay out both theoretical and empirical justifications for the context-independent approximation detailed in Section 3. ", + "bbox": [ + 173, + 217, + 825, + 247 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.1 THEORET ICAL ARGUMENT – DENS ITY RAT IO EST IMAT ION ", + "text_level": 1, + "bbox": [ + 174, + 262, + 642, + 279 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A principled justification for the approximation is best understood through the lens of density ratio estimation (Mohamed, 2018; Sugiyama et al., 2012). We denote the conditional density of each class as $p ( x | y = c )$ and assume equal a priori class probability $p ( y = c ) = 1 / C$ . Density ratio theory then uses Bayes’ theorem to show that the optimal softmax classifier can be expressed in terms of the conditional densities (Mohamed, 2018; Sugiyama et al., 2012): ", + "bbox": [ + 173, + 289, + 825, + 359 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/5b8e09d7f918dd3f35fe34ceb143a013181e41428343782d536977f94397a3d8.jpg", + "text": "$$\n\\mathrm { S o f t m a x } ( y = c | x ) = \\frac { \\exp ( h ( x ) ^ { \\top } w _ { c } ) } { \\sum _ { c ^ { \\prime } } \\exp ( h ( x ) ^ { \\top } w _ { c ^ { \\prime } } ) } = p ( y = c | x ) = \\frac { p ( x | y = c ) } { \\sum _ { c ^ { \\prime } } p ( x | y = c ^ { \\prime } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 246, + 366, + 750, + 410 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "This implies that the optimal classifier will construct estimators for the conditional density for each class, that is $\\mathrm { e x p } ( h ( x ) ^ { \\top } w _ { c } ) \\propto p ( x | y = c )$ . Importantly for our approximation, notice that these estimates are constructed independently for each class, similarly to training a naive Bayes classifier. VERSA mirrors this optimal form using: ", + "bbox": [ + 173, + 416, + 826, + 473 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/8c9a6dbce28b777539413e7ee08f9644837c2b159b951c3eabef558ffd3aa44a.jpg", + "text": "$$\n\\log p ( x | y = c ) \\propto h _ { \\theta } ( \\tilde { x } ) ^ { \\top } w _ { c } ,\n$$", + "text_format": "latex", + "bbox": [ + 401, + 478, + 596, + 498 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "where $w _ { c } \\sim q _ { \\phi } \\left( w | \\{ x _ { n } | y _ { n } = c \\} \\right)$ ) for each class in a given task. Under ideal conditions (i.e., if one could perfectly estimate $p ( \\tilde { x } | y = c )$ ), the context-independent assumption holds, further motivating our design. ", + "bbox": [ + 173, + 503, + 825, + 547 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.2 EMP IR ICAL JUST IFICAT ION ", + "text_level": 1, + "bbox": [ + 176, + 563, + 413, + 578 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Here we detail a simple experiment to evaluate the validity of the context-independent inference assumption. The goal of the experiment is to examine if weights may be context-independent without imposing the assumption on the amortization network. To see this, we randomly generate fifty tasks from a dataset, where classes may appear a number of times in different tasks. We then perform free-form (non-amortized) variational inference on the weights for each of the tasks, with a Gaussian variational distribution: ", + "bbox": [ + 173, + 588, + 825, + 672 + ], + "page_idx": 15 + }, + { + "type": "equation", + "img_path": "images/eb3287aac41fca226b6ac9c0bc0841a887d0303a81c7bcb1f33e9f6ded0b26b4.jpg", + "text": "$$\n\\begin{array} { r } { q _ { \\phi } \\left( W ^ { ( t ) } | \\mathcal { D } ^ { ( t ) } , \\theta \\right) = \\mathcal { N } \\left( W ^ { ( t ) } ; \\mu _ { \\phi } ^ { ( t ) } , \\sigma _ { \\phi } ^ { ( t ) 2 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 676, + 647, + 704 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "If the assumption is reasonable, we may expect the distribution of the weights of a specific class to be similar regardless of the additional classes in the task. ", + "bbox": [ + 171, + 717, + 823, + 746 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "We examine 5-way classification in the MNIST dataset. We randomly sample and fix fifty such tasks. We train the model twice using the same feature extraction network used in the few-shot classification experiments, and fix the $d _ { \\theta }$ to be 16 and 2. We then train the model in an episodic manner by minibatching tasks at each iteration. The model is trained to convergence, and achieves $9 9 \\%$ accuracy on held out test examples for the tasks. After training is complete we examine the optimized $\\mu _ { \\phi } ^ { ( t ) }$ for each class in each task. Fig. B.1a shows a t-SNE (Maaten and Hinton, 2008) plot for the 16-dimensional weights. We see that when reduced to 2-dimensions, the weights cluster according to class. Fig. B.1b visualizes the weights in their original space. In this plot, weights from the same class are grouped together, and clear similarity patterns are evident across the image, showing that weights from the same class have similar means across tasks. Fig. B.2 details the task weights in 2-dimensional space. Here, each pentagon represents the weight means learned for one training task, where the nodes of the pentagon are colored according to the class the weights represent. In Fig. B.2a we see that overall, the classes cluster in 2-dimensional space as well. However, there is some overlap (e.g., classes ‘1’ and $\\cdot _ { 2 } \\cdot ,$ ), and that for some tasks a class-weight may appear away from the cluster. Fig. B.2b shows the same plot, but only for tasks that contain both class ‘1’ and ‘2’. Here we can see that for these tasks, class $\\bullet _ { 2 } \\cdot$ weights are all located away from their cluster. ", + "bbox": [ + 173, + 751, + 826, + 924 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/07839a600a28d6118a99a997c66579d785371a45435e63c2ca530bf156ab5c22.jpg", + "image_caption": [ + "Figure B.1: Visualizing the learned weights for $d _ { \\theta } = 1 6$ . (a) Weight dimensionality is reduced using T-SNE (Maaten and Hinton, 2008). Weights are colored according to class. (b) Each weight represents one column of the image. Weights are grouped by class. " + ], + "image_footnote": [], + "bbox": [ + 287, + 154, + 709, + 512 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/6a6bb38f30c8cc0c065163afb7af7683b5fc6f6bb5a982c629606c9ac13733bb.jpg", + "image_caption": [ + "Figure B.2: Visualizing the task weights for $d _ { \\theta } = 2$ . (a) All training tasks. (b) Only training tasks containing both ‘1’s and $\\mathrm { { } ^ { \\circ } 2 \\mathrm { { } s } }$ . " + ], + "image_footnote": [], + "bbox": [ + 207, + 669, + 794, + 835 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "This implies that each class-weights are typically well-approximated as being independent of the task. However, if the model lacks capacity to properly assign each set of class weights to different regions of space, for tasks where classes from similar regions of space appear, the inference procedure will ‘move’ one of the class weights to an ‘empty’ region of the space. ", + "bbox": [ + 173, + 166, + 825, + 223 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C VAR IAT IONAL INFERENCE DER IVAT IONS FOR THE MODEL ", + "text_level": 1, + "bbox": [ + 176, + 242, + 707, + 258 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We derive a VI-based objective for our probabilistic model. By “amortized” VI we mean that $q _ { \\phi } ( \\psi | D ^ { ( t ) } , \\theta )$ is parameterized by a neural network with a fixed-sized $\\phi$ . Conversely, “non-amortized” VI refers to local parameters $\\phi ^ { ( t ) }$ that are optimized independently (at test time) for each new task $t$ , such that $\\begin{array} { r } { q ( \\psi | D ^ { ( t ) } , \\theta ) = \\mathcal { N } ( \\psi | \\mu _ { \\phi ^ { ( t ) } } , \\Sigma _ { \\phi ^ { ( t ) } } ) } \\end{array}$ . However, the derivation of the objective function does not change between these options. For a single task $t$ , an evidence lower bound (ELBO; (Wainwright and Jordan, 2008)) may be expressed as: ", + "bbox": [ + 174, + 272, + 826, + 362 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/e0c2f941288a7907937b9c542adfb439ef5b1d22645014837f2c1b33cd3524e8.jpg", + "text": "$$\n\\mathcal { L } _ { t } = \\mathbb { E } _ { q _ { \\phi } ( \\psi | D ^ { ( t ) } , \\theta ) } \\left[ \\sum _ { ( \\boldsymbol { x } , \\boldsymbol { y } ) \\in D ^ { ( t ) } } \\log p ( \\boldsymbol { y } | \\boldsymbol { x } , \\psi , \\theta ) \\right] - \\mathrm { K L } \\left[ q _ { \\phi } ( \\psi | D ^ { ( t ) } , \\theta ) \\| p ( \\psi | \\theta ) \\right] .\n$$", + "text_format": "latex", + "bbox": [ + 217, + 366, + 745, + 416 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "We can then derive a stochastic estimator to optimize Eq. (C.1) by sampling $D ^ { ( t ) } \\sim p ( D )$ (approximated with a training set of tasks) and simple Monte Carlo integration over $\\psi$ such that $\\mathsf { \\bar { \\psi } } ^ { ( l ) } \\sim q _ { \\phi } ( \\psi | D ^ { ( t ) } , \\theta )$ : ", + "bbox": [ + 176, + 421, + 823, + 465 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/cf34336365b2824d520485b41d203f622615d2638591095e0bece6f3584687a2.jpg", + "text": "$$\n\\hat { \\mathcal { L } } = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\left( \\sum _ { \\left( x , y \\right) \\in D ^ { \\left( t \\right) } } \\left( \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } \\log p ( y | x , \\psi ^ { \\left( l \\right) } , \\theta ) \\right) - \\mathrm { K L } \\left[ q _ { \\phi } ( \\psi | D ^ { \\left( t \\right) } , \\theta ) \\| p ( \\psi | \\theta ) \\right] \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 194, + 468, + 766, + 520 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Eq. (C.2) differs from our objective function in Eq. (4) in two important ways: (i) Eq. (4) does not contain a KL term for ${ q _ { \\phi } } ( \\psi | D ^ { ( t ) } , \\theta )$ (nor any other form of prior distribution over $\\psi$ , and (ii) Eq. (C.1) does not distinguish between training and test data within a task, and therefore does not explicitly encourage the model to generalize in any way. ", + "bbox": [ + 174, + 522, + 825, + 580 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D EXPER IMENTAT ION DETA ILS ", + "text_level": 1, + "bbox": [ + 174, + 599, + 459, + 616 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "In this section we provide comprehensive details on the few-shot classification experiments. ", + "bbox": [ + 173, + 631, + 772, + 645 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D.1 OMN IGLOT FEW-SHOT CLASS IFICAT ION TRA IN ING PROCEDURE ", + "text_level": 1, + "bbox": [ + 176, + 661, + 678, + 676 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Omniglot (Lake et al., 2011) is a few-shot learning dataset consisting of 1623 handwritten characters (each with 20 instances) derived from 50 alphabets. We follow a pre-processing and training procedure akin to that defined in (Vinyals et al., 2016). First the images are resized to $2 8 \\times 2 8$ pixels and then character classes are augmented with rotations of 90 degrees. The training, validation and test sets consist of a random split of 1100, 100, and 423 characters, respectively. When augmented this results in 4400 training, 400 validation, and 1292 test classes, each having 20 character instances. For $C$ -way, $k _ { c }$ -shot classification, training proceeds in an episodic manner. Each training iteration consists of a batch of one or more tasks. For each task $C$ classes are selected at random from the training set. During training, $k _ { c }$ character instances are used as training inputs and 15 character instances are used as test inputs. The validation set is used to monitor the progress of learning and to select the best model to test, but does not affect the training process. Final evaluation of the trained model is done on 600 randomly selected tasks from the test set. During evaluation, $k _ { c }$ character instances are used as training inputs and $k _ { c }$ character instances are used as test inputs. We use the Adam (Kingma and Ba, 2015) optimizer with a constant learning rate of 0.0001 with 16 tasks per batch to train all models. The 5-way - 5-shot and 5-way - 1-shot models are trained for 80,000 iterations while the 20-way - 5-shot model is trained for 60,000 iterations, and the 20-way - 1-shot model is trained for 100,000 iterations. In addition, we use a Gaussian form for $q$ and set the number of $\\psi$ samples to $L = 1 0$ . ", + "bbox": [ + 173, + 688, + 826, + 924 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "D.2 miniIMAGENET FEW-SHOT CLASS IFICAT ION TRA IN ING PROCEDURE ", + "text_level": 1, + "bbox": [ + 176, + 103, + 709, + 117 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "miniImageNet (Vinyals et al., 2016) is a dataset of 60,000 color images that is sub-divided into 100 classes, each with 600 instances. The images have dimensions of $8 4 \\times 8 4$ pixels. For our experiments, we use the 64 training, 16 validation, and 20 test class splits defined by (Ravi and Larochelle, 2017). Training proceeds in the same episodic manner as with Omniglot. We use the Adam (Kingma and Ba, 2015) optimizer and a Gaussian form for $q$ and set the number of $\\psi$ samples to $L = 1 0$ . For the 5-way - 5-shot model, we train using 4 tasks per batch for 100,000 iterations and use a constant learning rate of 0.0001. For the 5-way - 1-shot model, we train with 8 tasks per batch for 50,000 iterations and use a constant learning rate of 0.00025. ", + "bbox": [ + 173, + 128, + 826, + 241 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "D.3 FEW-SHOT CLASS IFICAT ION NETWORK ARCH ITECTURES ", + "text_level": 1, + "bbox": [ + 176, + 260, + 627, + 273 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Tables D.1 to D.4 detail the neural network architectures for the feature extractor $\\theta$ , amortization network $\\phi$ , and linear classifier $\\psi$ , respectively. The feature extraction network is very similar to that used in (Vinyals et al., 2016). The output of the amortization network yields mean-field Gaussian parameters for the weight distributions of the linear classifier $\\psi$ . When sampling from the weight distributions, we employ the local-reparameterization trick (Kingma et al., 2015), that is we sample from the implied distribution over the logits rather than directly from the variational distribution. To reduce the number of learned parameters, we share the feature extraction network $\\theta$ with the pre-processing phase of the amortizaion network $\\psi$ . ", + "bbox": [ + 174, + 285, + 825, + 397 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/3177433cee8fc09adf777344244cbd5fbb5f5ca7debd1c674497ad1238d3ce17.jpg", + "table_caption": [ + "Table D.1: Feature extraction network used for Omniglot few-shot learning. Batch Normalization and dropout with a keep probability of 0.9 used throughout. ", + "Omniglot Shared Feature Extraction Network $\\mathbf { \\eta } ^ { ( \\theta ) }$ : $\\tilde { x } h _ { \\theta } ( \\tilde { x } )$ " + ], + "table_footnote": [], + "table_body": "
Output sizeLayers
28×28×1Input image
14 ×14 × 64conv2d (3 × 3, stride 1, SAME,RELU),dropout, pool (2 × 2, stride 2, SAME)
7×7×64conv2d (3 × 3, stride 1, SAME,RELU),dropout, pool (2 × 2, stride 2, SAME)
4×4×64conv2d (3 × 3, stride 1, SAME,RELU),dropout, pool (2 × 2, stride 2, SAME)
2×2×64conv2d (3 × 3,stride 1, SAME,RELU),dropout,pool (2 × 2, stride 2, SAME)
256flatten
", + "bbox": [ + 176, + 463, + 821, + 577 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Table D.2: Feature extraction network used for miniImageNet few-shot learning. Batch Normalization and dropout with a keep probability of 0.5 used throughout. ", + "bbox": [ + 173, + 606, + 823, + 633 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/6d0de82b98209e4aea2763d33748a03d6ec0cdfed52e5945e805a7b3ff8fb018.jpg", + "table_caption": [ + "miniImageNet Shared Feature Extraction Network (θ): $\\tilde { x } h _ { \\theta } ( \\tilde { x } )$ " + ], + "table_footnote": [], + "table_body": "
Output sizeLayers
84×84×1Input image
42 × 42× 64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
21×21×64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
10 ×10×64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
5×5×64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
2×2×64 256conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID) flatten
", + "bbox": [ + 174, + 659, + 823, + 785 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E SHAPENET EXPER IMENTAT ION DETA ILS ", + "text_level": 1, + "bbox": [ + 173, + 821, + 558, + 839 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "E.1 VIEW RECONSTRUCT ION TRA IN ING PROCEDURE AND NETWORK ARCH ITECTURES", + "text_level": 1, + "bbox": [ + 173, + 854, + 810, + 869 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "ShapeNetCore v2 (Chang et al., 2015) is an annotated database of 3D objects covering 55 common object categories with ${ \\sim } 5 1 { , } 3 0 0$ unique objects. For our experiments, we use 12 of the largest object categories. Refer to Table E.1 for a complete list. We concatenate all instances from all 12 of the object categories together to obtain a dataset of 37,108 objects. This concatenated dataset is then randomly shuffled and we use $70 \\%$ of the objects (25,975 in total) for training, $10 \\%$ for validation (3,710 in total) , and $20 \\%$ (7423 in total) for testing. For each object, we generate $V = 3 6$ , $1 2 8 \\times 1 2 8$ pixel image views spaced evenly every 10 degrees in azimuth around the object. We then convert the rendered images to gray-scale and reduce their size to be $3 2 \\times 3 2$ pixels. Again, we train our model in an episodic manner. Each training iteration consists a batch of one or more tasks. For each task an object is selected at random from the training set. We train on a single view selected at random from the $V = 3 6$ views associated with each object and use the remaining 35 views to evaluate the objective function. We then generate 36 views of the object with a modified version of our amortization network which is shown diagrammatically in Fig. 3. To evaluate the system, we generate views and compute quantitative metrics over the entire test set. Tables E.2 to E.4 describe the network architectures for the encoder, amortization, and generator networks, respectively. To train, we use the Adam (Kingma and Ba, 2015) optimizer with a constant learning rate of 0.0001 with 24 tasks per batch for 500,000 training iterations. In addition, we set $d _ { \\phi } = 2 5 6$ , $d _ { \\psi } = 2 5 6$ and number of $\\psi$ samples to 1. ", + "bbox": [ + 176, + 881, + 823, + 924 + ], + "page_idx": 18 + }, + { + "type": "table", + "img_path": "images/9c3170d174774355a8ae09fd448f618557abdc96f8bfedd2e0fb50e68c6e551b.jpg", + "table_caption": [ + "Table D.3: Amortization network used for Omniglot and miniImageNet few-shot learning. " + ], + "table_footnote": [], + "table_body": "
PhaseOutput size Layers
feature extraction instance pooling weight distributionk×256shared feature network (0)
256mean
2562 × fully connected, ELU+
linear fully connected to μw(c), ² .2 w(c)
", + "bbox": [ + 235, + 125, + 756, + 228 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/290667502fde33abbc48fbdd7b3bea06c22ef764941daae22c83a407c26cead0.jpg", + "table_caption": [ + "Table D.4: Linear classifier used for Omniglot and miniImageNet few-shot learning. " + ], + "table_footnote": [], + "table_body": "
Linear Classifier (ψ): hθ(x) →p(y|𝑥,0,ψt)
Output sizeLayers
256Input features
Cfully connected, softmax
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Object Categorysysnet IDInstances
airplane026911564045
bench028288841813
cabinet029331121571
car029583433533
phone02992529831
chair030016276778
display032111171093
lamp036366492318
speaker036914591597
sofa042565203173
table043792438436
boat045305661939
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Output size Layers
32 ×32×1Input image
16 ×16×64conv2d (3 × 3, stride 1, SAME,RELU), pool (2 × 2, stride2,VALID)
8×8×64conv2d (3 × 3, stride 1, SAME,RELU),pool (2 × 2, stride 2, VALID)
4×4×64conv2d (3 × 3, stride 1, SAME,RELU), pool (2 × 2, stride 2,VALID)
2×2×64conv2d (3 × 3, stride 1, SAME,RELU),pool (2 × 2, stride 2, VALID)
dfully connected, RELU
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PhaseOutput sizeLayers
Φprek×dencoder network ()
concatenate h and Xk ×(dp+dx)concat(h, X)
Φmidkxd2 × 2 fully connected,ELU
instance pooling1×daverage
post1×d2× fully connected, ELU
distributiondfully connected linear layers to μψ,
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Output sizeLayers
d+dxconcat(y,x)
512fully connected, RELU
1024fully connected, RELU
2×2×256reshape
4×4×128deconv2d (3 × 3, stride 2, SAME, RELU)
8×8×64deconv2d (3 × 3, stride 2, SAME,RELU)
16 ×16 × 32deconv2d (3 × 3, stride 2, SAME,RELU)
32×32×1deconv2d (3 × 3, stride 2, SAME,sigmoid)
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We construct this by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 259, + 232 + ], + "score": 1.0, + "content": "amortizing the approximate posterior", + "type": "text" + }, + { + "bbox": [ + 259, + 219, + 295, + 231 + ], + "score": 0.92, + "content": "q _ { \\phi } ( \\psi | D )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "and then form the approximate posterior predictive", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 182, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 182, + 244 + ], + "score": 1.0, + "content": "distribution using:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 240, + 376, + 265 + ], + "lines": [ + { + "bbox": [ + 236, + 240, + 376, + 265 + ], + "spans": [ + { + "bbox": [ + 236, + 240, + 376, + 265 + ], + "score": 0.95, + "content": "q _ { \\phi } ( \\tilde { y } | D ) = \\int p ( \\tilde { y } | \\psi ) q _ { \\phi } ( \\psi | D ) \\mathrm { d } \\psi .", + "type": "interline_equation", + "image_path": "5fcbd72fe66f96372df83943188d1cb5d896503448e0fbbb27d455be38843bbb.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 236, + 240, + 376, + 265 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "This step may require additional approximation e.g. Monte Carlo sampling. The amortization will", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "enable fast predictions at test time. The form of these distributions is identical to those used in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "amortized variational inference (Edwards and Storkey, 2017; Kingma and Welling, 2014). 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The goal of learning will be", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 366, + 360, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 360, + 377 + ], + "score": 1.0, + "content": "to minimize the expected value of this KL averaged over tasks,", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 117, + 378, + 481, + 405 + ], + "lines": [ + { + "bbox": [ + 117, + 378, + 481, + 405 + ], + "spans": [ + { + "bbox": [ + 117, + 378, + 481, + 405 + ], + "score": 0.92, + "content": "\\phi ^ { * } = \\underset { \\phi } { \\operatorname { a r g m i n } } \\ \\underset { p ( D ) } { \\mathbb { E } } \\left[ \\mathrm { K L } \\left[ p ( \\tilde { y } | D ) \\lVert q _ { \\phi } ( \\tilde { y } | D ) \\right] \\right] = \\underset { \\phi } { \\operatorname { a r g m a x } } \\ \\underset { p ( \\tilde { y } , D ) } { \\mathbb { E } } \\left[ \\log \\int p ( \\tilde { y } | \\psi ) q _ { \\phi } ( \\psi | D ) \\mathrm { d } \\psi \\right] .", + "type": "interline_equation", + "image_path": "73402fc6a9affa5c4af8ffa0ef78bfb98365b92d1d80544d33b37e20aa546a9c.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 117, + 378, + 481, + 387.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 117, + 387.0, + 481, + 396.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 117, + 396.0, + 481, + 405.0 + ], + "spans": [], + "index": 25 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 407, + 505, + 474 + ], + "lines": [ + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 269, + 420 + ], + "score": 1.0, + "content": "Training will therefore return parameters", + "type": "text" + }, + { + "bbox": [ + 269, + 408, + 276, + 419 + ], + "score": 0.86, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 406, + 505, + 420 + ], + "score": 1.0, + "content": "that best approximate the posterior predictive distribution", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 417, + 505, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 328, + 431 + ], + "score": 1.0, + "content": "in an average KL sense. 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Repeating this process many", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 514, + 505, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 514, + 505, + 528 + ], + "score": 1.0, + "content": "times and averaging the results would provide an unbiased estimate of the objective which can then", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "be optimized. This perspective also makes it clear that the procedure is scoring the approximate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "inference procedure by simulating approximate Bayesian held-out log-likelihood evaluation. Im-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 546, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 104, + 546, + 506, + 561 + ], + "score": 1.0, + "content": "portantly, while an inference network is used to approximate posterior distributions, the training", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 571 + ], + "score": 1.0, + "content": "procedure differs significantly from standard variational inference. 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Our framework approxi-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 183, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 183, + 396, + 196 + ], + "score": 1.0, + "content": "mates the posterior predictive distribution by an amortized distribution", + "type": "text" + }, + { + "bbox": [ + 396, + 185, + 430, + 196 + ], + "score": 0.93, + "content": "q _ { \\phi } ( \\tilde { y } | D )", + "type": "inline_equation" + }, + { + "bbox": [ + 430, + 183, + 505, + 196 + ], + "score": 1.0, + "content": ". 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We construct this by", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 219, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 219, + 259, + 232 + ], + "score": 1.0, + "content": "amortizing the approximate posterior", + "type": "text" + }, + { + "bbox": [ + 259, + 219, + 295, + 231 + ], + "score": 0.92, + "content": "q _ { \\phi } ( \\psi | D )", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 219, + 505, + 232 + ], + "score": 1.0, + "content": "and then form the approximate posterior predictive", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 182, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 182, + 244 + ], + "score": 1.0, + "content": "distribution using:", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 172, + 506, + 244 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 236, + 240, + 376, + 265 + ], + "lines": [ + { + "bbox": [ + 236, + 240, + 376, + 265 + ], + "spans": [ + { + "bbox": [ + 236, + 240, + 376, + 265 + ], + "score": 0.95, + "content": "q _ { \\phi } ( \\tilde { y } | D ) = \\int p ( \\tilde { y } | \\psi ) q _ { \\phi } ( \\psi | D ) \\mathrm { d } \\psi .", + "type": "interline_equation", + "image_path": "5fcbd72fe66f96372df83943188d1cb5d896503448e0fbbb27d455be38843bbb.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 236, + 240, + 376, + 265 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 264, + 505, + 321 + ], + "lines": [ + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 106, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "This step may require additional approximation e.g. Monte Carlo sampling. The amortization will", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "enable fast predictions at test time. The form of these distributions is identical to those used in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 286, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 506, + 299 + ], + "score": 1.0, + "content": "amortized variational inference (Edwards and Storkey, 2017; Kingma and Welling, 2014). 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However, the training method described next is different.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 263, + 506, + 322 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 332, + 505, + 376 + ], + "lines": [ + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 345 + ], + "score": 1.0, + "content": "Meta-learning the approximate posterior predictive distribution. 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This type of training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "therefore uses episodic train / test splits at meta-train time. We have also approximated the integral", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 126, + 340 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 126, + 328, + 134, + 339 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 328, + 159, + 340 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 159, + 328, + 167, + 338 + ], + "score": 0.74, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "Monte Carlo samples. The local reparametrization (Kingma et al., 2015) trick enables", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 505, + 352 + ], + "score": 1.0, + "content": "optimization. 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A simple investigation of the inference method with synthetic data is provided in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 389, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 506, + 401 + ], + "score": 1.0, + "content": "Section 5.1. In Section 4 we will show that this formulation unifies a number of existing approaches,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 400, + 502, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 502, + 414 + ], + "score": 1.0, + "content": "but first we discuss a particular instance of the ML-PIP framework that supports versatile learning.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 108, + 429, + 319, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 320, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 320, + 443 + ], + "score": 1.0, + "content": "3 VERSAT ILE AMORT IZED INFERENCE", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 454, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "A versatile system is one that makes inferences both rapidly and flexibly. By rapidly we mean that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 466, + 504, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 504, + 477 + ], + "score": 1.0, + "content": "test-time inference involves only simple computation such as a feed-forward pass through a neural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "network. By flexibly we mean that the system supports a variety of tasks – including variable numbers", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "of shots or numbers of classes in classification problems – without retraining. Rapid inference comes", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "automatically with the use of a deep neural network to amortize the approximate posterior distribution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 107, + 511, + 113, + 521 + ], + "score": 0.65, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 510, + 506, + 523 + ], + "score": 1.0, + "content": ". However, it typically comes at the cost of flexibility: amortized inference is usually limited to a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 521, + 459, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 459, + 533 + ], + "score": 1.0, + "content": "single specific task. Below, we discuss design choices that enable us to retain flexibility.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 106, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "Inference with sets as inputs. The amortization network takes data sets of variable size as inputs", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "whose ordering we should be invariant to. We use permutation-invariant instance-pooling operations", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "to process these sets similarly to Qi et al. (2017) and as formalized in Zaheer et al. (2017). The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 579, + 503, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 503, + 590 + ], + "score": 1.0, + "content": "instance-pooling operation ensures that the network can process any number of training observations.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 602, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "VERSA for Few-Shot Image Classification. For few-shot image classification, our parameteriza-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "tion of the probabilistic model is inspired by early work from Heskes (2000); Bakker and Heskes", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "(2003) and recent extensions to deep learning (Bauer et al., 2017; Qiao et al., 2017). A feature ex-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 199, + 648 + ], + "score": 1.0, + "content": "tractor neural network", + "type": "text" + }, + { + "bbox": [ + 199, + 635, + 252, + 647 + ], + "score": 0.92, + "content": "h _ { \\theta } ( x ) \\in \\mathbb { R } ^ { d _ { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 634, + 506, + 648 + ], + "score": 1.0, + "content": ", shared across all tasks, feeds into a set of task-specific linear", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 645, + 460, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 329, + 661 + ], + "score": 1.0, + "content": "classifiers with softmax outputs and weights and biases", + "type": "text" + }, + { + "bbox": [ + 329, + 646, + 409, + 659 + ], + "score": 0.94, + "content": "\\psi ^ { ( t ) } = \\{ W ^ { ( t ) } , b ^ { ( t ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 645, + 460, + 661 + ], + "score": 1.0, + "content": "(see Fig. 2).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 664, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 335, + 678 + ], + "score": 1.0, + "content": "A naive amortization requires the approximate posterior", + "type": "text" + }, + { + "bbox": [ + 335, + 664, + 381, + 677 + ], + "score": 0.93, + "content": "q _ { \\phi } ( \\psi | D , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 663, + 506, + 678 + ], + "score": 1.0, + "content": "to model the distribution over", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 675, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 197, + 690 + ], + "score": 1.0, + "content": "full weight matrices in", + "type": "text" + }, + { + "bbox": [ + 198, + 676, + 226, + 687 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { d _ { \\theta } \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 675, + 506, + 690 + ], + "score": 1.0, + "content": "(and biases). This requires the specification of the number of few-shot", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 507, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 136, + 700 + ], + "score": 1.0, + "content": "classes", + "type": "text" + }, + { + "bbox": [ + 137, + 688, + 146, + 698 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 687, + 507, + 700 + ], + "score": 1.0, + "content": "ahead of time and limits inference to this chosen number. Moreover, it is difficult to meta-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "learn systems that directly output large matrices as the output dimensionality is high. 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This type of training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 329 + ], + "score": 1.0, + "content": "therefore uses episodic train / test splits at meta-train time. We have also approximated the integral", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 126, + 340 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 126, + 328, + 134, + 339 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 328, + 159, + 340 + ], + "score": 1.0, + "content": "using", + "type": "text" + }, + { + "bbox": [ + 159, + 328, + 167, + 338 + ], + "score": 0.74, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "Monte Carlo samples. 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A simple investigation of the inference method with synthetic data is provided in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 389, + 506, + 401 + ], + "spans": [ + { + "bbox": [ + 106, + 389, + 506, + 401 + ], + "score": 1.0, + "content": "Section 5.1. In Section 4 we will show that this formulation unifies a number of existing approaches,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 400, + 502, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 502, + 414 + ], + "score": 1.0, + "content": "but first we discuss a particular instance of the ML-PIP framework that supports versatile learning.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 368, + 506, + 414 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 429, + 319, + 442 + ], + "lines": [ + { + "bbox": [ + 105, + 428, + 320, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 320, + 443 + ], + "score": 1.0, + "content": "3 VERSAT ILE AMORT IZED INFERENCE", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 454, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 505, + 468 + ], + "score": 1.0, + "content": "A versatile system is one that makes inferences both rapidly and flexibly. By rapidly we mean that", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 466, + 504, + 477 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 504, + 477 + ], + "score": 1.0, + "content": "test-time inference involves only simple computation such as a feed-forward pass through a neural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "network. By flexibly we mean that the system supports a variety of tasks – including variable numbers", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 505, + 500 + ], + "score": 1.0, + "content": "of shots or numbers of classes in classification problems – without retraining. Rapid inference comes", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "spans": [ + { + "bbox": [ + 106, + 498, + 505, + 511 + ], + "score": 1.0, + "content": "automatically with the use of a deep neural network to amortize the approximate posterior distribution", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 107, + 510, + 506, + 523 + ], + "spans": [ + { + "bbox": [ + 107, + 511, + 113, + 521 + ], + "score": 0.65, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 510, + 506, + 523 + ], + "score": 1.0, + "content": ". However, it typically comes at the cost of flexibility: amortized inference is usually limited to a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 521, + 459, + 533 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 459, + 533 + ], + "score": 1.0, + "content": "single specific task. Below, we discuss design choices that enable us to retain flexibility.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 455, + 506, + 533 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 106, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 106, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "Inference with sets as inputs. The amortization network takes data sets of variable size as inputs", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "whose ordering we should be invariant to. We use permutation-invariant instance-pooling operations", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 579 + ], + "score": 1.0, + "content": "to process these sets similarly to Qi et al. (2017) and as formalized in Zaheer et al. (2017). The", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 579, + 503, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 503, + 590 + ], + "score": 1.0, + "content": "instance-pooling operation ensures that the network can process any number of training observations.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 544, + 505, + 590 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 602, + 505, + 659 + ], + "lines": [ + { + "bbox": [ + 106, + 602, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 506, + 615 + ], + "score": 1.0, + "content": "VERSA for Few-Shot Image Classification. For few-shot image classification, our parameteriza-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 613, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 506, + 626 + ], + "score": 1.0, + "content": "tion of the probabilistic model is inspired by early work from Heskes (2000); Bakker and Heskes", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "(2003) and recent extensions to deep learning (Bauer et al., 2017; Qiao et al., 2017). A feature ex-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 199, + 648 + ], + "score": 1.0, + "content": "tractor neural network", + "type": "text" + }, + { + "bbox": [ + 199, + 635, + 252, + 647 + ], + "score": 0.92, + "content": "h _ { \\theta } ( x ) \\in \\mathbb { R } ^ { d _ { \\theta } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 634, + 506, + 648 + ], + "score": 1.0, + "content": ", shared across all tasks, feeds into a set of task-specific linear", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 645, + 460, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 329, + 661 + ], + "score": 1.0, + "content": "classifiers with softmax outputs and weights and biases", + "type": "text" + }, + { + "bbox": [ + 329, + 646, + 409, + 659 + ], + "score": 0.94, + "content": "\\psi ^ { ( t ) } = \\{ W ^ { ( t ) } , b ^ { ( t ) } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 645, + 460, + 661 + ], + "score": 1.0, + "content": "(see Fig. 2).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 602, + 506, + 661 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 664, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 663, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 335, + 678 + ], + "score": 1.0, + "content": "A naive amortization requires the approximate posterior", + "type": "text" + }, + { + "bbox": [ + 335, + 664, + 381, + 677 + ], + "score": 0.93, + "content": "q _ { \\phi } ( \\psi | D , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 663, + 506, + 678 + ], + "score": 1.0, + "content": "to model the distribution over", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 104, + 675, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 197, + 690 + ], + "score": 1.0, + "content": "full weight matrices in", + "type": "text" + }, + { + "bbox": [ + 198, + 676, + 226, + 687 + ], + "score": 0.9, + "content": "\\mathbb { R } ^ { d _ { \\theta } \\times C }", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 675, + 506, + 690 + ], + "score": 1.0, + "content": "(and biases). This requires the specification of the number of few-shot", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 687, + 507, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 136, + 700 + ], + "score": 1.0, + "content": "classes", + "type": "text" + }, + { + "bbox": [ + 137, + 688, + 146, + 698 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 146, + 687, + 507, + 700 + ], + "score": 1.0, + "content": "ahead of time and limits inference to this chosen number. Moreover, it is difficult to meta-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "learn systems that directly output large matrices as the output dimensionality is high. We therefore", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 708, + 504, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 187, + 723 + ], + "score": 1.0, + "content": "propose specifying", + "type": "text" + }, + { + "bbox": [ + 187, + 710, + 233, + 722 + ], + "score": 0.93, + "content": "q _ { \\phi } ( \\psi | D , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 708, + 492, + 723 + ], + "score": 1.0, + "content": "in a context independent manner such that each weight vector", + "type": "text" + }, + { + "bbox": [ + 492, + 710, + 504, + 721 + ], + "score": 0.87, + "content": "\\psi _ { c }", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 720, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 261, + 733 + ], + "score": 1.0, + "content": "depends only on examples from class", + "type": "text" + }, + { + "bbox": [ + 261, + 725, + 266, + 730 + ], + "score": 0.78, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 720, + 506, + 733 + ], + "score": 1.0, + "content": ", by amortizing individual weight vectors associated with a", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "single softmax output instead of the entire weight matrix directly. 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Left: A set of training images and", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 204, + 507, + 221 + ], + "spans": [ + { + "bbox": [ + 104, + 204, + 131, + 221 + ], + "score": 1.0, + "content": "angles", + "type": "text" + }, + { + "bbox": [ + 132, + 205, + 195, + 218 + ], + "score": 0.93, + "content": "\\{ ( y _ { n } ^ { ( t ) } , x _ { n } ^ { ( t ) } ) \\} _ { n = 1 } ^ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 204, + 311, + 221 + ], + "score": 1.0, + "content": "are mapped to a stochastic input", + "type": "text" + }, + { + "bbox": [ + 311, + 205, + 327, + 218 + ], + "score": 0.91, + "content": "\\psi ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 204, + 448, + 221 + ], + "score": 1.0, + "content": "through the amortization network", + "type": "text" + }, + { + "bbox": [ + 448, + 207, + 458, + 218 + ], + "score": 0.4, + "content": "q _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 204, + 461, + 221 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 462, + 205, + 478, + 218 + ], + "score": 0.72, + "content": "\\psi ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 204, + 507, + 221 + ], + "score": 1.0, + "content": "is then", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 216, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 218, + 228 + ], + "score": 1.0, + "content": "concatenated with a test angle", + "type": "text" + }, + { + "bbox": [ + 218, + 218, + 225, + 226 + ], + "score": 0.77, + "content": "\\tilde { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 216, + 420, + 228 + ], + "score": 1.0, + "content": "and mapped onto a new image through the generator", + "type": "text" + }, + { + "bbox": [ + 420, + 218, + 426, + 226 + ], + "score": 0.63, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 216, + 505, + 228 + ], + "score": 1.0, + "content": ". Right: Amortization", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 227, + 504, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 227, + 174, + 239 + ], + "score": 1.0, + "content": "network that maps", + "type": "text" + }, + { + "bbox": [ + 174, + 227, + 180, + 236 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 227, + 504, + 239 + ], + "score": 1.0, + "content": "image/angle examples of a particular object-instance to the corresponding stochastic input.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 105, + 270, + 505, + 293 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 283 + ], + "score": 1.0, + "content": "single softmax output instead of the entire weight matrix directly. 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Here", + "type": "text" + }, + { + "bbox": [ + 223, + 372, + 234, + 383 + ], + "score": 0.88, + "content": "k _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 370, + 418, + 385 + ], + "score": 1.0, + "content": "is the number of observed examples in class", + "type": "text" + }, + { + "bbox": [ + 418, + 374, + 424, + 382 + ], + "score": 0.7, + "content": "c", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 370, + 443, + 385 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 443, + 372, + 504, + 384 + ], + "score": 0.93, + "content": "\\psi _ { c } = \\{ w _ { c } , b _ { c } \\}", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 383, + 505, + 395 + ], + "score": 1.0, + "content": "denotes the weight vector and bias of the linear classifier associated with that class. Thus, we construct", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 393, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 206, + 408 + ], + "score": 1.0, + "content": "the classification matrix", + "type": "text" + }, + { + "bbox": [ + 206, + 394, + 223, + 407 + ], + "score": 0.92, + "content": "\\psi ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 224, + 393, + 286, + 408 + ], + "score": 1.0, + "content": "by performing", + "type": "text" + }, + { + "bbox": [ + 286, + 396, + 295, + 405 + ], + "score": 0.81, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 393, + 506, + 408 + ], + "score": 1.0, + "content": "feed-forward passes through the inference network", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 107, + 405, + 204, + 419 + ], + "spans": [ + { + "bbox": [ + 107, + 406, + 153, + 418 + ], + "score": 0.92, + "content": "q _ { \\phi } ( \\psi | D , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 405, + 204, + 419 + ], + "score": 1.0, + "content": "(see Fig. 2).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 522 + ], + "lines": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "The assumption of context independent inference is an approximation. In Appendix B, we provide", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "theoretical and empirical justification for its validity. Our theoretical arguments use insights from", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 445, + 505, + 457 + ], + "score": 1.0, + "content": "Density Ratio Estimation (Mohamed, 2018; Sugiyama et al., 2012), and we empirically demonstrate", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 456, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 506, + 468 + ], + "score": 1.0, + "content": "that full approximate posterior distributions are close to their context independent counterparts. Crit-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 479 + ], + "score": 1.0, + "content": "ically, the context independent approximation addresses all the limitations of a naive amortization", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 478, + 504, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 504, + 489 + ], + "score": 1.0, + "content": "mentioned above: (i) the inference network needs to amortize far fewer parameters whose number", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 489, + 505, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 258, + 501 + ], + "score": 1.0, + "content": "does not scale with number of classes", + "type": "text" + }, + { + "bbox": [ + 258, + 489, + 267, + 499 + ], + "score": 0.8, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 489, + 505, + 501 + ], + "score": 1.0, + "content": "(a single weight vector instead of the entire matrix); (ii) the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 499, + 505, + 512 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 505, + 512 + ], + "score": 1.0, + "content": "amortization network can be meta-trained with different numbers of classes per task, and (iii) the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 511, + 276, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 180, + 523 + ], + "score": 1.0, + "content": "number of classes", + "type": "text" + }, + { + "bbox": [ + 181, + 511, + 190, + 520 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 190, + 511, + 276, + 523 + ], + "score": 1.0, + "content": "can vary at test-time.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 547, + 505, + 602 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 560 + ], + "score": 1.0, + "content": "VERSA for Few-Shot Image Reconstruction (Regression). 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We frame this as a multi-output regression task from a set of training images with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 590, + 365, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 365, + 604 + ], + "score": 1.0, + "content": "known orientations to output images with specified orientations.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 547, + 505, + 604 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 505, + 619 + ], + "score": 1.0, + "content": "Our generative model is similar to the generator of a GAN or the decoder of a VAE: A latent vector", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 107, + 617, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 107, + 619, + 153, + 631 + ], + "score": 0.92, + "content": "\\psi ^ { ( t ) } \\in \\mathbb { R } ^ { d _ { \\psi } }", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 617, + 506, + 633 + ], + "score": 1.0, + "content": ", which acts as an object-instance level input to the generator, is concatenated with an angle", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "representation and mapped through the generator to produce an image at the specified orientation. In", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 642, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 244, + 655 + ], + "score": 1.0, + "content": "this setting, we treat all parameters", + "type": "text" + }, + { + "bbox": [ + 244, + 642, + 250, + 652 + ], + "score": 0.81, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 642, + 505, + 655 + ], + "score": 1.0, + "content": "of the generator network as global parameters (see Appendix E.1", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 653, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 653, + 343, + 667 + ], + "score": 1.0, + "content": "for full details of the architecture), whereas the latent inputs", + "type": "text" + }, + { + "bbox": [ + 343, + 654, + 360, + 665 + ], + "score": 0.92, + "content": "\\psi ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 653, + 506, + 667 + ], + "score": 1.0, + "content": "are the task-specific parameters. We", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 505, + 677 + ], + "score": 1.0, + "content": "use a Gaussian likelihood in pixel space for the outputs of the generator. To ensure that the output", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 675, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 428, + 689 + ], + "score": 1.0, + "content": "means are between zero and one, we use a sigmoid activation after the final layer.", + "type": "text" + }, + { + "bbox": [ + 428, + 676, + 435, + 687 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 675, + 505, + 689 + ], + "score": 1.0, + "content": "parameterizes an", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "amortization network that first processes the image representations of an object, concatenates them", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 697, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 505, + 711 + ], + "score": 1.0, + "content": "with their associated view orientations, and processes them further before instance-pooling. From", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 707, + 506, + 724 + ], + "spans": [ + { + "bbox": [ + 104, + 707, + 219, + 724 + ], + "score": 1.0, + "content": "the pooled representations,", + "type": "text" + }, + { + "bbox": [ + 220, + 711, + 265, + 722 + ], + "score": 0.93, + "content": "q _ { \\phi } ( \\psi | D , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 707, + 418, + 724 + ], + "score": 1.0, + "content": "produces a distribution over vectors", + "type": "text" + }, + { + "bbox": [ + 418, + 709, + 435, + 721 + ], + "score": 0.91, + "content": "\\psi ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 436, + 707, + 506, + 724 + ], + "score": 1.0, + "content": ". This process is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 720, + 187, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 187, + 734 + ], + "score": 1.0, + "content": "illustrated in Fig. 3.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35, + "bbox_fs": [ + 104, + 608, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 371, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 372, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 372, + 96 + ], + "score": 1.0, + "content": "4 ML-PIP UN IFIES DISPARATE RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 104, + 506, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 507, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 507, + 119 + ], + "score": 1.0, + "content": "In this section, we continue in the spirit of Grant et al. (2018), and recast a broader class of meta-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 507, + 129 + ], + "score": 1.0, + "content": "learning approaches as approximate inference in hierarchical models. We show that ML-PIP unifies a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 507, + 140 + ], + "score": 1.0, + "content": "number of important approaches to meta-learning, including both gradient and metric based variants,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 507, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 507, + 152 + ], + "score": 1.0, + "content": "as well as amortized MAP inference and conditional modelling approaches (Garnelo et al., 2018a).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "We lay out these connections, most of which rely on point estimates for the task-specific parame-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 159, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 104, + 159, + 195, + 176 + ], + "score": 1.0, + "content": "ters corresponding to", + "type": "text" + }, + { + "bbox": [ + 196, + 160, + 361, + 174 + ], + "score": 0.92, + "content": "q ( \\psi ^ { ( t ) } | D ^ { ( t ) } , \\theta ) = \\delta \\left( \\psi ^ { ( t ) } \\bar { - } \\psi ^ { * } ( D ^ { ( t ) } , \\theta ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 159, + 506, + 176 + ], + "score": 1.0, + "content": ". In addition, we compare previous", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 200, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 200, + 185 + ], + "score": 1.0, + "content": "approaches to VERSA.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 196, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 195, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 383, + 210 + ], + "score": 1.0, + "content": "Gradient-Based Meta-Learning. Let the task-specific parameters", + "type": "text" + }, + { + "bbox": [ + 383, + 196, + 400, + 208 + ], + "score": 0.92, + "content": "\\psi ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 195, + 506, + 210 + ], + "score": 1.0, + "content": "be all the parameters in a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "neural network. Consider a point estimate formed by taking a step of gradient ascent of the training", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 293, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 179, + 232 + ], + "score": 1.0, + "content": "loss, initialized at", + "type": "text" + }, + { + "bbox": [ + 179, + 219, + 191, + 230 + ], + "score": 0.89, + "content": "\\psi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 218, + 282, + 232 + ], + "score": 1.0, + "content": "and with learning rate", + "type": "text" + }, + { + "bbox": [ + 282, + 221, + 289, + 230 + ], + "score": 0.77, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 218, + 293, + 232 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 233, + 416, + 267 + ], + "lines": [ + { + "bbox": [ + 195, + 233, + 416, + 267 + ], + "spans": [ + { + "bbox": [ + 195, + 233, + 416, + 267 + ], + "score": 0.94, + "content": "\\psi ^ { * } ( D ^ { ( t ) } , \\theta ) = \\psi _ { 0 } + \\eta \\frac { \\partial } { \\partial \\psi } \\log \\sum _ { n = 1 } ^ { N _ { t } } p ( y _ { n } ^ { ( t ) } | x _ { n } ^ { ( t ) } , \\psi , \\theta ) \\bigg | _ { \\psi _ { 0 } } .", + "type": "interline_equation", + "image_path": "7d060f52c6d2a1ee859d2459d8dfab8fa2d3add7c90d8311a5cfc786c6ecf8de.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 233, + 416, + 250.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 195, + 250.0, + 416, + 267.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 506, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "This is an example of semi-amortized inference (Kim et al., 2018b), as the only shared inference", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 291 + ], + "score": 1.0, + "content": "parameters are the initialization and learning rate, and optimization is required for each task (albeit", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 506, + 304 + ], + "score": 1.0, + "content": "only for one step). Importantly, Eq. (6) recovers Model-agnostic meta-learning (Finn et al., 2017),", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "providing a perspective as semi-amortized ML-PIP. This perspective is complementary to that of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "Grant et al. (2018) who justify the one-step gradient parameter update employed by MAML through", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 271, + 336 + ], + "score": 1.0, + "content": "MAP inference and the form of the prior", + "type": "text" + }, + { + "bbox": [ + 272, + 324, + 299, + 335 + ], + "score": 0.91, + "content": "p ( \\psi | \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 322, + 506, + 336 + ], + "score": 1.0, + "content": ". Note that the episodic meta-train / meta-test splits", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "do not fall out of this perspective. Instead we view the update choice as one of amortization which is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "trained using the predictive KL and naturally recovers the test-train splits. More generally, multiple", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 507, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 316, + 369 + ], + "score": 1.0, + "content": "gradient steps could be fed into an RNN to compute", + "type": "text" + }, + { + "bbox": [ + 316, + 357, + 328, + 367 + ], + "score": 0.89, + "content": "\\psi ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 356, + 507, + 369 + ], + "score": 1.0, + "content": "which recovers Ravi and Larochelle (2017).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 376, + 379 + ], + "score": 1.0, + "content": "In comparison to these methods, besides being distributional over", + "type": "text" + }, + { + "bbox": [ + 377, + 367, + 384, + 379 + ], + "score": 0.83, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 367, + 506, + 379 + ], + "score": 1.0, + "content": ", VERSA relieves the need to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "back-propagate through gradient based updates during training and compute gradients at test time, as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 478, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 478, + 402 + ], + "score": 1.0, + "content": "well as enables the treatment of both local and global parameters which simplifies inference.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "Metric-Based Few-Shot Learning. Let the task-specific parameters be the top layer softmax", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 421, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 264, + 441 + ], + "score": 1.0, + "content": "weights and biases of a neural network", + "type": "text" + }, + { + "bbox": [ + 264, + 423, + 355, + 438 + ], + "score": 0.93, + "content": "\\boldsymbol { \\psi } ^ { ( t ) } = \\{ w _ { c } ^ { ( t ) } , b _ { c } ^ { ( t ) } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 421, + 505, + 441 + ], + "score": 1.0, + "content": ". The shared parameters are the lower", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "layer weights. 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In cases where a point estimate of the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 696, + 335, + 709 + ], + "spans": [ + { + "bbox": [ + 106, + 696, + 335, + 709 + ], + "score": 1.0, + "content": "task-specific parameters are used the predictive becomes", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + }, + { + "type": "interline_equation", + "bbox": [ + 181, + 710, + 430, + 736 + ], + "lines": [ + { + "bbox": [ + 181, + 710, + 430, + 736 + ], + "spans": [ + { + "bbox": [ + 181, + 710, + 430, + 736 + ], + "score": 0.92, + "content": "q _ { \\phi } ( \\tilde { y } | D , \\theta ) = \\int p ( \\tilde { y } | \\psi , \\theta ) q _ { \\phi } ( \\psi | D , \\theta ) \\mathrm { d } \\psi = p ( \\tilde { y } | \\psi ^ { * } ( D , \\theta ) , \\theta ) .", + "type": "interline_equation", + "image_path": "fc06a6df173a21c00e71f5e7b83cece086ccd15fa74546f7f29eafe70805ec8a.jpg" + } + ] + } + ], + "index": 49, + "virtual_lines": [ + { + "bbox": [ + 181, + 710, + 430, + 736 + ], + "spans": [], + "index": 49 + } + ] + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 310, + 761 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 106, + 81, + 371, + 94 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 372, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 372, + 96 + ], + "score": 1.0, + "content": "4 ML-PIP UN IFIES DISPARATE RELATED WORK", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 106, + 104, + 506, + 185 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 507, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 507, + 119 + ], + "score": 1.0, + "content": "In this section, we continue in the spirit of Grant et al. (2018), and recast a broader class of meta-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 507, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 507, + 129 + ], + "score": 1.0, + "content": "learning approaches as approximate inference in hierarchical models. We show that ML-PIP unifies a", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 128, + 507, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 507, + 140 + ], + "score": 1.0, + "content": "number of important approaches to meta-learning, including both gradient and metric based variants,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 138, + 507, + 152 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 507, + 152 + ], + "score": 1.0, + "content": "as well as amortized MAP inference and conditional modelling approaches (Garnelo et al., 2018a).", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 163 + ], + "score": 1.0, + "content": "We lay out these connections, most of which rely on point estimates for the task-specific parame-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 159, + 506, + 176 + ], + "spans": [ + { + "bbox": [ + 104, + 159, + 195, + 176 + ], + "score": 1.0, + "content": "ters corresponding to", + "type": "text" + }, + { + "bbox": [ + 196, + 160, + 361, + 174 + ], + "score": 0.92, + "content": "q ( \\psi ^ { ( t ) } | D ^ { ( t ) } , \\theta ) = \\delta \\left( \\psi ^ { ( t ) } \\bar { - } \\psi ^ { * } ( D ^ { ( t ) } , \\theta ) \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 159, + 506, + 176 + ], + "score": 1.0, + "content": ". In addition, we compare previous", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 173, + 200, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 200, + 185 + ], + "score": 1.0, + "content": "approaches to VERSA.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4, + "bbox_fs": [ + 104, + 104, + 507, + 185 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 196, + 505, + 230 + ], + "lines": [ + { + "bbox": [ + 105, + 195, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 383, + 210 + ], + "score": 1.0, + "content": "Gradient-Based Meta-Learning. Let the task-specific parameters", + "type": "text" + }, + { + "bbox": [ + 383, + 196, + 400, + 208 + ], + "score": 0.92, + "content": "\\psi ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 401, + 195, + 506, + 210 + ], + "score": 1.0, + "content": "be all the parameters in a", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 506, + 221 + ], + "score": 1.0, + "content": "neural network. Consider a point estimate formed by taking a step of gradient ascent of the training", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 218, + 293, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 179, + 232 + ], + "score": 1.0, + "content": "loss, initialized at", + "type": "text" + }, + { + "bbox": [ + 179, + 219, + 191, + 230 + ], + "score": 0.89, + "content": "\\psi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 218, + 282, + 232 + ], + "score": 1.0, + "content": "and with learning rate", + "type": "text" + }, + { + "bbox": [ + 282, + 221, + 289, + 230 + ], + "score": 0.77, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 218, + 293, + 232 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 195, + 506, + 232 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 195, + 233, + 416, + 267 + ], + "lines": [ + { + "bbox": [ + 195, + 233, + 416, + 267 + ], + "spans": [ + { + "bbox": [ + 195, + 233, + 416, + 267 + ], + "score": 0.94, + "content": "\\psi ^ { * } ( D ^ { ( t ) } , \\theta ) = \\psi _ { 0 } + \\eta \\frac { \\partial } { \\partial \\psi } \\log \\sum _ { n = 1 } ^ { N _ { t } } p ( y _ { n } ^ { ( t ) } | x _ { n } ^ { ( t ) } , \\psi , \\theta ) \\bigg | _ { \\psi _ { 0 } } .", + "type": "interline_equation", + "image_path": "7d060f52c6d2a1ee859d2459d8dfab8fa2d3add7c90d8311a5cfc786c6ecf8de.jpg" + } + ] + } + ], + "index": 11.5, + "virtual_lines": [ + { + "bbox": [ + 195, + 233, + 416, + 250.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 195, + 250.0, + 416, + 267.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 267, + 506, + 401 + ], + "lines": [ + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 506, + 281 + ], + "score": 1.0, + "content": "This is an example of semi-amortized inference (Kim et al., 2018b), as the only shared inference", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 506, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 506, + 291 + ], + "score": 1.0, + "content": "parameters are the initialization and learning rate, and optimization is required for each task (albeit", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 290, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 506, + 304 + ], + "score": 1.0, + "content": "only for one step). Importantly, Eq. (6) recovers Model-agnostic meta-learning (Finn et al., 2017),", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 105, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "providing a perspective as semi-amortized ML-PIP. This perspective is complementary to that of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "Grant et al. (2018) who justify the one-step gradient parameter update employed by MAML through", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 322, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 104, + 322, + 271, + 336 + ], + "score": 1.0, + "content": "MAP inference and the form of the prior", + "type": "text" + }, + { + "bbox": [ + 272, + 324, + 299, + 335 + ], + "score": 0.91, + "content": "p ( \\psi | \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 322, + 506, + 336 + ], + "score": 1.0, + "content": ". Note that the episodic meta-train / meta-test splits", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "do not fall out of this perspective. Instead we view the update choice as one of amortization which is", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 345, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 106, + 345, + 506, + 357 + ], + "score": 1.0, + "content": "trained using the predictive KL and naturally recovers the test-train splits. More generally, multiple", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 356, + 507, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 316, + 369 + ], + "score": 1.0, + "content": "gradient steps could be fed into an RNN to compute", + "type": "text" + }, + { + "bbox": [ + 316, + 357, + 328, + 367 + ], + "score": 0.89, + "content": "\\psi ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 356, + 507, + 369 + ], + "score": 1.0, + "content": "which recovers Ravi and Larochelle (2017).", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 367, + 506, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 376, + 379 + ], + "score": 1.0, + "content": "In comparison to these methods, besides being distributional over", + "type": "text" + }, + { + "bbox": [ + 377, + 367, + 384, + 379 + ], + "score": 0.83, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 367, + 506, + 379 + ], + "score": 1.0, + "content": ", VERSA relieves the need to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 391 + ], + "score": 1.0, + "content": "back-propagate through gradient based updates during training and compute gradients at test time, as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 388, + 478, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 478, + 402 + ], + "score": 1.0, + "content": "well as enables the treatment of both local and global parameters which simplifies inference.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 267, + 507, + 402 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 411, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 424 + ], + "score": 1.0, + "content": "Metric-Based Few-Shot Learning. Let the task-specific parameters be the top layer softmax", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 421, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 104, + 421, + 264, + 441 + ], + "score": 1.0, + "content": "weights and biases of a neural network", + "type": "text" + }, + { + "bbox": [ + 264, + 423, + 355, + 438 + ], + "score": 0.93, + "content": "\\boldsymbol { \\psi } ^ { ( t ) } = \\{ w _ { c } ^ { ( t ) } , b _ { c } ^ { ( t ) } \\} _ { c = 1 } ^ { C }", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 421, + 505, + 441 + ], + "score": 1.0, + "content": ". The shared parameters are the lower", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "layer weights. Consider amortized point estimates for these parameters constructed by averaging the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 448, + 249, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 249, + 460 + ], + "score": 1.0, + "content": "top-layer activations for each class,", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5, + "bbox_fs": [ + 104, + 411, + 505, + 460 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 129, + 461, + 481, + 496 + ], + "lines": [ + { + "bbox": [ + 129, + 461, + 481, + 496 + ], + "spans": [ + { + "bbox": [ + 129, + 461, + 481, + 496 + ], + "score": 0.9, + "content": "\\psi ^ { * } ( D ^ { ( t ) } , \\theta ) = \\big \\{ w _ { c } ^ { * } , b _ { c } ^ { * } \\big \\} _ { c = 1 } ^ { C } = \\Big \\{ \\mu _ { c } ^ { ( t ) } , - \\| \\mu _ { c } ^ { ( t ) } \\| ^ { 2 } / 2 \\Big \\} _ { c = 1 } ^ { C } \\mathrm { ~ w h e r e ~ } \\mu _ { c } ^ { ( t ) } = \\frac { 1 } { k _ { c } } \\sum _ { n = 1 } ^ { k _ { c } } h _ { \\theta } ( x _ { n } ^ { ( c ) } )", + "type": "interline_equation", + "image_path": "ee67c7c3e5a42acf9658aaa2eb7880f88fc72ba536326f5c3d8f5c7e566667f0.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 129, + 461, + 481, + 472.6666666666667 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 129, + 472.6666666666667, + 481, + 484.33333333333337 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 129, + 484.33333333333337, + 481, + 496.00000000000006 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 497, + 342, + 509 + ], + "lines": [ + { + "bbox": [ + 106, + 497, + 342, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 342, + 510 + ], + "score": 1.0, + "content": "These choices lead to the following predictive distribution:", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32, + "bbox_fs": [ + 106, + 497, + 342, + 510 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 132, + 511, + 478, + 538 + ], + "lines": [ + { + "bbox": [ + 132, + 511, + 478, + 538 + ], + "spans": [ + { + "bbox": [ + 132, + 511, + 478, + 538 + ], + "score": 0.9, + "content": "p ( \\tilde { y } ^ { ( t ) } = c | \\tilde { x } ^ { ( t ) } , \\theta ) \\propto \\exp \\left( - d ( h _ { \\theta } ( \\tilde { x } ^ { ( t ) } ) , \\mu _ { c } ^ { ( t ) } ) \\right) = \\exp \\left( h _ { \\theta } ( \\tilde { x } ^ { ( t ) } ) ^ { T } \\mu _ { c } ^ { ( t ) } - \\frac { 1 } { 2 } \\| \\mu _ { c } ^ { ( t ) } \\| ^ { 2 } \\right) ,", + "type": "interline_equation", + "image_path": "23010b57029acf028828c91fca0a3880fe24aa126dbbc6e728251582b8b4aed2.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 132, + 511, + 478, + 520.0 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 132, + 520.0, + 478, + 529.0 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 132, + 529.0, + 478, + 538.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 540, + 504, + 574 + ], + "lines": [ + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 477, + 554 + ], + "score": 1.0, + "content": "which recovers prototypical networks (Snell et al., 2017) using a Euclidean distance function", + "type": "text" + }, + { + "bbox": [ + 478, + 541, + 484, + 550 + ], + "score": 0.82, + "content": "d", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 540, + 505, + 554 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "the final hidden layer being the embedding space. 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Standard application of amortized VI (Kingma and", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 151, + 506, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 151, + 430, + 163 + ], + "score": 1.0, + "content": "Welling, 2014; Rezende et al., 2014; Kingma et al., 2015; Blundell et al., 2015) for", + "type": "text" + }, + { + "bbox": [ + 430, + 152, + 438, + 162 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 438, + 151, + 506, + 163 + ], + "score": 1.0, + "content": "in the multi-task", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 466, + 174 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 430, + 174 + ], + "score": 1.0, + "content": "discriminative model optimizes the Monte Carlo approximated free-energy w.r.t.", + "type": "text" + }, + { + "bbox": [ + 430, + 163, + 437, + 173 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 161, + 455, + 174 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 455, + 163, + 461, + 172 + ], + "score": 0.83, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 161, + 466, + 174 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 178, + 501, + 217 + ], + "lines": [ + { + "bbox": [ + 111, + 178, + 501, + 217 + ], + "spans": [ + { + "bbox": [ + 111, + 178, + 501, + 217 + ], + "score": 0.93, + "content": "\\hat { \\mathcal { L } } ( \\boldsymbol { \\theta } , \\boldsymbol { \\phi } ) = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\left( \\sum _ { ( \\boldsymbol { x } , \\boldsymbol { y } ) \\in D ^ { ( t ) } } \\left( \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } \\log p ( \\boldsymbol { y } ^ { ( t ) } | \\boldsymbol { x } ^ { ( t ) } , \\boldsymbol { \\psi } _ { l } ^ { ( t ) } , \\boldsymbol { \\theta } ) \\right) - \\mathrm { K L } \\left[ q _ { \\boldsymbol { \\phi } } ( \\boldsymbol { \\psi } | D ^ { ( t ) } , \\boldsymbol { \\theta } ) \\| p ( \\boldsymbol { \\psi } | \\boldsymbol { \\theta } ) \\right] \\right) ,", + "type": "interline_equation", + "image_path": "9a7a2798835e23ac425f38572b265a4f2e6ee3f798c6481b9b91502faf72a79d.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 111, + 178, + 501, + 191.0 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 111, + 191.0, + 501, + 204.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 111, + 204.0, + 501, + 217.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 272 + ], + "lines": [ + { + "bbox": [ + 104, + 225, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 104, + 227, + 134, + 241 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 225, + 222, + 241 + ], + "score": 0.94, + "content": "\\psi _ { l } ^ { ( t ) } \\sim q _ { \\phi } ( \\psi | D ^ { ( t ) } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 227, + 506, + 241 + ], + "score": 1.0, + "content": ". In addition to the conceptual difference from ML-PIP (discussed in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "Section 2.1), this differs from the ML-PIP objective by i) not employing meta train / test splits, and ii)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "including the KL for regularization instead. In Section 5, we show that VERSA significantly improves", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "over standard VI in the few-shot classification case and compare to recent VI/meta-learning hybrids.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "title", + "bbox": [ + 108, + 288, + 280, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 281, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 281, + 302 + ], + "score": 1.0, + "content": "5 EXPER IMENTS AND RESULTS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "We evaluate VERSA on several few-shot learning tasks. We begin with toy experiments to investigate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "the properties of the amortized posterior inference achieved by VERSA. We then report few-shot", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "classification results using the Omniglot and miniImageNet datasets in Section 5.2, and demonstrate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "score": 1.0, + "content": "VERSA’s ability to retain high accuracy as the shot and way are varied at test time. 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We introduce the inference network", + "type": "text" + }, + { + "bbox": [ + 374, + 467, + 502, + 482 + ], + "score": 0.91, + "content": "q _ { \\phi } ( \\psi | D ^ { ( t ) } ) = \\mathcal { N } ( \\psi ; \\mu _ { q } ^ { ( t ) } , \\sigma _ { q } ^ { ( t ) 2 } )", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 481, + 205, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 205, + 493 + ], + "score": 1.0, + "content": "amortizing inference as:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 497, + 431, + 531 + ], + "lines": [ + { + "bbox": [ + 180, + 497, + 431, + 531 + ], + "spans": [ + { + "bbox": [ + 180, + 497, + 431, + 531 + ], + "score": 0.96, + "content": "\\mu _ { q } ^ { ( t ) } = w _ { \\mu } \\sum _ { n = 1 } ^ { N } y _ { n } ^ { ( t ) } + b _ { \\mu } , \\quad \\sigma _ { q } ^ { ( t ) 2 } = \\exp \\left( w _ { \\sigma } \\sum _ { n = 1 } ^ { N } y _ { n } ^ { ( t ) } + b _ { \\sigma } \\right) .", + "type": "interline_equation", + "image_path": "3c616af307bd84b679e0cbd405f9716ac21b732b7ad1b18487cb88fb01843057.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 180, + 497, + 431, + 508.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 180, + 508.3333333333333, + 431, + 519.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 180, + 519.6666666666666, + 431, + 531.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 535, + 506, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 207, + 550 + ], + "score": 1.0, + "content": "The learnable parameters", + "type": "text" + }, + { + "bbox": [ + 207, + 536, + 294, + 548 + ], + "score": 0.93, + "content": "\\phi = \\{ w _ { \\boldsymbol { \\mu } } , b _ { \\boldsymbol { \\mu } } , w _ { \\sigma } , b _ { \\sigma } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 535, + 506, + 550 + ], + "score": 1.0, + "content": "are trained with the objective function in Eq. 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Specif-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "ically, we consider the Omniglot (Lake et al., 2011) and miniImageNet (Ravi and Larochelle, 2017)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 680, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 184, + 692 + ], + "score": 1.0, + "content": "datasets which are", + "type": "text" + }, + { + "bbox": [ + 184, + 680, + 193, + 690 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 680, + 314, + 692 + ], + "score": 1.0, + "content": "-way classification tasks with", + "type": "text" + }, + { + "bbox": [ + 314, + 680, + 325, + 691 + ], + "score": 0.87, + "content": "k _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 680, + 506, + 692 + ], + "score": 1.0, + "content": "examples per class. 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In addition to the conceptual difference from ML-PIP (discussed in", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 252 + ], + "score": 1.0, + "content": "Section 2.1), this differs from the ML-PIP objective by i) not employing meta train / test splits, and ii)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 263 + ], + "score": 1.0, + "content": "including the KL for regularization instead. In Section 5, we show that VERSA significantly improves", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "over standard VI in the few-shot classification case and compare to recent VI/meta-learning hybrids.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 104, + 225, + 506, + 273 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 288, + 280, + 301 + ], + "lines": [ + { + "bbox": [ + 105, + 288, + 281, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 288, + 281, + 302 + ], + "score": 1.0, + "content": "5 EXPER IMENTS AND RESULTS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 313, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 106, + 313, + 505, + 326 + ], + "score": 1.0, + "content": "We evaluate VERSA on several few-shot learning tasks. We begin with toy experiments to investigate", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "the properties of the amortized posterior inference achieved by VERSA. We then report few-shot", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 506, + 348 + ], + "score": 1.0, + "content": "classification results using the Omniglot and miniImageNet datasets in Section 5.2, and demonstrate", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 505, + 358 + ], + "score": 1.0, + "content": "VERSA’s ability to retain high accuracy as the shot and way are varied at test time. In Section 5.3, we", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 357, + 491, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 491, + 370 + ], + "score": 1.0, + "content": "examine VERSA’s performance on a one-shot view reconstruction task with ShapeNet objects.2", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 313, + 506, + 370 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 382, + 311, + 393 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 312, + 395 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 312, + 395 + ], + "score": 1.0, + "content": "5.1 POSTER IOR INFERENCE W ITH TOY DATA", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "To investigate the approximate inference performed by our training procedure, we run the following", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 426 + ], + "score": 1.0, + "content": "experiment. 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We introduce the inference network", + "type": "text" + }, + { + "bbox": [ + 374, + 467, + 502, + 482 + ], + "score": 0.91, + "content": "q _ { \\phi } ( \\psi | D ^ { ( t ) } ) = \\mathcal { N } ( \\psi ; \\mu _ { q } ^ { ( t ) } , \\sigma _ { q } ^ { ( t ) 2 } )", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 481, + 205, + 493 + ], + "spans": [ + { + "bbox": [ + 106, + 481, + 205, + 493 + ], + "score": 1.0, + "content": "amortizing inference as:", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 103, + 455, + 505, + 493 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 497, + 431, + 531 + ], + "lines": [ + { + "bbox": [ + 180, + 497, + 431, + 531 + ], + "spans": [ + { + "bbox": [ + 180, + 497, + 431, + 531 + ], + "score": 0.96, + "content": "\\mu _ { q } ^ { ( t ) } = w _ { \\mu } \\sum _ { n = 1 } ^ { N } y _ { n } ^ { ( t ) } + b _ { \\mu } , \\quad \\sigma _ { q } ^ { ( t ) 2 } = \\exp \\left( w _ { \\sigma } \\sum _ { n = 1 } ^ { N } y _ { n } ^ { ( t ) } + b _ { \\sigma } \\right) .", + "type": "interline_equation", + "image_path": "3c616af307bd84b679e0cbd405f9716ac21b732b7ad1b18487cb88fb01843057.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 180, + 497, + 431, + 508.3333333333333 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 180, + 508.3333333333333, + 431, + 519.6666666666666 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 180, + 519.6666666666666, + 431, + 531.0 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 535, + 506, + 624 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 207, + 550 + ], + "score": 1.0, + "content": "The learnable parameters", + "type": "text" + }, + { + "bbox": [ + 207, + 536, + 294, + 548 + ], + "score": 0.93, + "content": "\\phi = \\{ w _ { \\boldsymbol { \\mu } } , b _ { \\boldsymbol { \\mu } } , w _ { \\sigma } , b _ { \\sigma } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 535, + 506, + 550 + ], + "score": 1.0, + "content": "are trained with the objective function in Eq. 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Then, a separate set of tasks is generated from the same generative process, and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 568, + 504, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 158, + 582 + ], + "score": 1.0, + "content": "the posterior", + "type": "text" + }, + { + "bbox": [ + 159, + 570, + 195, + 581 + ], + "score": 0.93, + "content": "q _ { \\phi } ( \\psi | D )", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 568, + 496, + 582 + ], + "score": 1.0, + "content": "is inferred with the learned amortization parameters. The true posterior over", + "type": "text" + }, + { + "bbox": [ + 496, + 570, + 504, + 580 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 592 + ], + "score": 1.0, + "content": "is Gaussian with a mean that depends on the task, and may be computed analytically. 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Specif-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "ically, we consider the Omniglot (Lake et al., 2011) and miniImageNet (Ravi and Larochelle, 2017)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 680, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 184, + 692 + ], + "score": 1.0, + "content": "datasets which are", + "type": "text" + }, + { + "bbox": [ + 184, + 680, + 193, + 690 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 680, + 314, + 692 + ], + "score": 1.0, + "content": "-way classification tasks with", + "type": "text" + }, + { + "bbox": [ + 314, + 680, + 325, + 691 + ], + "score": 0.87, + "content": "k _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 680, + 506, + 692 + ], + "score": 1.0, + "content": "examples per class. 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OmniglotminilmageNet
5-way NLL20-way NLL5-way NLL
Method1-shot5-shot1-shot5-shot1-shot5-shot
Amortized VI0.179 ± 0.0090.137 ± 0.0040.456± 0.0100.253 ± 0.0041.328 ± 0.0241.165 ± 0.010
Non-Amortized VI0.144 ± 0.0050.025 ± 0.0010.393 ± 0.0050.078 ± 0.002
VERSA0.010± 0.0050.007 ± 0.0030.079 ± 0.0090.031 ± 0.0041.183 ± 0.0230.859 ± 0.015
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While most of the methods evaluated in Table 3 adapt all of the learned", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 104, + 433, + 506, + 448 + ], + "score": 1.0, + "content": "parameters for new tasks, VERSA is able to achieve state-of-the-art performance despite adapting", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 445, + 277, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 277, + 457 + ], + "score": 1.0, + "content": "only the weights of the top-level classifier.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 345, + 506, + 457 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "Comparison to standard and amortized VI. To investigate the performance of our inference", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 485, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 500 + ], + "score": 1.0, + "content": "procedure, we compare it in terms of log-likelihood (Table 1) and accuracy (Table 3) to training", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "the same model using both amortized and non-amortized VI (i.e., Eq. (10)). Derivations and further", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 508, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 520 + ], + "score": 1.0, + "content": "experimental details are provided in Appendix C. VERSA improves substantially over amortized VI", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 519, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 506, + 531 + ], + "score": 1.0, + "content": "even though the same amortization network is used for both. This is due to VI’s tendency to under-fit,", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 529, + 507, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 507, + 542 + ], + "score": 1.0, + "content": "especially for small numbers of data points (Trippe and Turner, 2018; Turner and Sahani, 2011)", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 554 + ], + "score": 1.0, + "content": "which is compounded when using inference networks (Cremer et al., 2018). Using non-amortized VI", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "improves performance substantially, but does not reach the level of VERSA and forming the posterior", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "is significantly slower as it requires many forward / backward passes through the network. This is", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "similar in spirit to MAML (Finn et al., 2017), though MAML dramatically reduces the number of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "required iterations by finding good global initializations e.g., five gradient steps for miniImageNet.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 720, + 369, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 369, + 734 + ], + "score": 1.0, + "content": "This is in contrast to the single forward pass required by VERSA.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 29, + "bbox_fs": [ + 105, + 475, + 507, + 554 + ] + }, + { + "type": "table", + "bbox": [ + 111, + 592, + 500, + 654 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 106, + 567, + 504, + 589 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 568, + 506, + 579 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 506, + 579 + ], + "score": 1.0, + "content": "Table 1: Negative Log-likelihood (NLL) results for different few-shot settings on Omniglot and miniImageNet.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 578, + 493, + 589 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 122, + 589 + ], + "score": 1.0, + "content": "The", + "type": "text" + }, + { + "bbox": [ + 122, + 578, + 132, + 588 + ], + "score": 0.52, + "content": "\\pm", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 578, + 196, + 589 + ], + "score": 1.0, + "content": "sign indicates the", + "type": "text" + }, + { + "bbox": [ + 196, + 578, + 214, + 587 + ], + "score": 0.87, + "content": "9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 578, + 493, + 589 + ], + "score": 1.0, + "content": "confidence interval over tasks using a Student’s t-distribution approximation.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 33.5 + }, + { + "type": "table_body", + "bbox": [ + 111, + 592, + 500, + 654 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 592, + 500, + 654 + ], + "spans": [ + { + "bbox": [ + 111, + 592, + 500, + 654 + ], + "score": 0.978, + "html": "
OmniglotminilmageNet
5-way NLL20-way NLL5-way NLL
Method1-shot5-shot1-shot5-shot1-shot5-shot
Amortized VI0.179 ± 0.0090.137 ± 0.0040.456± 0.0100.253 ± 0.0041.328 ± 0.0241.165 ± 0.010
Non-Amortized VI0.144 ± 0.0050.025 ± 0.0010.393 ± 0.0050.078 ± 0.002
VERSA0.010± 0.0050.007 ± 0.0030.079 ± 0.0090.031 ± 0.0041.183 ± 0.0230.859 ± 0.015
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Fig. 5a shows that a model trained for a particular", + "type": "text" + }, + { + "bbox": [ + 371, + 94, + 380, + 104 + ], + "score": 0.8, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 92, + 506, + 108 + ], + "score": 1.0, + "content": "-way retains very high accuracy", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 117, + 117 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 105, + 127, + 114 + ], + "score": 0.82, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "is varied. For example, when VERSA is trained for the 20-Way, 5-Shot condition, at test-time", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 159, + 128 + ], + "score": 1.0, + "content": "it can handle", + "type": "text" + }, + { + "bbox": [ + 160, + 115, + 198, + 126 + ], + "score": 0.9, + "content": "C = 1 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 114, + 423, + 128 + ], + "score": 1.0, + "content": "way conditions and retain an accuracy of approximately", + "type": "text" + }, + { + "bbox": [ + 424, + 115, + 443, + 126 + ], + "score": 0.85, + "content": "94 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 114, + 506, + 128 + ], + "score": 1.0, + "content": ". Fig. 5b shows", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 273, + 138 + ], + "score": 1.0, + "content": "similar robustness as the number of shots", + "type": "text" + }, + { + "bbox": [ + 273, + 127, + 284, + 137 + ], + "score": 0.88, + "content": "k _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 126, + 505, + 138 + ], + "score": 1.0, + "content": "is varied. VERSA therefore demonstrates considerable", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "flexibility and robustness to the test-time conditions, but at the same time it is efficient as it only", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 146, + 507, + 163 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 507, + 163 + ], + "score": 1.0, + "content": "requires forward passes through the network. The time taken to evaluate 1000 test tasks with a 5-way,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "5-shot miniImageNet trained model using MAML (https://github.com/cbfinn/maml) is", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 170, + 505, + 183 + ], + "score": 1.0, + "content": "302.9 seconds whereas VERSA took 53.5 seconds on a NVIDIA Tesla P100-PCIE-16GB GPU. This", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 181, + 504, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 157, + 194 + ], + "score": 1.0, + "content": "is more than", + "type": "text" + }, + { + "bbox": [ + 158, + 182, + 172, + 192 + ], + "score": 0.86, + "content": "5 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 181, + 471, + 194 + ], + "score": 1.0, + "content": "speed advantage in favor of VERSA while bettering MAML in accuracy by", + "type": "text" + }, + { + "bbox": [ + 472, + 181, + 499, + 192 + ], + "score": 0.87, + "content": "4 . 2 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 181, + 504, + 194 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "image", + "bbox": [ + 133, + 208, + 485, + 325 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 133, + 208, + 485, + 325 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 133, + 208, + 485, + 325 + ], + "spans": [ + { + "bbox": [ + 133, + 208, + 485, + 325 + ], + "score": 0.973, + "type": "image", + "image_path": "69a7c60c0513a773fe18756537a4337d39a44e0da27404102321c16b9171f824.jpg" + } + ] + } + ], + "index": 11, + "virtual_lines": [ + { + "bbox": [ + 133, + 208, + 485, + 247.0 + ], + "spans": [], + "index": 10 + }, + { + "bbox": [ + 133, + 247.0, + 485, + 286.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 133, + 286.0, + 485, + 325.0 + ], + "spans": [], + "index": 12 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 334, + 506, + 365 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "Figure 5: Test accuracy on Omniglot when varying (a) way (fixing shot to be that used for training) and (b)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "shot. In Fig. 5b, all models are evaluated on 5-way classification. Colors indicate models trained with different", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 354, + 226, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 226, + 366 + ], + "score": 1.0, + "content": "way-shot episodic combinations.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14 + }, + { + "type": "image_caption", + "bbox": [ + 308, + 653, + 505, + 704 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 308, + 653, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 308, + 653, + 505, + 664 + ], + "score": 1.0, + "content": "Table 2: View reconstruction test results. Mean squared", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 308, + 663, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 308, + 663, + 506, + 673 + ], + "score": 1.0, + "content": "error (MSE – lower is better) and the structural similar-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 309, + 673, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 309, + 673, + 505, + 684 + ], + "score": 1.0, + "content": "ity index (SSIM - higher is better) (Wang et al., 2004)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 308, + 684, + 505, + 694 + ], + "spans": [ + { + "bbox": [ + 308, + 684, + 505, + 694 + ], + "score": 1.0, + "content": "are measured between the generated and ground truth", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 308, + 693, + 505, + 705 + ], + "spans": [ + { + "bbox": [ + 308, + 693, + 505, + 705 + ], + "score": 1.0, + "content": "images. Error bars not shown as they are insignificant.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40 + } + ], + "index": 14 + }, + { + "type": "title", + "bbox": [ + 108, + 400, + 293, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 295, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 295, + 413 + ], + "score": 1.0, + "content": "5.3 SHAPENET VIEW RECONSTRUCT ION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 422, + 505, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "score": 1.0, + "content": "ShapeNetCore v2 (Chang et al., 2015) is a database of 3D objects covering 55 common object", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 434, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 167, + 446 + ], + "score": 1.0, + "content": "categories with", + "type": "text" + }, + { + "bbox": [ + 167, + 434, + 203, + 444 + ], + "score": 0.82, + "content": "{ \\sim } 5 1 { , } 3 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 434, + 506, + 446 + ], + "score": 1.0, + "content": "unique objects. For our experiments, we use 12 of the largest object categories.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "We concatenate all instances from all 12 of the object categories together to obtain a dataset of 37,108", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 456, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 342, + 468 + ], + "score": 1.0, + "content": "objects. This dataset is then randomly shuffled and we use", + "type": "text" + }, + { + "bbox": [ + 342, + 456, + 362, + 466 + ], + "score": 0.88, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 456, + 470, + 468 + ], + "score": 1.0, + "content": "of the objects for training,", + "type": "text" + }, + { + "bbox": [ + 470, + 456, + 489, + 466 + ], + "score": 0.85, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 490, + 456, + 505, + 468 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 167, + 479 + ], + "score": 1.0, + "content": "validation, and", + "type": "text" + }, + { + "bbox": [ + 167, + 467, + 187, + 477 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 467, + 415, + 479 + ], + "score": 1.0, + "content": "for testing. For each object, we generate 36 views of size", + "type": "text" + }, + { + "bbox": [ + 415, + 467, + 449, + 477 + ], + "score": 0.89, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "pixels spaced", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 478, + 324, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 324, + 491 + ], + "score": 1.0, + "content": "evenly every 10 degrees in azimuth around the object.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 506, + 637 + ], + "lines": [ + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 494, + 506, + 508 + ], + "score": 1.0, + "content": "We evaluate VERSA by comparing it to a conditional variational autoencoder (C-VAE) with view", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "angles as labels (Kingma et al., 2014; Narayanaswamy et al., 2017) and identical architectures. We", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 515, + 507, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 507, + 529 + ], + "score": 1.0, + "content": "train VERSA in an episodic manner and the C-VAE in batch-mode on all 12 object classes at once.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "We train on a single view selected at random and use the remaining views to evaluate the objective", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "function. For full experimentation details see Appendix E. Fig. 6 shows views of unseen objects", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "score": 1.0, + "content": "from the test set generated from a single shot with VERSA as well as a C-VAE and compares both to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 506, + 573 + ], + "score": 1.0, + "content": "ground truth views. Both VERSA and the C-VAE capture the correct orientation of the object in the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "generated images. 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ModelMSESSIM
C-VAE 1-shot0.02690.5705
VERSA 1-shot0.01080.7893
VERSA 5-shot0.00690.8483
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We", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 515, + 507, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 507, + 529 + ], + "score": 1.0, + "content": "train VERSA in an episodic manner and the C-VAE in batch-mode on all 12 object classes at once.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "We train on a single view selected at random and use the remaining views to evaluate the objective", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 551 + ], + "score": 1.0, + "content": "function. For full experimentation details see Appendix E. 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ModelMSESSIM
C-VAE 1-shot0.02690.5705
VERSA 1-shot0.01080.7893
VERSA 5-shot0.00690.8483
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OmniglotminilmageNet
Method5-way accuracy (%)5-shot20-way accuracy(%) 1-shot5-shot5-way accuracy (%) 1-shot 5-shot
1-shot98.488.197.0
Siamese Nets (Koch et al., 2015)97.398.946.6
Matching Nets (Vinyals et al.,2016)98.193.898.560.0
Neural Statistician (Edwards and Storkey, 2017)98.199.593.298.1
Memory Mod (Kaiser et al.,2017)98.499.695.098.6
Meta LSTM (Ravi and Larochelle, 2017)43.44 ± 0.77 60.60± 0.71
MAML (Finn et al., 2017)98.7 ± 0.499.9 ± 0.195.8 ± 0.398.9± 0.248.7 ± 1.84 63.11 ± 0.92
Prototypical Netsa (Snell et al., 2017)97.499.395.498.7 46.61 ± 0.7865.77 ± 0.70
mAP-SSVM (Triantafillou et al., 2017)98.699.695.298.650.32 ± 0.80 63.94 ± 0.72
mAP-DLM (Triantafillou et al., 2017)98.899.695.498.6 50.28 ± 0.8063.70± 0.70
LLAMA (Grant et al., 2018)49.40 ± 1.83
PLATIPUS (Finn et al., 2018)50.13 ± 1.86
Meta-SGD (Li et al., 2017)99.53± 0.2699.93 ± 0.0995.93 ± 0.3898.97 ± 0.1950.47 ± 1.87 64.03 ± 0.94
SNAIL (Mishra et al., 2018)99.07 ± 0.1699.78 ± 0.0997.64 ± 0.3099.36 ± 0.18 45.155.2
Relation Net (Yang et al.,2018)99.6 ± 0.299.8 ± 0.197.6 ± 0.299.1 ± 0.1 50.44 ± 0.8265.32 ± 0.70
Reptile (Nichol and Schulman,2018)97.68 ± 0.0499.48 ± 0.0689.43 ± 0.1497.12 ± 0.32 49.97 ± 0.3265.99 ± 0.58
BMAML (Kim et al., 2018a)53.8 ± 1.46
Amortized VI98.71 ± 0.2290.56 ± 0.5444.13 ± 1.78
Non-Amortized VI97.77 ± 0.55 98.77 ± 0.1899.74 ± 0.0695.28 ± 0.1996.12 ± 0.23 98.84± 0.0955.68 ± 0.91
VERSA (Ours)99.70 ± 0.2099.75 ± 0.1397.66 ± 0.2998.77 ± 0.1853.40 ± 1.82 67.37 ± 0.86
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OmniglotminilmageNet
Method5-way accuracy (%)5-shot20-way accuracy(%) 1-shot5-shot5-way accuracy (%) 1-shot 5-shot
1-shot98.488.197.0
Siamese Nets (Koch et al., 2015)97.398.946.6
Matching Nets (Vinyals et al.,2016)98.193.898.560.0
Neural Statistician (Edwards and Storkey, 2017)98.199.593.298.1
Memory Mod (Kaiser et al.,2017)98.499.695.098.6
Meta LSTM (Ravi and Larochelle, 2017)43.44 ± 0.77 60.60± 0.71
MAML (Finn et al., 2017)98.7 ± 0.499.9 ± 0.195.8 ± 0.398.9± 0.248.7 ± 1.84 63.11 ± 0.92
Prototypical Netsa (Snell et al., 2017)97.499.395.498.7 46.61 ± 0.7865.77 ± 0.70
mAP-SSVM (Triantafillou et al., 2017)98.699.695.298.650.32 ± 0.80 63.94 ± 0.72
mAP-DLM (Triantafillou et al., 2017)98.899.695.498.6 50.28 ± 0.8063.70± 0.70
LLAMA (Grant et al., 2018)49.40 ± 1.83
PLATIPUS (Finn et al., 2018)50.13 ± 1.86
Meta-SGD (Li et al., 2017)99.53± 0.2699.93 ± 0.0995.93 ± 0.3898.97 ± 0.1950.47 ± 1.87 64.03 ± 0.94
SNAIL (Mishra et al., 2018)99.07 ± 0.1699.78 ± 0.0997.64 ± 0.3099.36 ± 0.18 45.155.2
Relation Net (Yang et al.,2018)99.6 ± 0.299.8 ± 0.197.6 ± 0.299.1 ± 0.1 50.44 ± 0.8265.32 ± 0.70
Reptile (Nichol and Schulman,2018)97.68 ± 0.0499.48 ± 0.0689.43 ± 0.1497.12 ± 0.32 49.97 ± 0.3265.99 ± 0.58
BMAML (Kim et al., 2018a)53.8 ± 1.46
Amortized VI98.71 ± 0.2290.56 ± 0.5444.13 ± 1.78
Non-Amortized VI97.77 ± 0.55 98.77 ± 0.1899.74 ± 0.0695.28 ± 0.1996.12 ± 0.23 98.84± 0.0955.68 ± 0.91
VERSA (Ours)99.70 ± 0.2099.75 ± 0.1397.66 ± 0.2998.77 ± 0.1853.40 ± 1.82 67.37 ± 0.86
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Let", + "type": "text" + }, + { + "bbox": [ + 382, + 429, + 389, + 440 + ], + "score": 0.85, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 428, + 505, + 440 + ], + "score": 1.0, + "content": "be a set of shared variational", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 439, + 507, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 189, + 452 + ], + "score": 1.0, + "content": "parameters such that", + "type": "text" + }, + { + "bbox": [ + 190, + 439, + 255, + 452 + ], + "score": 0.92, + "content": "q ( \\tilde { y } ) = q _ { \\phi } \\overline { { ( \\tilde { y } | D ) } }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 439, + 271, + 452 + ], + "score": 1.0, + "content": "(or", + "type": "text" + }, + { + "bbox": [ + 271, + 441, + 281, + 452 + ], + "score": 0.85, + "content": "q _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 439, + 507, + 452 + ], + "score": 1.0, + "content": "for shorthand). 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Note that Eq. 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Critically, this", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 574 + ], + "score": 1.0, + "content": "does not require computation of the true predictive distribution. 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Exploration of alternative proper", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 703 + ], + "score": 1.0, + "content": "scoring rules (Dawid, 2007) and more task-specific losses (Lacoste-Julien et al., 2011) is left for", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 703, + 158, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 703, + 158, + 713 + ], + "score": 1.0, + "content": "future work.", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 104, + 646, + 506, + 713 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 506, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 473, + 95 + ], + "score": 1.0, + "content": "Specification of the approximate predictive distribution. Next, we consider the form of", + "type": "text" + }, + { + "bbox": [ + 473, + 84, + 484, + 95 + ], + "score": 0.84, + "content": "q _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 82, + 506, + 95 + ], + "score": 1.0, + "content": ". Mo-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 491, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 491, + 106 + ], + "score": 1.0, + "content": "tivated by the optimal predictive distribution, we replace the true posterior by an approximation:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 110, + 375, + 136 + ], + "lines": [ + { + "bbox": [ + 235, + 110, + 375, + 136 + ], + "spans": [ + { + "bbox": [ + 235, + 110, + 375, + 136 + ], + "score": 0.95, + "content": "q _ { \\phi } ( \\tilde { y } | D ) = \\int p ( \\tilde { y } | \\psi ) q _ { \\phi } ( \\psi | D ) \\mathrm { d } \\psi .", + "type": "interline_equation", + "image_path": "8e8f447d5b00fb9d4b78abf81dd402ee7405ed4a1ad19a7279713f0aad5632d1.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 235, + 110, + 375, + 136 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "title", + "bbox": [ + 105, + 147, + 461, + 161 + ], + "lines": [ + { + "bbox": [ + 105, + 147, + 462, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 462, + 162 + ], + "score": 1.0, + "content": "B JUST IFICAT ION FOR CONTEXT-INDEPENDENT APPROX IMAT ION", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 505, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "In this section we lay out both theoretical and empirical justifications for the context-independent", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 184, + 254, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 254, + 196 + ], + "score": 1.0, + "content": "approximation detailed in Section 3.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5 + }, + { + "type": "title", + "bbox": [ + 107, + 208, + 393, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 394, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 394, + 222 + ], + "score": 1.0, + "content": "B.1 THEORET ICAL ARGUMENT – DENS ITY RAT IO EST IMAT ION", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 229, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "score": 1.0, + "content": "A principled justification for the approximation is best understood through the lens of density ratio", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "estimation (Mohamed, 2018; Sugiyama et al., 2012). We denote the conditional density of each class", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 251, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 118, + 266 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 252, + 165, + 264 + ], + "score": 0.91, + "content": "p ( x | y = c )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 251, + 345, + 266 + ], + "score": 1.0, + "content": "and assume equal a priori class probability", + "type": "text" + }, + { + "bbox": [ + 345, + 251, + 417, + 264 + ], + "score": 0.93, + "content": "p ( y = c ) = 1 / C", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 251, + 506, + 266 + ], + "score": 1.0, + "content": ". Density ratio theory", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "then uses Bayes’ theorem to show that the optimal softmax classifier can be expressed in terms of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 273, + 360, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 360, + 286 + ], + "score": 1.0, + "content": "conditional densities (Mohamed, 2018; Sugiyama et al., 2012):", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 290, + 459, + 325 + ], + "lines": [ + { + "bbox": [ + 151, + 290, + 459, + 325 + ], + "spans": [ + { + "bbox": [ + 151, + 290, + 459, + 325 + ], + "score": 0.93, + "content": "\\mathrm { S o f t m a x } ( y = c | x ) = \\frac { \\exp ( h ( x ) ^ { \\top } w _ { c } ) } { \\sum _ { c ^ { \\prime } } \\exp ( h ( x ) ^ { \\top } w _ { c ^ { \\prime } } ) } = p ( y = c | x ) = \\frac { p ( x | y = c ) } { \\sum _ { c ^ { \\prime } } p ( x | y = c ^ { \\prime } ) } ,", + "type": "interline_equation", + "image_path": "5b8e09d7f918dd3f35fe34ceb143a013181e41428343782d536977f94397a3d8.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 151, + 290, + 459, + 301.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 151, + 301.6666666666667, + 459, + 313.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 151, + 313.33333333333337, + 459, + 325.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 506, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "This implies that the optimal classifier will construct estimators for the conditional density for each", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 160, + 354 + ], + "score": 1.0, + "content": "class, that is", + "type": "text" + }, + { + "bbox": [ + 160, + 340, + 285, + 353 + ], + "score": 0.9, + "content": "\\mathrm { e x p } ( h ( x ) ^ { \\top } w _ { c } ) \\propto p ( x | y = c )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 341, + 506, + 354 + ], + "score": 1.0, + "content": ". Importantly for our approximation, notice that these", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 353, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 506, + 364 + ], + "score": 1.0, + "content": "estimates are constructed independently for each class, similarly to training a naive Bayes classifier.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 362, + 270, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 270, + 376 + ], + "score": 1.0, + "content": "VERSA mirrors this optimal form using:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 379, + 365, + 395 + ], + "lines": [ + { + "bbox": [ + 246, + 379, + 365, + 395 + ], + "spans": [ + { + "bbox": [ + 246, + 379, + 365, + 395 + ], + "score": 0.92, + "content": "\\log p ( x | y = c ) \\propto h _ { \\theta } ( \\tilde { x } ) ^ { \\top } w _ { c } ,", + "type": "interline_equation", + "image_path": "8c9a6dbce28b777539413e7ee08f9644837c2b159b951c3eabef558ffd3aa44a.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 246, + 379, + 365, + 395 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 133, + 412 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 399, + 239, + 412 + ], + "score": 0.91, + "content": "w _ { c } \\sim q _ { \\phi } \\left( w | \\{ x _ { n } | y _ { n } = c \\} \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 398, + 505, + 412 + ], + "score": 1.0, + "content": ") for each class in a given task. Under ideal conditions (i.e., if one", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 409, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 205, + 424 + ], + "score": 1.0, + "content": "could perfectly estimate", + "type": "text" + }, + { + "bbox": [ + 205, + 411, + 250, + 423 + ], + "score": 0.89, + "content": "p ( \\tilde { x } | y = c )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 409, + 506, + 424 + ], + "score": 1.0, + "content": "), the context-independent assumption holds, further motivating", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 420, + 154, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 154, + 435 + ], + "score": 1.0, + "content": "our design.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 108, + 446, + 253, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 254, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 254, + 459 + ], + "score": 1.0, + "content": "B.2 EMP IR ICAL JUST IFICAT ION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "Here we detail a simple experiment to evaluate the validity of the context-independent inference", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "assumption. The goal of the experiment is to examine if weights may be context-independent without", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "imposing the assumption on the amortization network. To see this, we randomly generate fifty tasks", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 499, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 504, + 513 + ], + "score": 1.0, + "content": "from a dataset, where classes may appear a number of times in different tasks. We then perform", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "free-form (non-amortized) variational inference on the weights for each of the tasks, with a Gaussian", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 522, + 202, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 202, + 534 + ], + "score": 1.0, + "content": "variational distribution:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 536, + 396, + 558 + ], + "lines": [ + { + "bbox": [ + 214, + 536, + 396, + 558 + ], + "spans": [ + { + "bbox": [ + 214, + 536, + 396, + 558 + ], + "score": 0.93, + "content": "\\begin{array} { r } { q _ { \\phi } \\left( W ^ { ( t ) } | \\mathcal { D } ^ { ( t ) } , \\theta \\right) = \\mathcal { N } \\left( W ^ { ( t ) } ; \\mu _ { \\phi } ^ { ( t ) } , \\sigma _ { \\phi } ^ { ( t ) 2 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "eb3287aac41fca226b6ac9c0bc0841a887d0303a81c7bcb1f33e9f6ded0b26b4.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 214, + 536, + 396, + 558 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 568, + 504, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "If the assumption is reasonable, we may expect the distribution of the weights of a specific class to be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 579, + 323, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 323, + 591 + ], + "score": 1.0, + "content": "similar regardless of the additional classes in the task.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 595, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "We examine 5-way classification in the MNIST dataset. We randomly sample and fix fifty such tasks.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "We train the model twice using the same feature extraction network used in the few-shot classification", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 205, + 631 + ], + "score": 1.0, + "content": "experiments, and fix the", + "type": "text" + }, + { + "bbox": [ + 205, + 619, + 216, + 629 + ], + "score": 0.87, + "content": "d _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "to be 16 and 2. We then train the model in an episodic manner by mini-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 434, + 642 + ], + "score": 1.0, + "content": "batching tasks at each iteration. The model is trained to convergence, and achieves", + "type": "text" + }, + { + "bbox": [ + 434, + 629, + 454, + 640 + ], + "score": 0.87, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "accuracy on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 103, + 638, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 103, + 638, + 453, + 659 + ], + "score": 1.0, + "content": "held out test examples for the tasks. After training is complete we examine the optimized", + "type": "text" + }, + { + "bbox": [ + 453, + 640, + 470, + 656 + ], + "score": 0.91, + "content": "\\mu _ { \\phi } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 638, + 505, + 659 + ], + "score": 1.0, + "content": "for each", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "class in each task. Fig. B.1a shows a t-SNE (Maaten and Hinton, 2008) plot for the 16-dimensional", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "weights. We see that when reduced to 2-dimensions, the weights cluster according to class. Fig. B.1b", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "visualizes the weights in their original space. In this plot, weights from the same class are grouped", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "together, and clear similarity patterns are evident across the image, showing that weights from the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 698, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 507, + 712 + ], + "score": 1.0, + "content": "same class have similar means across tasks. Fig. B.2 details the task weights in 2-dimensional space.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "Here, each pentagon represents the weight means learned for one training task, where the nodes of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 734 + ], + "score": 1.0, + "content": "the pentagon are colored according to the class the weights represent. In Fig. B.2a we see that overall,", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38.5 + } + ], + "page_idx": 15, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 26, + 293, + 38 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "16", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 506, + 106 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 473, + 95 + ], + "score": 1.0, + "content": "Specification of the approximate predictive distribution. Next, we consider the form of", + "type": "text" + }, + { + "bbox": [ + 473, + 84, + 484, + 95 + ], + "score": 0.84, + "content": "q _ { \\phi }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 82, + 506, + 95 + ], + "score": 1.0, + "content": ". Mo-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 491, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 491, + 106 + ], + "score": 1.0, + "content": "tivated by the optimal predictive distribution, we replace the true posterior by an approximation:", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 82, + 506, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 110, + 375, + 136 + ], + "lines": [ + { + "bbox": [ + 235, + 110, + 375, + 136 + ], + "spans": [ + { + "bbox": [ + 235, + 110, + 375, + 136 + ], + "score": 0.95, + "content": "q _ { \\phi } ( \\tilde { y } | D ) = \\int p ( \\tilde { y } | \\psi ) q _ { \\phi } ( \\psi | D ) \\mathrm { d } \\psi .", + "type": "interline_equation", + "image_path": "8e8f447d5b00fb9d4b78abf81dd402ee7405ed4a1ad19a7279713f0aad5632d1.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 235, + 110, + 375, + 136 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "title", + "bbox": [ + 105, + 147, + 461, + 161 + ], + "lines": [ + { + "bbox": [ + 105, + 147, + 462, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 147, + 462, + 162 + ], + "score": 1.0, + "content": "B JUST IFICAT ION FOR CONTEXT-INDEPENDENT APPROX IMAT ION", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 106, + 172, + 505, + 196 + ], + "lines": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 105, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "In this section we lay out both theoretical and empirical justifications for the context-independent", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 184, + 254, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 184, + 254, + 196 + ], + "score": 1.0, + "content": "approximation detailed in Section 3.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 173, + 505, + 196 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 208, + 393, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 208, + 394, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 394, + 222 + ], + "score": 1.0, + "content": "B.1 THEORET ICAL ARGUMENT – DENS ITY RAT IO EST IMAT ION", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 229, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 505, + 242 + ], + "score": 1.0, + "content": "A principled justification for the approximation is best understood through the lens of density ratio", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 240, + 505, + 252 + ], + "score": 1.0, + "content": "estimation (Mohamed, 2018; Sugiyama et al., 2012). We denote the conditional density of each class", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 251, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 118, + 266 + ], + "score": 1.0, + "content": "as", + "type": "text" + }, + { + "bbox": [ + 118, + 252, + 165, + 264 + ], + "score": 0.91, + "content": "p ( x | y = c )", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 251, + 345, + 266 + ], + "score": 1.0, + "content": "and assume equal a priori class probability", + "type": "text" + }, + { + "bbox": [ + 345, + 251, + 417, + 264 + ], + "score": 0.93, + "content": "p ( y = c ) = 1 / C", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 251, + 506, + 266 + ], + "score": 1.0, + "content": ". Density ratio theory", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "then uses Bayes’ theorem to show that the optimal softmax classifier can be expressed in terms of the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 273, + 360, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 360, + 286 + ], + "score": 1.0, + "content": "conditional densities (Mohamed, 2018; Sugiyama et al., 2012):", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 230, + 506, + 286 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 290, + 459, + 325 + ], + "lines": [ + { + "bbox": [ + 151, + 290, + 459, + 325 + ], + "spans": [ + { + "bbox": [ + 151, + 290, + 459, + 325 + ], + "score": 0.93, + "content": "\\mathrm { S o f t m a x } ( y = c | x ) = \\frac { \\exp ( h ( x ) ^ { \\top } w _ { c } ) } { \\sum _ { c ^ { \\prime } } \\exp ( h ( x ) ^ { \\top } w _ { c ^ { \\prime } } ) } = p ( y = c | x ) = \\frac { p ( x | y = c ) } { \\sum _ { c ^ { \\prime } } p ( x | y = c ^ { \\prime } ) } ,", + "type": "interline_equation", + "image_path": "5b8e09d7f918dd3f35fe34ceb143a013181e41428343782d536977f94397a3d8.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 151, + 290, + 459, + 301.6666666666667 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 151, + 301.6666666666667, + 459, + 313.33333333333337 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 151, + 313.33333333333337, + 459, + 325.00000000000006 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 330, + 506, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "This implies that the optimal classifier will construct estimators for the conditional density for each", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 341, + 160, + 354 + ], + "score": 1.0, + "content": "class, that is", + "type": "text" + }, + { + "bbox": [ + 160, + 340, + 285, + 353 + ], + "score": 0.9, + "content": "\\mathrm { e x p } ( h ( x ) ^ { \\top } w _ { c } ) \\propto p ( x | y = c )", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 341, + 506, + 354 + ], + "score": 1.0, + "content": ". Importantly for our approximation, notice that these", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 353, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 506, + 364 + ], + "score": 1.0, + "content": "estimates are constructed independently for each class, similarly to training a naive Bayes classifier.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 362, + 270, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 270, + 376 + ], + "score": 1.0, + "content": "VERSA mirrors this optimal form using:", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 330, + 506, + 376 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 246, + 379, + 365, + 395 + ], + "lines": [ + { + "bbox": [ + 246, + 379, + 365, + 395 + ], + "spans": [ + { + "bbox": [ + 246, + 379, + 365, + 395 + ], + "score": 0.92, + "content": "\\log p ( x | y = c ) \\propto h _ { \\theta } ( \\tilde { x } ) ^ { \\top } w _ { c } ,", + "type": "interline_equation", + "image_path": "8c9a6dbce28b777539413e7ee08f9644837c2b159b951c3eabef558ffd3aa44a.jpg" + } + ] + } + ], + "index": 19, + "virtual_lines": [ + { + "bbox": [ + 246, + 379, + 365, + 395 + ], + "spans": [], + "index": 19 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 399, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 133, + 412 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 399, + 239, + 412 + ], + "score": 0.91, + "content": "w _ { c } \\sim q _ { \\phi } \\left( w | \\{ x _ { n } | y _ { n } = c \\} \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 398, + 505, + 412 + ], + "score": 1.0, + "content": ") for each class in a given task. Under ideal conditions (i.e., if one", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 409, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 205, + 424 + ], + "score": 1.0, + "content": "could perfectly estimate", + "type": "text" + }, + { + "bbox": [ + 205, + 411, + 250, + 423 + ], + "score": 0.89, + "content": "p ( \\tilde { x } | y = c )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 409, + 506, + 424 + ], + "score": 1.0, + "content": "), the context-independent assumption holds, further motivating", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 420, + 154, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 154, + 435 + ], + "score": 1.0, + "content": "our design.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 398, + 506, + 435 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 446, + 253, + 458 + ], + "lines": [ + { + "bbox": [ + 105, + 445, + 254, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 254, + 459 + ], + "score": 1.0, + "content": "B.2 EMP IR ICAL JUST IFICAT ION", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 466, + 505, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 505, + 480 + ], + "score": 1.0, + "content": "Here we detail a simple experiment to evaluate the validity of the context-independent inference", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 491 + ], + "score": 1.0, + "content": "assumption. The goal of the experiment is to examine if weights may be context-independent without", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "imposing the assumption on the amortization network. To see this, we randomly generate fifty tasks", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 499, + 504, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 499, + 504, + 513 + ], + "score": 1.0, + "content": "from a dataset, where classes may appear a number of times in different tasks. We then perform", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 511, + 505, + 523 + ], + "score": 1.0, + "content": "free-form (non-amortized) variational inference on the weights for each of the tasks, with a Gaussian", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 522, + 202, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 202, + 534 + ], + "score": 1.0, + "content": "variational distribution:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 467, + 505, + 534 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 214, + 536, + 396, + 558 + ], + "lines": [ + { + "bbox": [ + 214, + 536, + 396, + 558 + ], + "spans": [ + { + "bbox": [ + 214, + 536, + 396, + 558 + ], + "score": 0.93, + "content": "\\begin{array} { r } { q _ { \\phi } \\left( W ^ { ( t ) } | \\mathcal { D } ^ { ( t ) } , \\theta \\right) = \\mathcal { N } \\left( W ^ { ( t ) } ; \\mu _ { \\phi } ^ { ( t ) } , \\sigma _ { \\phi } ^ { ( t ) 2 } \\right) . } \\end{array}", + "type": "interline_equation", + "image_path": "eb3287aac41fca226b6ac9c0bc0841a887d0303a81c7bcb1f33e9f6ded0b26b4.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 214, + 536, + 396, + 558 + ], + "spans": [], + "index": 30 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 568, + 504, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 581 + ], + "score": 1.0, + "content": "If the assumption is reasonable, we may expect the distribution of the weights of a specific class to be", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 579, + 323, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 323, + 591 + ], + "score": 1.0, + "content": "similar regardless of the additional classes in the task.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 567, + 505, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 595, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 506, + 609 + ], + "score": 1.0, + "content": "We examine 5-way classification in the MNIST dataset. We randomly sample and fix fifty such tasks.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 619 + ], + "score": 1.0, + "content": "We train the model twice using the same feature extraction network used in the few-shot classification", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 618, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 205, + 631 + ], + "score": 1.0, + "content": "experiments, and fix the", + "type": "text" + }, + { + "bbox": [ + 205, + 619, + 216, + 629 + ], + "score": 0.87, + "content": "d _ { \\theta }", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 618, + 506, + 631 + ], + "score": 1.0, + "content": "to be 16 and 2. We then train the model in an episodic manner by mini-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 434, + 642 + ], + "score": 1.0, + "content": "batching tasks at each iteration. The model is trained to convergence, and achieves", + "type": "text" + }, + { + "bbox": [ + 434, + 629, + 454, + 640 + ], + "score": 0.87, + "content": "9 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "accuracy on", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 103, + 638, + 505, + 659 + ], + "spans": [ + { + "bbox": [ + 103, + 638, + 453, + 659 + ], + "score": 1.0, + "content": "held out test examples for the tasks. After training is complete we examine the optimized", + "type": "text" + }, + { + "bbox": [ + 453, + 640, + 470, + 656 + ], + "score": 0.91, + "content": "\\mu _ { \\phi } ^ { ( t ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 638, + 505, + 659 + ], + "score": 1.0, + "content": "for each", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "class in each task. Fig. B.1a shows a t-SNE (Maaten and Hinton, 2008) plot for the 16-dimensional", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 677 + ], + "score": 1.0, + "content": "weights. We see that when reduced to 2-dimensions, the weights cluster according to class. Fig. B.1b", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "visualizes the weights in their original space. In this plot, weights from the same class are grouped", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "together, and clear similarity patterns are evident across the image, showing that weights from the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 104, + 698, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 698, + 507, + 712 + ], + "score": 1.0, + "content": "same class have similar means across tasks. Fig. B.2 details the task weights in 2-dimensional space.", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "Here, each pentagon represents the weight means learned for one training task, where the nodes of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 506, + 734 + ], + "score": 1.0, + "content": "the pentagon are colored according to the class the weights represent. In Fig. B.2a we see that overall,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 506, + 95 + ], + "score": 1.0, + "content": "the classes cluster in 2-dimensional space as well. 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Here we can see that for these", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 356, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 154, + 127 + ], + "score": 1.0, + "content": "tasks, class", + "type": "text" + }, + { + "bbox": [ + 154, + 116, + 166, + 126 + ], + "score": 0.59, + "content": "\\bullet _ { 2 } \\cdot", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 115, + 356, + 127 + ], + "score": 1.0, + "content": "weights are all located away from their cluster.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 507, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 507, + 145 + ], + "score": 1.0, + "content": "This implies that each class-weights are typically well-approximated as being independent of the task.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "However, if the model lacks capacity to properly assign each set of class weights to different regions", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "of space, for tasks where classes from similar regions of space appear, the inference procedure will", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 164, + 370, + 178 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 370, + 178 + ], + "score": 1.0, + "content": "‘move’ one of the class weights to an ‘empty’ region of the space.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 108, + 192, + 433, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 436, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 436, + 208 + ], + "score": 1.0, + "content": "C VAR IAT IONAL INFERENCE DER IVAT IONS FOR THE MODEL", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 506, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "score": 1.0, + "content": "We derive a VI-based objective for our probabilistic model. 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However, the derivation of the objective function does", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 310, + 277 + ], + "score": 1.0, + "content": "not change between these options. For a single task", + "type": "text" + }, + { + "bbox": [ + 311, + 266, + 315, + 275 + ], + "score": 0.77, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 264, + 506, + 277 + ], + "score": 1.0, + "content": ", an evidence lower bound (ELBO; (Wainwright", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 275, + 271, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 271, + 289 + ], + "score": 1.0, + "content": "and Jordan, 2008)) may be expressed as:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11.5 + }, + { + "type": "interline_equation", + "bbox": [ + 133, + 290, + 456, + 330 + ], + "lines": [ + { + "bbox": [ + 133, + 290, + 456, + 330 + ], + "spans": [ + { + "bbox": [ + 133, + 290, + 456, + 330 + ], + "score": 0.95, + "content": "\\mathcal { L } _ { t } = \\mathbb { E } _ { q _ { \\phi } ( \\psi | D ^ { ( t ) } , \\theta ) } \\left[ \\sum _ { ( \\boldsymbol { x } , \\boldsymbol { y } ) \\in D ^ { ( t ) } } \\log p ( \\boldsymbol { y } | \\boldsymbol { x } , \\psi , \\theta ) \\right] - \\mathrm { K L } \\left[ q _ { \\phi } ( \\psi | D ^ { ( t ) } , \\theta ) \\| p ( \\psi | \\theta ) \\right] .", + "type": "interline_equation", + "image_path": "e0c2f941288a7907937b9c542adfb439ef5b1d22645014837f2c1b33cd3524e8.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 133, + 290, + 456, + 303.3333333333333 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 133, + 303.3333333333333, + 456, + 316.66666666666663 + ], + "spans": [], + "index": 16 + }, + { + "bbox": [ + 133, + 316.66666666666663, + 456, + 329.99999999999994 + ], + "spans": [], + "index": 17 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 334, + 504, + 369 + ], + "lines": [ + { + "bbox": [ + 104, + 330, + 506, + 348 + ], + "spans": [ + { + "bbox": [ + 104, + 330, + 428, + 348 + ], + "score": 1.0, + "content": "We can then derive a stochastic estimator to optimize Eq. 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(C.2) differs from our objective function in Eq. (4) in two important ways: (i) Eq. (4) does not", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 193, + 439 + ], + "score": 1.0, + "content": "contain a KL term for", + "type": "text" + }, + { + "bbox": [ + 193, + 425, + 249, + 439 + ], + "score": 0.93, + "content": "{ q _ { \\phi } } ( \\psi | D ^ { ( t ) } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 425, + 425, + 439 + ], + "score": 1.0, + "content": "(nor any other form of prior distribution over", + "type": "text" + }, + { + "bbox": [ + 426, + 427, + 433, + 438 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 425, + 506, + 439 + ], + "score": 1.0, + "content": ", and (ii) Eq. (C.1)", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "does not distinguish between training and test data within a task, and therefore does not explicitly", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 448, + 294, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 294, + 462 + ], + "score": 1.0, + "content": "encourage the model to generalize in any way.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 25.5 + }, + { + "type": "title", + "bbox": [ + 107, + 475, + 281, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 282, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 282, + 489 + ], + "score": 1.0, + "content": "D EXPER IMENTAT ION DETA ILS", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 500, + 473, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 498, + 475, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 475, + 514 + ], + "score": 1.0, + "content": "In this section we provide comprehensive details on the few-shot classification experiments.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 108, + 524, + 415, + 536 + ], + "lines": [ + { + "bbox": [ + 106, + 523, + 416, + 537 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 416, + 537 + ], + "score": 1.0, + "content": "D.1 OMN IGLOT FEW-SHOT CLASS IFICAT ION TRA IN ING PROCEDURE", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 545, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 506, + 558 + ], + "score": 1.0, + "content": "Omniglot (Lake et al., 2011) is a few-shot learning dataset consisting of 1623 handwritten characters", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "(each with 20 instances) derived from 50 alphabets. We follow a pre-processing and training procedure", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 407, + 580 + ], + "score": 1.0, + "content": "akin to that defined in (Vinyals et al., 2016). First the images are resized to", + "type": "text" + }, + { + "bbox": [ + 407, + 567, + 441, + 578 + ], + "score": 0.9, + "content": "2 8 \\times 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 566, + 506, + 580 + ], + "score": 1.0, + "content": "pixels and then", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "score": 1.0, + "content": "character classes are augmented with rotations of 90 degrees. The training, validation and test sets", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "consist of a random split of 1100, 100, and 423 characters, respectively. When augmented this results", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 475, + 613 + ], + "score": 1.0, + "content": "in 4400 training, 400 validation, and 1292 test classes, each having 20 character instances. For", + "type": "text" + }, + { + "bbox": [ + 475, + 601, + 484, + 610 + ], + "score": 0.8, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "-way,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 117, + 622 + ], + "score": 0.87, + "content": "k _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "-shot classification, training proceeds in an episodic manner. Each training iteration consists of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 621, + 507, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 621, + 284, + 635 + ], + "score": 1.0, + "content": "a batch of one or more tasks. For each task", + "type": "text" + }, + { + "bbox": [ + 284, + 622, + 293, + 632 + ], + "score": 0.78, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 621, + 507, + 635 + ], + "score": 1.0, + "content": "classes are selected at random from the training set.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 171, + 646 + ], + "score": 1.0, + "content": "During training,", + "type": "text" + }, + { + "bbox": [ + 172, + 633, + 182, + 644 + ], + "score": 0.87, + "content": "k _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "character instances are used as training inputs and 15 character instances are used", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "as test inputs. The validation set is used to monitor the progress of learning and to select the best", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "model to test, but does not affect the training process. Final evaluation of the trained model is done on", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 369, + 678 + ], + "score": 1.0, + "content": "600 randomly selected tasks from the test set. During evaluation,", + "type": "text" + }, + { + "bbox": [ + 369, + 666, + 379, + 677 + ], + "score": 0.88, + "content": "k _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 379, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "character instances are used as", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 677, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 183, + 689 + ], + "score": 1.0, + "content": "training inputs and", + "type": "text" + }, + { + "bbox": [ + 183, + 677, + 194, + 688 + ], + "score": 0.87, + "content": "k _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 194, + 677, + 506, + 689 + ], + "score": 1.0, + "content": "character instances are used as test inputs. We use the Adam (Kingma and Ba,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 507, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 507, + 700 + ], + "score": 1.0, + "content": "2015) optimizer with a constant learning rate of 0.0001 with 16 tasks per batch to train all models.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 697, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 507, + 712 + ], + "score": 1.0, + "content": "The 5-way - 5-shot and 5-way - 1-shot models are trained for 80,000 iterations while the 20-way -", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "5-shot model is trained for 60,000 iterations, and the 20-way - 1-shot model is trained for 100,000", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 494, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 309, + 732 + ], + "score": 1.0, + "content": "iterations. In addition, we use a Gaussian form for", + "type": "text" + }, + { + "bbox": [ + 309, + 723, + 315, + 732 + ], + "score": 0.78, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 721, + 405, + 732 + ], + "score": 1.0, + "content": "and set the number of", + "type": "text" + }, + { + "bbox": [ + 405, + 721, + 413, + 732 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 721, + 459, + 732 + ], + "score": 1.0, + "content": "samples to", + "type": "text" + }, + { + "bbox": [ + 459, + 721, + 490, + 731 + ], + "score": 0.88, + "content": "L = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 721, + 494, + 732 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 39 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 127 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 82, + 506, + 127 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 507, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 507, + 145 + ], + "score": 1.0, + "content": "This implies that each class-weights are typically well-approximated as being independent of the task.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "However, if the model lacks capacity to properly assign each set of class weights to different regions", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 168 + ], + "score": 1.0, + "content": "of space, for tasks where classes from similar regions of space appear, the inference procedure will", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 164, + 370, + 178 + ], + "spans": [ + { + "bbox": [ + 104, + 164, + 370, + 178 + ], + "score": 1.0, + "content": "‘move’ one of the class weights to an ‘empty’ region of the space.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5.5, + "bbox_fs": [ + 104, + 131, + 507, + 178 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 192, + 433, + 205 + ], + "lines": [ + { + "bbox": [ + 105, + 191, + 436, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 436, + 208 + ], + "score": 1.0, + "content": "C VAR IAT IONAL INFERENCE DER IVAT IONS FOR THE MODEL", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 506, + 287 + ], + "lines": [ + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "spans": [ + { + "bbox": [ + 105, + 216, + 506, + 230 + ], + "score": 1.0, + "content": "We derive a VI-based objective for our probabilistic model. 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However, the derivation of the objective function does", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 310, + 277 + ], + "score": 1.0, + "content": "not change between these options. 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(C.2) differs from our objective function in Eq. (4) in two important ways: (i) Eq. (4) does not", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 193, + 439 + ], + "score": 1.0, + "content": "contain a KL term for", + "type": "text" + }, + { + "bbox": [ + 193, + 425, + 249, + 439 + ], + "score": 0.93, + "content": "{ q _ { \\phi } } ( \\psi | D ^ { ( t ) } , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 425, + 425, + 439 + ], + "score": 1.0, + "content": "(nor any other form of prior distribution over", + "type": "text" + }, + { + "bbox": [ + 426, + 427, + 433, + 438 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 425, + 506, + 439 + ], + "score": 1.0, + "content": ", and (ii) Eq. 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We follow a pre-processing and training procedure", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 566, + 506, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 407, + 580 + ], + "score": 1.0, + "content": "akin to that defined in (Vinyals et al., 2016). First the images are resized to", + "type": "text" + }, + { + "bbox": [ + 407, + 567, + 441, + 578 + ], + "score": 0.9, + "content": "2 8 \\times 2 8", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 566, + 506, + 580 + ], + "score": 1.0, + "content": "pixels and then", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 579, + 505, + 590 + ], + "score": 1.0, + "content": "character classes are augmented with rotations of 90 degrees. The training, validation and test sets", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 602 + ], + "score": 1.0, + "content": "consist of a random split of 1100, 100, and 423 characters, respectively. When augmented this results", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 475, + 613 + ], + "score": 1.0, + "content": "in 4400 training, 400 validation, and 1292 test classes, each having 20 character instances. For", + "type": "text" + }, + { + "bbox": [ + 475, + 601, + 484, + 610 + ], + "score": 0.8, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "-way,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 117, + 622 + ], + "score": 0.87, + "content": "k _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "-shot classification, training proceeds in an episodic manner. Each training iteration consists of", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 104, + 621, + 507, + 635 + ], + "spans": [ + { + "bbox": [ + 104, + 621, + 284, + 635 + ], + "score": 1.0, + "content": "a batch of one or more tasks. For each task", + "type": "text" + }, + { + "bbox": [ + 284, + 622, + 293, + 632 + ], + "score": 0.78, + "content": "C", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 621, + 507, + 635 + ], + "score": 1.0, + "content": "classes are selected at random from the training set.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 633, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 171, + 646 + ], + "score": 1.0, + "content": "During training,", + "type": "text" + }, + { + "bbox": [ + 172, + 633, + 182, + 644 + ], + "score": 0.87, + "content": "k _ { c }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 633, + 506, + 646 + ], + "score": 1.0, + "content": "character instances are used as training inputs and 15 character instances are used", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 506, + 657 + ], + "score": 1.0, + "content": "as test inputs. 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We use the Adam (Kingma and Ba,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 687, + 507, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 507, + 700 + ], + "score": 1.0, + "content": "2015) optimizer with a constant learning rate of 0.0001 with 16 tasks per batch to train all models.", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 104, + 697, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 697, + 507, + 712 + ], + "score": 1.0, + "content": "The 5-way - 5-shot and 5-way - 1-shot models are trained for 80,000 iterations while the 20-way -", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "5-shot model is trained for 60,000 iterations, and the 20-way - 1-shot model is trained for 100,000", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 721, + 494, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 309, + 732 + ], + "score": 1.0, + "content": "iterations. 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The images have dimensions of", + "type": "text" + }, + { + "bbox": [ + 361, + 114, + 394, + 125 + ], + "score": 0.9, + "content": "8 4 \\times 8 4", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 114, + 507, + 127 + ], + "score": 1.0, + "content": "pixels. For our experiments,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 507, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 138 + ], + "score": 1.0, + "content": "we use the 64 training, 16 validation, and 20 test class splits defined by (Ravi and Larochelle, 2017).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 507, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 507, + 148 + ], + "score": 1.0, + "content": "Training proceeds in the same episodic manner as with Omniglot. We use the Adam (Kingma and Ba,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 146, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 267, + 160 + ], + "score": 1.0, + "content": "2015) optimizer and a Gaussian form for", + "type": "text" + }, + { + "bbox": [ + 268, + 149, + 274, + 159 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 146, + 361, + 160 + ], + "score": 1.0, + "content": "and set the number of", + "type": "text" + }, + { + "bbox": [ + 361, + 148, + 369, + 159 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 146, + 414, + 160 + ], + "score": 1.0, + "content": "samples to", + "type": "text" + }, + { + "bbox": [ + 414, + 147, + 445, + 158 + ], + "score": 0.9, + "content": "L = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 146, + 506, + 160 + ], + "score": 1.0, + "content": ". 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The feature extraction network is very similar to that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "used in (Vinyals et al., 2016). The output of the amortization network yields mean-field Gaussian", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 357, + 272 + ], + "score": 1.0, + "content": "parameters for the weight distributions of the linear classifier", + "type": "text" + }, + { + "bbox": [ + 358, + 260, + 366, + 271 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 259, + 506, + 272 + ], + "score": 1.0, + "content": ". 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The images have dimensions of", + "type": "text" + }, + { + "bbox": [ + 361, + 114, + 394, + 125 + ], + "score": 0.9, + "content": "8 4 \\times 8 4", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 114, + 507, + 127 + ], + "score": 1.0, + "content": "pixels. For our experiments,", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 126, + 507, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 507, + 138 + ], + "score": 1.0, + "content": "we use the 64 training, 16 validation, and 20 test class splits defined by (Ravi and Larochelle, 2017).", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 136, + 507, + 148 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 507, + 148 + ], + "score": 1.0, + "content": "Training proceeds in the same episodic manner as with Omniglot. We use the Adam (Kingma and Ba,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 146, + 506, + 160 + ], + "spans": [ + { + "bbox": [ + 104, + 146, + 267, + 160 + ], + "score": 1.0, + "content": "2015) optimizer and a Gaussian form for", + "type": "text" + }, + { + "bbox": [ + 268, + 149, + 274, + 159 + ], + "score": 0.8, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 146, + 361, + 160 + ], + "score": 1.0, + "content": "and set the number of", + "type": "text" + }, + { + "bbox": [ + 361, + 148, + 369, + 159 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 146, + 414, + 160 + ], + "score": 1.0, + "content": "samples to", + "type": "text" + }, + { + "bbox": [ + 414, + 147, + 445, + 158 + ], + "score": 0.9, + "content": "L = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 146, + 506, + 160 + ], + "score": 1.0, + "content": ". For the 5-way", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 157, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 104, + 157, + 505, + 171 + ], + "score": 1.0, + "content": "- 5-shot model, we train using 4 tasks per batch for 100,000 iterations and use a constant learning rate", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 506, + 182 + ], + "score": 1.0, + "content": "of 0.0001. 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The feature extraction network is very similar to that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 505, + 261 + ], + "score": 1.0, + "content": "used in (Vinyals et al., 2016). The output of the amortization network yields mean-field Gaussian", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 357, + 272 + ], + "score": 1.0, + "content": "parameters for the weight distributions of the linear classifier", + "type": "text" + }, + { + "bbox": [ + 358, + 260, + 366, + 271 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 259, + 506, + 272 + ], + "score": 1.0, + "content": ". 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We then convert the", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 334, + 357 + ], + "score": 1.0, + "content": "rendered images to gray-scale and reduce their size to be", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 335, + 343, + 368, + 354 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 369, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "pixels. Again, we train our model", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "in an episodic manner. Each training iteration consists a batch of one or more tasks. For each task an", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "score": 1.0, + "content": "object is selected at random from the training set. We train on a single view selected at random from", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 120, + 388 + ], + "score": 1.0, + "content": "the", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 120, + 376, + 153, + 386 + ], + "score": 0.9, + "content": "V = 3 6", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 154, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "views associated with each object and use the remaining 35 views to evaluate the objective", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "function. We then generate 36 views of the object with a modified version of our amortization network", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "score": 1.0, + "content": "which is shown diagrammatically in Fig. 3. To evaluate the system, we generate views and compute", + "type": "text", + "cross_page": true + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "quantitative metrics over the entire test set. Tables E.2 to E.4 describe the network architectures for", + "type": "text", + "cross_page": true + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "the encoder, amortization, and generator networks, respectively. To train, we use the Adam (Kingma", + "type": "text", + "cross_page": true + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "and Ba, 2015) optimizer with a constant learning rate of 0.0001 with 24 tasks per batch for 500,000", + "type": "text", + "cross_page": true + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 442, + 469, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 258, + 455 + ], + "score": 1.0, + "content": "training iterations. In addition, we set", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 258, + 442, + 298, + 454 + ], + "score": 0.91, + "content": "d _ { \\phi } = 2 5 6", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 298, + 442, + 302, + 455 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 302, + 442, + 343, + 454 + ], + "score": 0.91, + "content": "d _ { \\psi } = 2 5 6", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 343, + 442, + 404, + 455 + ], + "score": 1.0, + "content": "and number of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 405, + 442, + 412, + 453 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 413, + 442, + 469, + 455 + ], + "score": 1.0, + "content": "samples to 1.", + "type": "text", + "cross_page": true + } + ], + "index": 22 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 698, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 144, + 99, + 463, + 181 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 142, + 80, + 468, + 91 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 141, + 78, + 469, + 93 + ], + "spans": [ + { + "bbox": [ + 141, + 78, + 469, + 93 + ], + "score": 1.0, + "content": "Table D.3: Amortization network used for Omniglot and miniImageNet few-shot learning.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 144, + 99, + 463, + 181 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 99, + 463, + 181 + ], + "spans": [ + { + "bbox": [ + 144, + 99, + 463, + 181 + ], + "score": 0.977, + "html": "
PhaseOutput size Layers
feature extraction instance pooling weight distributionk×256shared feature network (0)
256mean
2562 × fully connected, ELU+
linear fully connected to μw(c), ² .2 w(c)
", + "type": "table", + "image_path": "9c3170d174774355a8ae09fd448f618557abdc96f8bfedd2e0fb50e68c6e551b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 144, + 99, + 463, + 126.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 144, + 126.33333333333333, + 463, + 153.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 144, + 153.66666666666666, + 463, + 181.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "table", + "bbox": [ + 205, + 216, + 402, + 274 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 152, + 196, + 457, + 208 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 151, + 194, + 459, + 210 + ], + "spans": [ + { + "bbox": [ + 151, + 194, + 459, + 210 + ], + "score": 1.0, + "content": "Table D.4: Linear classifier used for Omniglot and miniImageNet few-shot learning.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "table_body", + "bbox": [ + 205, + 216, + 402, + 274 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 205, + 216, + 402, + 274 + ], + "spans": [ + { + "bbox": [ + 205, + 216, + 402, + 274 + ], + "score": 0.958, + "html": "
Linear Classifier (ψ): hθ(x) →p(y|𝑥,0,ψt)
Output sizeLayers
256Input features
Cfully connected, softmax
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This concatenated dataset is then", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 310, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 104, + 310, + 230, + 322 + ], + "score": 1.0, + "content": "randomly shuffled and we use", + "type": "text" + }, + { + "bbox": [ + 230, + 310, + 250, + 321 + ], + "score": 0.87, + "content": "70 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 310, + 427, + 322 + ], + "score": 1.0, + "content": "of the objects (25,975 in total) for training,", + "type": "text" + }, + { + "bbox": [ + 428, + 310, + 447, + 320 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 310, + 506, + 322 + ], + "score": 1.0, + "content": "for validation", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 321, + 504, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 187, + 334 + ], + "score": 1.0, + "content": "(3,710 in total) , and", + "type": "text" + }, + { + "bbox": [ + 188, + 321, + 208, + 332 + ], + "score": 0.88, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 321, + 425, + 334 + ], + "score": 1.0, + "content": "(7423 in total) for testing. For each object, we generate", + "type": "text" + }, + { + "bbox": [ + 425, + 321, + 458, + 332 + ], + "score": 0.84, + "content": "V = 3 6", + "type": "inline_equation" + }, + { + "bbox": [ + 458, + 321, + 461, + 334 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 461, + 321, + 504, + 332 + ], + "score": 0.86, + "content": "1 2 8 \\times 1 2 8", + "type": "inline_equation" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 505, + 344 + ], + "score": 1.0, + "content": "pixel image views spaced evenly every 10 degrees in azimuth around the object. We then convert the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 343, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 334, + 357 + ], + "score": 1.0, + "content": "rendered images to gray-scale and reduce their size to be", + "type": "text" + }, + { + "bbox": [ + 335, + 343, + 368, + 354 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 343, + 506, + 357 + ], + "score": 1.0, + "content": "pixels. Again, we train our model", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "in an episodic manner. Each training iteration consists a batch of one or more tasks. For each task an", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 106, + 366, + 505, + 377 + ], + "score": 1.0, + "content": "object is selected at random from the training set. We train on a single view selected at random from", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 120, + 388 + ], + "score": 1.0, + "content": "the", + "type": "text" + }, + { + "bbox": [ + 120, + 376, + 153, + 386 + ], + "score": 0.9, + "content": "V = 3 6", + "type": "inline_equation" + }, + { + "bbox": [ + 154, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "views associated with each object and use the remaining 35 views to evaluate the objective", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "spans": [ + { + "bbox": [ + 106, + 387, + 506, + 399 + ], + "score": 1.0, + "content": "function. We then generate 36 views of the object with a modified version of our amortization network", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 397, + 505, + 411 + ], + "score": 1.0, + "content": "which is shown diagrammatically in Fig. 3. To evaluate the system, we generate views and compute", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 506, + 421 + ], + "score": 1.0, + "content": "quantitative metrics over the entire test set. Tables E.2 to E.4 describe the network architectures for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 506, + 433 + ], + "score": 1.0, + "content": "the encoder, amortization, and generator networks, respectively. To train, we use the Adam (Kingma", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 506, + 443 + ], + "score": 1.0, + "content": "and Ba, 2015) optimizer with a constant learning rate of 0.0001 with 24 tasks per batch for 500,000", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 442, + 469, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 258, + 455 + ], + "score": 1.0, + "content": "training iterations. In addition, we set", + "type": "text" + }, + { + "bbox": [ + 258, + 442, + 298, + 454 + ], + "score": 0.91, + "content": "d _ { \\phi } = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 298, + 442, + 302, + 455 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 302, + 442, + 343, + 454 + ], + "score": 0.91, + "content": "d _ { \\psi } = 2 5 6", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 442, + 404, + 455 + ], + "score": 1.0, + "content": "and number of", + "type": "text" + }, + { + "bbox": [ + 405, + 442, + 412, + 453 + ], + "score": 0.84, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 442, + 469, + 455 + ], + "score": 1.0, + "content": "samples to 1.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 15.5 + }, + { + "type": "table", + "bbox": [ + 209, + 484, + 398, + 639 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 138, + 464, + 471, + 475 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 137, + 463, + 473, + 477 + ], + "spans": [ + { + "bbox": [ + 137, + 463, + 473, + 477 + ], + "score": 1.0, + "content": "Table E.1: List of ShapeNet categories used in the VERSA view reconstruction experiments.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "table_body", + "bbox": [ + 209, + 484, + 398, + 639 + ], + "group_id": 2, + "lines": [ + { + "bbox": [ + 209, + 484, + 398, + 639 + ], + "spans": [ + { + "bbox": [ + 209, + 484, + 398, + 639 + ], + "score": 0.977, + "html": "
Object Categorysysnet IDInstances
airplane026911564045
bench028288841813
cabinet029331121571
car029583433533
phone02992529831
chair030016276778
display032111171093
lamp036366492318
speaker036914591597
sofa042565203173
table043792438436
boat045305661939
", + "type": "table", + "image_path": "c7565ab2841dbbbc629656a1648aabd524e0f2b57998a92a2c26feb5670f5397.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 209, + 484, + 398, + 498.09090909090907 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 209, + 498.09090909090907, + 398, + 512.1818181818181 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 209, + 512.1818181818181, + 398, + 526.2727272727273 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 209, + 526.2727272727273, + 398, + 540.3636363636364 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 209, + 540.3636363636364, + 398, + 554.4545454545455 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 209, + 554.4545454545455, + 398, + 568.5454545454546 + ], + "spans": [], + "index": 29 + }, + { + "bbox": [ + 209, + 568.5454545454546, + 398, + 582.6363636363637 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 209, + 582.6363636363637, + 398, + 596.7272727272729 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 209, + 596.7272727272729, + 398, + 610.818181818182 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 209, + 610.818181818182, + 398, + 624.9090909090911 + ], + "spans": [], + "index": 33 + }, + { + "bbox": [ + 209, + 624.9090909090911, + 398, + 639.0000000000002 + ], + "spans": [], + "index": 34 + } + ] + } + ], + "index": 26.0 + } + ], + "page_idx": 19, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 293, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 298, + 749, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 15, + "width": 15 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 144, + 99, + 463, + 181 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 142, + 80, + 468, + 91 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 141, + 78, + 469, + 93 + ], + "spans": [ + { + "bbox": [ + 141, + 78, + 469, + 93 + ], + "score": 1.0, + "content": "Table D.3: Amortization network used for Omniglot and miniImageNet few-shot learning.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 144, + 99, + 463, + 181 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 144, + 99, + 463, + 181 + ], + "spans": [ + { + "bbox": [ + 144, + 99, + 463, + 181 + ], + "score": 0.977, + "html": "
PhaseOutput size Layers
feature extraction instance pooling weight distributionk×256shared feature network (0)
256mean
2562 × fully connected, ELU+
linear fully connected to μw(c), ² .2 w(c)
", + "type": "table", + "image_path": "9c3170d174774355a8ae09fd448f618557abdc96f8bfedd2e0fb50e68c6e551b.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 144, + 99, + 463, + 126.33333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 144, + 126.33333333333333, + 463, + 153.66666666666666 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 144, + 153.66666666666666, + 463, + 181.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 1.0 + }, + { + "type": "table", + "bbox": [ + 205, + 216, + 402, + 274 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 152, + 196, + 457, + 208 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 151, + 194, + 459, + 210 + ], + "spans": [ + { + "bbox": [ + 151, + 194, + 459, + 210 + ], + "score": 1.0, + "content": "Table D.4: Linear classifier used for Omniglot and miniImageNet few-shot learning.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "table_body", + "bbox": [ + 205, + 216, + 402, + 274 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 205, + 216, + 402, + 274 + ], + "spans": [ + { + "bbox": [ + 205, + 216, + 402, + 274 + ], + "score": 0.958, + "html": "
Linear Classifier (ψ): hθ(x) →p(y|𝑥,0,ψt)
Output sizeLayers
256Input features
Cfully connected, softmax
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Object Categorysysnet IDInstances
airplane026911564045
bench028288841813
cabinet029331121571
car029583433533
phone02992529831
chair030016276778
display032111171093
lamp036366492318
speaker036914591597
sofa042565203173
table043792438436
boat045305661939
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Output size Layers
32 ×32×1Input image
16 ×16×64conv2d (3 × 3, stride 1, SAME,RELU), pool (2 × 2, stride2,VALID)
8×8×64conv2d (3 × 3, stride 1, SAME,RELU),pool (2 × 2, stride 2, VALID)
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PhaseOutput sizeLayers
Φprek×dencoder network ()
concatenate h and Xk ×(dp+dx)concat(h, X)
Φmidkxd2 × 2 fully connected,ELU
instance pooling1×daverage
post1×d2× fully connected, ELU
distributiondfully connected linear layers to μψ,
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Output sizeLayers
d+dxconcat(y,x)
512fully connected, RELU
1024fully connected, RELU
2×2×256reshape
4×4×128deconv2d (3 × 3, stride 2, SAME, RELU)
8×8×64deconv2d (3 × 3, stride 2, SAME,RELU)
16 ×16 × 32deconv2d (3 × 3, stride 2, SAME,RELU)
32×32×1deconv2d (3 × 3, stride 2, SAME,sigmoid)
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Output size Layers
32 ×32×1Input image
16 ×16×64conv2d (3 × 3, stride 1, SAME,RELU), pool (2 × 2, stride2,VALID)
8×8×64conv2d (3 × 3, stride 1, SAME,RELU),pool (2 × 2, stride 2, VALID)
4×4×64conv2d (3 × 3, stride 1, SAME,RELU), pool (2 × 2, stride 2,VALID)
2×2×64conv2d (3 × 3, stride 1, SAME,RELU),pool (2 × 2, stride 2, VALID)
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PhaseOutput sizeLayers
Φprek×dencoder network ()
concatenate h and Xk ×(dp+dx)concat(h, X)
Φmidkxd2 × 2 fully connected,ELU
instance pooling1×daverage
post1×d2× fully connected, ELU
distributiondfully connected linear layers to μψ,
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Output sizeLayers
d+dxconcat(y,x)
512fully connected, RELU
1024fully connected, RELU
2×2×256reshape
4×4×128deconv2d (3 × 3, stride 2, SAME, RELU)
8×8×64deconv2d (3 × 3, stride 2, SAME,RELU)
16 ×16 × 32deconv2d (3 × 3, stride 2, SAME,RELU)
32×32×1deconv2d (3 × 3, stride 2, SAME,sigmoid)
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OmniglotminilmageNet
5-way NLL20-way NLL5-way NLL
Method1-shot5-shot1-shot5-shot1-shot5-shot
Amortized VI0.179 ± 0.0090.137 ± 0.0040.456± 0.0100.253 ± 0.0041.328 ± 0.0241.165 ± 0.010
Non-Amortized VI0.144 ± 0.0050.025 ± 0.0010.393 ± 0.0050.078 ± 0.002
VERSA0.010± 0.0050.007 ± 0.0030.079 ± 0.0090.031 ± 0.0041.183 ± 0.0230.859 ± 0.015
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OmniglotminilmageNet
Method5-way accuracy (%)5-shot20-way accuracy(%) 1-shot5-shot5-way accuracy (%) 1-shot 5-shot
1-shot98.488.197.0
Siamese Nets (Koch et al., 2015)97.398.946.6
Matching Nets (Vinyals et al.,2016)98.193.898.560.0
Neural Statistician (Edwards and Storkey, 2017)98.199.593.298.1
Memory Mod (Kaiser et al.,2017)98.499.695.098.6
Meta LSTM (Ravi and Larochelle, 2017)43.44 ± 0.77 60.60± 0.71
MAML (Finn et al., 2017)98.7 ± 0.499.9 ± 0.195.8 ± 0.398.9± 0.248.7 ± 1.84 63.11 ± 0.92
Prototypical Netsa (Snell et al., 2017)97.499.395.498.7 46.61 ± 0.7865.77 ± 0.70
mAP-SSVM (Triantafillou et al., 2017)98.699.695.298.650.32 ± 0.80 63.94 ± 0.72
mAP-DLM (Triantafillou et al., 2017)98.899.695.498.6 50.28 ± 0.8063.70± 0.70
LLAMA (Grant et al., 2018)49.40 ± 1.83
PLATIPUS (Finn et al., 2018)50.13 ± 1.86
Meta-SGD (Li et al., 2017)99.53± 0.2699.93 ± 0.0995.93 ± 0.3898.97 ± 0.1950.47 ± 1.87 64.03 ± 0.94
SNAIL (Mishra et al., 2018)99.07 ± 0.1699.78 ± 0.0997.64 ± 0.3099.36 ± 0.18 45.155.2
Relation Net (Yang et al.,2018)99.6 ± 0.299.8 ± 0.197.6 ± 0.299.1 ± 0.1 50.44 ± 0.8265.32 ± 0.70
Reptile (Nichol and Schulman,2018)97.68 ± 0.0499.48 ± 0.0689.43 ± 0.1497.12 ± 0.32 49.97 ± 0.3265.99 ± 0.58
BMAML (Kim et al., 2018a)53.8 ± 1.46
Amortized VI98.71 ± 0.2290.56 ± 0.5444.13 ± 1.78
Non-Amortized VI97.77 ± 0.55 98.77 ± 0.1899.74 ± 0.0695.28 ± 0.1996.12 ± 0.23 98.84± 0.0955.68 ± 0.91
VERSA (Ours)99.70 ± 0.2099.75 ± 0.1397.66 ± 0.2998.77 ± 0.1853.40 ± 1.82 67.37 ± 0.86
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Output sizeLayers
28×28×1Input image
14 ×14 × 64conv2d (3 × 3, stride 1, SAME,RELU),dropout, pool (2 × 2, stride 2, SAME)
7×7×64conv2d (3 × 3, stride 1, SAME,RELU),dropout, pool (2 × 2, stride 2, SAME)
4×4×64conv2d (3 × 3, stride 1, SAME,RELU),dropout, pool (2 × 2, stride 2, SAME)
2×2×64conv2d (3 × 3,stride 1, SAME,RELU),dropout,pool (2 × 2, stride 2, SAME)
256flatten
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Output sizeLayers
84×84×1Input image
42 × 42× 64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
21×21×64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
10 ×10×64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
5×5×64conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID)
2×2×64 256conv2d (3 × 3, stride 1, SAME,RELU), dropout, pool (2 × 2, stride 2, VALID) flatten
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Object Categorysysnet IDInstances
airplane026911564045
bench028288841813
cabinet029331121571
car029583433533
phone02992529831
chair030016276778
display032111171093
lamp036366492318
speaker036914591597
sofa042565203173
table043792438436
boat045305661939
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PhaseOutput size Layers
feature extraction instance pooling weight distributionk×256shared feature network (0)
256mean
2562 × fully connected, ELU+
linear fully connected to μw(c), ² .2 w(c)
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Linear Classifier (ψ): hθ(x) →p(y|𝑥,0,ψt)
Output sizeLayers
256Input features
Cfully connected, softmax
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Output size Layers
32 ×32×1Input image
16 ×16×64conv2d (3 × 3, stride 1, SAME,RELU), pool (2 × 2, stride2,VALID)
8×8×64conv2d (3 × 3, stride 1, SAME,RELU),pool (2 × 2, stride 2, VALID)
4×4×64conv2d (3 × 3, stride 1, SAME,RELU), pool (2 × 2, stride 2,VALID)
2×2×64conv2d (3 × 3, stride 1, SAME,RELU),pool (2 × 2, stride 2, VALID)
dfully connected, RELU
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PhaseOutput sizeLayers
Φprek×dencoder network ()
concatenate h and Xk ×(dp+dx)concat(h, X)
Φmidkxd2 × 2 fully connected,ELU
instance pooling1×daverage
post1×d2× fully connected, ELU
distributiondfully connected linear layers to μψ,
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Output sizeLayers
d+dxconcat(y,x)
512fully connected, RELU
1024fully connected, RELU
2×2×256reshape
4×4×128deconv2d (3 × 3, stride 2, SAME, RELU)
8×8×64deconv2d (3 × 3, stride 2, SAME,RELU)
16 ×16 × 32deconv2d (3 × 3, stride 2, SAME,RELU)
32×32×1deconv2d (3 × 3, stride 2, SAME,sigmoid)
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0000000000000000000000000000000000000000..ceae5a790fa0f5bb98c571a2ac1d64f3b125fbca --- /dev/null +++ b/parse/train/HwGNkx1WcIs/HwGNkx1WcIs.md @@ -0,0 +1,363 @@ +# Object DGCNN: 3D Object Detection using Dynamic Graphs + +Yue Wang Massachusetts Institute of Technology yuewang@csail.mit.edu + +Justin Solomon Massachusetts Institute of Technology jsolomon@mit.edu + +# Abstract + +3D object detection often involves complicated training and testing pipelines, which require substantial domain knowledge about individual datasets. Inspired by recent non-maximum suppression-free 2D object detection models, we propose a 3D object detection architecture on point clouds. Our method models 3D object detection as message passing on a dynamic graph, generalizing the DGCNN framework to predict a set of objects. In our construction, we remove the necessity of post-processing via object confidence aggregation or non-maximum suppression. To facilitate object detection from sparse point clouds, we also propose a set-to-set distillation approach customized to 3D detection. This approach aligns the outputs of the teacher model and the student model in a permutation-invariant fashion, significantly simplifying knowledge distillation for the 3D detection task. Our method achieves state-of-the-art performance on autonomous driving benchmarks. We also provide abundant analysis of the detection model and distillation framework. + +Methods for 3D object detection have progressed rapidly, yielding deployable autonomous driving perception systems. Following common practice in 2D vision, 3D object detection often employs complex training and testing pipelines including many post-processing operations to achieve superior performance. These operations are typically non-parallelizable and inefficient even with modern deep learning frameworks, implying a steep trade-off between between efficiency and effectiveness. + +Modern methods usually employ two stages [1, 2], including a region proposal network [3] that can introduce significant training overhead. Subsequent efforts simplify this pipeline for 3D object detection. PointPillars [4] introduces a one-stage anchor-based design, simplfying training. PillarOD [5] and CenterPoint [6] improve the one-stage model by making per-pillar predictions, that is, one prediction per point on the ground plane. They assign ground-truth bounding boxes to multiple outputs while training to ease optimization. However, they predict redundant boxes, which can overlap in the same positions; extra boxes are eliminated a posteriori using non-maximum suppression (NMS). It remains elusive to remove hand-designed components like NMS in training and testing. + +We introduce Object DGCNN, a streamlined architecture for 3D object detection from point clouds. Like DETR for 2D object detection [7], we predict a set of bounding boxes from the raw data, enabling an NMS-free pipeline that achieves real-time performance. A critical new component is to treat each object query as a point in a set whose embedding is learned using DGCNN [8]. Compared to the self-attention module [9] in DETR, DGCNN leverages a sparse set of object relations, which reflects the real object distribution in the scene. In contrast to PointPillars [4], PillarOD [5], and CenterPoint [6], our method does not require post-processing. + +We also provide a knowledge distillation approach customized to 3D object detection. Existing methods typically distill dense feature maps from a teacher model to a student model, whose training objective does not necessarily capture 3D object detection performance [10]. In contrast, we propose set-to-set distillation training that aligns the outputs of the teacher and the student in a permutationinvariant fashion. This process is enabled by the unified Object DGCNN architecture. In addition to obtaining better performance, through this process our model can benefit from privileged information (e.g., dense point clouds) only available at training time. + +Contributions. We summarize our key contributions as follows: + +• We propose a post-processing-free 3D object detection model achieving state-of-the-art performance. To our knowledge, this is the first NMS-free 3D object detector. +• We generalize DGCNN to model objects as a point set. The DGCNN module outperforms its self-attention counterpart thanks to its sparse structure. +• We propose a set-to-set distillation method for 3D object detection. In our construction, knowledge distillation on object detection simply penalizes differences between the outputs of the teacher model and the student model. +• We show our model can use privileged information (such as dense point clouds) that is naturally available at training time to improve the model performance at inference time. +• We release our code to promote reproducibility and future research. 1 + +# 1 Related Work + +2D object detection. Object recognition research has been transitioning from models with handcrafted components to models with limited post-processing. One-stage detectors [11–13] remove the complicated region proposal networks in two-stage objectors [3, 14], yielding more efficient training and testing. Anchor-free methods [15, 16] further simplify the one-stage pipeline by shifting from per-anchor prediction to per-pixel prediction. However, these methods still make dense predictions and rely on NMS to reduce redundancy. To alleviate this issue, DETR [7] formulates object detection as a set-to-set prediction problem. It introduces a set-to-set loss that implicitly penalizes redundant boxes, removing the necessity of post-processing. To accelerate convergence, Deformable DETR [17] proposes deformable self-attention and streamlines the optimization process. Our method also formulates 3D object detection as set prediction, but with a customized design for 3D. + +3D object detection. VoxelNet [18] generalizes one-stage object detection to 3D. It uses 3D dense convolutions to learn representations on voxelized point clouds, which is too inefficient to capture fine-grained features. To address that, PIXOR [19] and PointPillars [4] project points to a birds-eye view (BEV) and operate on 2D feature maps; PointNet [20] aggregates features within each BEV pixel. We use a variant of PointPillars [4] for 3D detection (§3). These methods are efficient but drop information along the vertical axis. To accompany the BEV projection, MVF [21] adds a spherical projection. PillarOd [5] and CenterPoint [6] use pillar-centric object detection, making predictions per BEV pixel (pillar) rather than per anchor. These anchor-free methods simplify 3D object detection while maintaining efficiency. Beyond SSD-style [11] one-stage models, Complex-YOLO [22] extends YOLO to 3D for real-time perception. PointRCNN [23] employs a two-stage architecture for high-quality detection. To improve representations of two-stage models, PVRCNN [2] proposes a point-voxel feature set abstraction layer to leverage the flexible receptive fields of PointNet-based networks. Unlike works on point clouds, LaserNet [24] operates on raw range scans with comparable performance. [25–27] combine point clouds with camera images. Frustum-PointNet [28] leverages 2D object detectors to form a frustum crop of points and then uses PointNet to aggregate features. [29] describes an end-to-end learnable architecture that exploits continuous convolutions to fuse feature maps. VoteNet [30, 31] generalizes Hough voting [32] for 3D object detection in point clouds. DOPS [33] extends VoteNet and predicts 3D object shapes. In addition to visual input, [34] shows that high-definition (HD) maps can boost performance of 3D object detectors. [35] argues that multi-tasking can learn better representations than single-tasking. Beyond supervised learning, [36] learns a perception model for unknown classes. + +DGCNN. DGCNN [8] pioneered learning point cloud representations via dynamic graphs. It models point clouds as connected graphs, which are dynamically built using $k$ -nearest neighbors in the latent space. DGCNN learns per-point features through message passing. However, it operates on point clouds for single object recognition and semantic segmentation. One of our key contributions is to generalize DGCNN to model scene-level object relations for 3D detection. + +Knowledge distillation (KD). KD compresses knowledge from an ensemble of models into a single smaller model [37]. [38] generalizes this idea and combines it with deep learning. KD transfers knowledge from a teacher model to a student model by minimizing a loss, in which the target is the distribution of class probabilities induced by the teacher. [39–50] improve knowledge distillation for classification. Beyond image classification, KD has been extended to improve object detection. [51] leverages FitNets for object detection, addressing obstacles such as class imbalance, loss instability, and feature distribution mismatch. [52] distills between region proposals, accelerating training with added instability. To address this issue, [53] uses fine-grained representation imitation using object masks. [54] uses KD to tackle a continual learning problem. + +Privileged information. [55] introduces the framework of learning with privileged information in the context of support vector machines (SVMs), wherein additional information is accessible during training but not testing. [56] unifies KD and learning using privileged information theoretically. [57] identifies practical applications, e.g., transferring knowledge from localized data to non-localized data, from high resolution to low resolution, from color images to edge images, and from regular images to distorted images. To mediate uncertainty and improve training efficiency, [58] makes the variance of Dropout [59] a function of privileged information. We extend these methods to 3D data, in which privileged information consists of dense point clouds aggregated from LiDAR sequences. + +# 2 Overview + +Our target application of object detection differs from the recognition and segmentation tasks considered for DGCNN. Our point clouds typically contain too many points to apply DGCNN and its peers directly to the entire scene. Moreover, the size of our output set, a small set of bounding boxes, differs from the size of our input set, a huge set of points in $\mathbb { R } ^ { 3 }$ . + +Following state-of-the-art in large-scale object detection, our pipeline learns a grid-based intermediate representation to capture local features (§3). We test two standard learning-based methods for collecting local point cloud features on a birds-eye view (BEV) grid. While in principle it might be possible to avoid grids entirely in our pipeline, this BEV representation is far more efficient and—as observed in previous work—is sufficient to find objects reliably in autonomous driving, where there is likely only one object above any given grid cell on the ground plane. + +Our main architecture contribution is the Object DGCNN pipeline (§4), which transitions from this BEV grid of features to a set of object bounding boxes. Object DGCNN draws inspiration from the DGCNN architecture; its layers alternate between local feature transformations and $k$ - nearest neighbor aggregation to capture relationships between objects. Unlike conventional DGCNN, however, Object DGCNN incorporates features from the BEV grid in each of its layers; each layer incorporates several queries into the BEV to refine object position estimates. The output of Object DGCNN is a set of objects in the scene. We use a permutation-invariant loss (10) to measure divergence from the ground truth set of objects. + +The pipeline above does not require hand-designed post-processing like NMS; our output boxes are usable directly for object detection. Beyond simplifying the object detection pipeline, this allows us to propose object detection-specific distillation procedures $( \ S 6 . 3 )$ that further improve performance. These use one network to train another, e.g., to train a network operating on sparse point clouds to output features that imitate those learned by a network trained on denser, more detailed point clouds. + +# 3 Local Features + +We begin with a point cloud $\mathcal { X } \ = \ \{ \pmb { x } _ { 1 } , \dots , \pmb { x } _ { i } , \dots , \pmb { x } _ { N } \} \ \subset \ \mathbb { R } ^ { 3 }$ with per-point features ${ \mathcal { F } } =$ $\{ \pmb { f } _ { 1 } , \ldots , \pmb { f } _ { i } , \ldots , \pmb { \bar { f } } _ { N } \} \subset \mathbb { R } ^ { K }$ , ground-truth bounding boxes $\boldsymbol { B } = \{ b _ { 1 } , \dots , \bar { b } _ { j } , \dots , b _ { M } \} \subset \mathbb { R } ^ { 9 }$ , and categorical labels $\bar { \mathcal { C } } \overset { \cdot } { = } \{ c _ { j } , \dotsc , c _ { j } , \dotsc , c _ { M } \} \subset \mathbb { Z }$ . Each $b _ { j }$ contains position, size, heading angle, and velocity in the birds-eye view (BEV); our architecture aims to predict these boxes and their labels from the point cloud and its features. + +As an initial step, modern 3D object detection models scatter points into either BEV pillars or 3D voxels and then use convolutional neural networks to extract features on a grid. This strategy accelerates object detection for large point clouds. We test two neural network architectures for BEV feature extraction, detailed below. + +PointPillars [4] maps sparse point clouds onto a dense BEV pillar map on which 2D convolutions can be applied. Suppose $F _ { P } ( i )$ returns the points in pillar $i$ , that is, the set of points in a vertical column above point $i$ on the ground. When collecting features from points to pillars, multiple points can fall into the same pillar. In this case, PointNet [20] (PN) is used to obtain pillar-wise features: + +![](images/4961b1385646350b25971984d745696f6c00c96db1551f123510f35addb612d1.jpg) +Figure 1: Overview. Point cloud features are learned in BEV, followed by $L$ DGCNNs to model object relations. We predict a set of bounding boxes and compute loss in a one-to-one manner. + +$$ +\pmb { f } _ { i } ^ { \mathrm { p i l l a r } } = \operatorname { P N } ( \{ \pmb { f } _ { j } | \pmb { x } _ { j } \in F _ { P } ( \pmb { p } _ { i } ) \} ) , +$$ + +where $f _ { i } ^ { \mathrm { p i l l a r } }$ is the feature of pillar $\mathbf { \nabla } _ { \pmb { p } _ { i } }$ . We set the features for empty pillars to 0. This results in a dense 2D grid $\mathcal { F } ^ { \mathrm { p i l l a r } } \subset \mathbb { R } ^ { H ^ { \mathrm { p } } \times W ^ { \mathrm { p } } \times C ^ { \mathrm { p } } }$ , where $H ^ { \mathrm { p } } , W ^ { \mathrm { p } }$ and $C ^ { \mathrm { p } }$ are the height, width, number of channels of this 2D pillar map, respectively. Multiple stacked convolutional layers further embed the pillar features to the final feature map F d ⊂ RHd×W d×Cd . + +An alternative BEV embedding is SparseConv [60]. If $F _ { V } ( i )$ returns the set of points in voxel $i$ SparseConv collects point-wise features into voxel-wise features by + +$$ +\pmb { f } _ { i } ^ { \mathrm { v o x e l } } = \mathrm { P N } ( \{ \pmb { f } _ { j } | \pmb { x } _ { j } \in F _ { V } ( i ) \} ) , +$$ + +where $f _ { i } ^ { \mathrm { v o x e l } }$ contains the features of voxel $i$ . In contrast to PointPillars, SparseConv conducts 3D sparse convolutions to refine the voxel-wise features. Finally, we compress these sparse voxels to a BEV 2D grid by filling empty voxels with zeros and averaging along the $z$ -axis. For ease of notation, we also denote the resulting 2D grid F d ⊂ RHd×W d×Cd . + +# 4 Object DGCNN + +After obtaining the BEV features ${ \mathcal { F } } ^ { \mathrm { d } }$ using one of the architectures above, we predict a set of bounding boxes as well as a label for each box. The key difference between our architecture and most recent 3D object detection methods is that ours produces a set of bounding boxes rather than a box per grid cell followed by NMS, as in [5, 6]. Hence, we need to transition from a grid of per-pillar features to an unordered set of objects; we detail our approach below. We address two key issues: prediction of the bounding boxes and evaluation of the loss. + +Desiderata. Object DGCNN uses a DGCNN-inspired architecture but incorporates grid-based BEV features, built on the philosophy that local features (§3) are reasonable to store on a dense grid, but object predictions are better modeled using sets. Hence, we require a new architecture and set-to-set loss that encourage bounding box diversity. + +Object DGCNN uses $L$ layers that follow a series of set-based computations to produce bounding box predictions from the BEV feature maps. Each layer employs the following steps (Figure 1): + +1. predict a set of query points and attention weights; +2. collect BEV features from keypoints determined by the queries; and +3. model object-object interactions via DGCNN. + +Each layer results in a more refined set of bounding box predictions, one per query. At the end of these layers, we match the prediction set with the ground-truth set in a one-to-one fashion and evaluate a set-to-set object detection loss. + +Single layer. Inspired by DETR [7], each layer $\ell \in \{ 0 , \ldots , L - 1 \}$ of Object DGCNN operates on a set of object queries $\mathcal { Q } _ { \ell } = \{ \pmb q _ { \ell 1 } , \dots , \pmb q _ { \ell M ^ { * } } \} \subset \mathbb { R } ^ { Q }$ , producing a new set $\mathcal { Q } _ { \ell + 1 }$ . Although queries are fully learnable, our intuition is that they represent progressively refined object positions. + +The initial set of object queries $\mathcal { Q } _ { 0 }$ is learned jointly with the neural network weights, yielding a dataset-specific prior. Beyond this fixed initial set, below we detail how to incorporate scene information to obtain $\mathcal { Q } _ { \ell + 1 }$ from $\mathcal { Q } _ { \ell }$ using an approach inspired by DGCNN [8] and deformable self-attention [17]. For notational convenience, we drop the $\ell$ subscript. + +Starting from each query $\pmb q _ { i }$ (or, without the index dropped, $\mathbf { \Delta } q _ { \ell i } .$ ), we decode a reference point $\pmb { p } _ { i } \in \mathbb { R } ^ { 2 }$ , a set of offsets $\{ \delta _ { i 0 } , \dots , \delta _ { i K } \} \subset \mathbb { R } ^ { 2 }$ , and a set of attention weights $\{ w _ { i 0 } , \dots , w _ { i K } \} \subset \mathbb { R }$ : + +$$ +\begin{array} { c } { p _ { i } = \Phi _ { \mathrm { r e f } } ( { \boldsymbol { q } } _ { i } ) , \qquad \{ \delta _ { i } ^ { 0 } , \ldots , \delta _ { i } ^ { k } , \ldots \delta _ { i } ^ { K } \} = \Phi _ { \mathrm { n e i g h b o r } } ( { \boldsymbol { q } } _ { i } ) , } \\ { \{ w _ { i } ^ { 0 } , \ldots , w _ { i } ^ { k } , \ldots w _ { i } ^ { K } \} = \Phi _ { \mathrm { a t t e n } } ( { \boldsymbol { q } } _ { i } ) , } \end{array} +$$ + +where $\Phi _ { \mathrm { r e f } }$ , $\Phi _ { \mathrm { n e i g h b o r } }$ , and $\Phi _ { \mathrm { a t t e n } }$ are shared neural networks among the queries. We think of $\mathbf { \nabla } _ { \pmb { p } _ { i } }$ as a hypothesis for the center of the $i$ -th object; the $\delta$ ’s represent the positions of $K$ informative points relative to the position of the object that determine its geometry. + +Next, we collect a BEV feature $\mathbf { \boldsymbol { f } } _ { i k }$ associated to each neighbor point $\pmb { p } _ { i k } = \pmb { p } _ { i } + \pmb { \delta } _ { i k }$ + +$$ +\pmb { f } _ { i k } = f _ { \mathrm { b i l i n e a r } } ( \mathcal { F } ^ { d } , \pmb { p } _ { i } + \delta _ { i k } ) , +$$ + +where $f _ { \mathrm { b i l i n e a r } }$ bilinearly interpolates the BEV feature map ${ \mathcal { F } } ^ { d }$ . Note this step is the interaction between our set-based architecture manipulating query points $\pmb q _ { i }$ and the grid-based feature map ${ \mathcal { F } } ^ { d }$ . We then aggregate a single object query feature $\pmb { f } _ { i } ^ { o }$ from the $\pmb { f } _ { i k } \mathbf { s }$ : + +$$ +f _ { i } ^ { o } = \sum _ { k } { \frac { e ^ { w _ { i k } } } { \sum _ { k } { e ^ { w _ { i k } } } } } { f _ { i k } } . +$$ + +This generates scene-aware features; each object query “attends” to a certain area in the scene. + +In the current layer $\ell$ , the queries have not yet interacted with each other. To incorporate neighborhood information in object detection estimates, we use DGCNN-style operations to model a sparse set of relations. We construct a graph between the queries using a nearest neighbor search in feature space. In particular, we connect each query feature $\pmb { f } _ { i } ^ { o }$ to its 16 nearest neighbors as ablated in Table 7. Identically to DGCNN, we learn a feature per edge $e _ { i j }$ and then aggregate back to the vertices $i$ to produce the new set of object queries. In detail, we write: + +$$ +{ \pmb q } _ { ( \ell + 1 ) i } = \operatorname* { m a x } _ { \mathrm { e d g e s } ~ e _ { i j } } \Phi _ { \mathrm { e d g e } } ( { \pmb f } _ { i } ^ { o } , { \pmb f } _ { j } ^ { o } ) , +$$ + +where max denotes a channel-wise maximum and $\Phi _ { \mathrm { e d g e } }$ is a neural network for computing edge features. This completes our layer for computing $\mathcal { Q } _ { \ell + 1 }$ from $\mathcal { Q } _ { \ell }$ . Optionally, we repeat this last step multiple times, in effect applying DGCNN to the features $\pmb { f } _ { i } ^ { o }$ to get the point set $\mathcal { Q } _ { \ell + 1 }$ . + +Set-to-set loss. After $L$ Object DGCNN layers as described above, we are left with a set of $M ^ { * }$ queries $\mathcal { Q } _ { L }$ used to predict our bounding boxes. For each query $\mathbf { \nabla } _ { \mathbf { q } _ { L i } }$ , we use a classification network to predict a categorical label $\hat { c _ { i } }$ and a regression network to predict bounding box parameters $\hat { b _ { i } }$ . Our final task is to assign the predictions to the ground-truth boxes and compute a set-to-set loss. + +Most object detection models minimize a loss $\mathcal { L } _ { \mathrm { o d } }$ given by + +$$ +\mathcal { L } _ { \mathrm { o d } } = \sum _ { j = 1 } ^ { \hat { M } } - \log \hat { p } _ { \hat { \sigma } ( j ) } ( \hat { c } _ { j } ) + 1 _ { \{ c _ { \hat { \sigma } ( j ) } \neq \emptyset \} } \mathcal { L } _ { \mathrm { b o x } } ( \hat { b } _ { j } , b _ { \hat { \sigma } ( j ) } ) , +$$ + +where $\hat { M } = H ^ { \mathrm { d } } * W ^ { \mathrm { d } }$ , $\hat { \sigma } ( * )$ returns the corresponding index of the ground-truth bounding box, $\hat { p } _ { \hat { \sigma } ( j ) } ( c _ { j } )$ is the probability of class $c _ { \hat { \sigma } ( j ) }$ for the prediction with index $\sigma ( j )$ , $\mathcal { D }$ denotes an invalid box, and $\mathcal { L } _ { \mathrm { b o x } }$ is typically the $\mathcal { L } _ { 1 }$ distance. Different matchings $\hat { \sigma }$ yield different optimization landscapes and hence different prediction models. Pillar-OD [5] and CenterPoint [6] employ a simple $\hat { \sigma }$ to determine the ground-truth box used to evaluate the box predicted at BEV pixel $j$ : + +$$ +\begin{array} { r } { \hat { \sigma } _ { \mathrm { o v e r l a p } } ( j ) = \left\{ \begin{array} { l l } { j ^ { ' } , \quad } & { \mathrm { i f } \ b _ { j ^ { ' } } \ \mathrm { o v e r l a p s \ w i t h \ B E V \ p i x e l } \ j ; } \\ { \varnothing , \quad } & { \mathrm { o t h e r w i s e } . } \end{array} \right. } \end{array} +$$ + +This strategy can assign a box to multiple nearby BEV pixels. This one-to-many assignment provides dense supervision for the object detector and eases optimization. Since the training objective encourages each BEV pixel to predict the same surrounding box, however, redundant boxes are inevitable. So, NMS is usually required to remove redundant boxes at inference time. + +![](images/b3a034d74f018210cdbafa53766ce355aeb8c11224c01f00432b81db0fb7d96f.jpg) +Figure 2: The set-to-set distillation pipeline. The student network is trained with the ground-truth supervision as well as with the supervision from a fixed teacher network. + +Rather than performing dense predictions in the BEV, we make per-query predictions. Typically, $M ^ { * }$ is much larger than the number of ground-truth boxes $M$ . To account for this difference, we pad the set of ground-truth boxes with $\mathcal { O } \mathbf { s }$ (no object) up to $M ^ { * }$ . Following [7], we use an objective built on an optimal matching between these two sets. We define the optimal bipartite matching as + +$$ +\boldsymbol { \sigma } ^ { * } = \underset { \boldsymbol { \sigma } \in \mathcal { P } } { \arg \operatorname* { m i n } } \sum _ { j = 1 } ^ { M } - 1 _ { \{ c _ { j } \neq \emptyset \} } \hat { p } _ { \boldsymbol { \sigma } ( j ) } ( c _ { j } ) + 1 _ { \{ c _ { j } = \emptyset \} } \mathcal { L } _ { \mathrm { b o x } } ( \boldsymbol { b } _ { j } , \hat { \boldsymbol { b } } _ { \boldsymbol { \sigma } ( j ) } ) , +$$ + +where $\mathcal { P }$ denotes the set of permutations, $\hat { p } _ { \sigma ( j ) } ( \boldsymbol { c } _ { j } )$ is the probability of class $c _ { j }$ for the prediction with index $\sigma ( j )$ , and $\mathcal { L } _ { \mathrm { b o x } }$ is the $\mathcal { L } _ { 1 }$ loss for bounding box parameters. We use the Hungarian algorithm [61] to solve this assignment problem, as in [62, 7]. Our final set-to-set loss adapts (7): + +$$ +\mathcal { L } _ { \mathrm { s u p } } = \sum _ { j = 1 } ^ { N } - \log \hat { p } _ { \sigma ^ { * } ( j ) } ( c _ { j } ) + 1 _ { \{ c _ { j } \neq \emptyset \} } \mathcal { L } _ { \mathrm { b o x } } \bigl ( b _ { j } , \hat { b } _ { \sigma ^ { * } ( j ) } \bigr ) . +$$ + +# 5 Distillation + +Object DGCNN enables a new set-to-set knowledge distillation (KD) pipeline. KD usually involves a teacher model $\tau$ and a student model $s$ . The common practice is to align the outputs of the student with those of the teacher using $\mathcal { L } _ { 2 }$ distance or KL-divergence. In past 3D object detection methods, since final performance heavily relies on NMS and the predictions are post-processed to be a smaller set, distilling the teacher to the student is neither efficient nor effective. Since our set-based detection model is NMS-free, we can easily distill the information between models with homogeneous detection heads (per-query object detection head in our case). First, we train a teacher $\tau$ using the method above with the loss in (10). Then, we train a student $s$ with supervision given by $\tau$ and the ground-truth. The class label and box parameters predicted by the teacher for each object query are $c _ { j } ^ { \mathcal { T } }$ and $b _ { j } ^ { \mathcal { T } }$ , respectively. The corresponding student outputs are $c _ { j } ^ { S }$ and $b _ { j } ^ { S }$ . We find an optimal matching between the output set of the teacher and that of the student: + +$$ +\sigma _ { d } ^ { * } = \underset { \sigma _ { d } \in \mathcal { P } } { \arg \operatorname* { m i n } } \sum _ { j } ^ { N } - \log p _ { \sigma _ { d } ( j ) } ( c _ { j } ^ { \mathcal { T } } ) + \mathcal { L } _ { \mathrm { b o x } } ( { b } _ { j } ^ { { \mathcal { T } } } , { b } _ { \sigma _ { d } ( j ) } ^ { S } ) . +$$ + +Then, the optimal matching’s KD loss is given by + +$$ +\mathcal { L } _ { \mathrm { d i s t i l l } } = \sum _ { j } ^ { N } - \log \hat { p } _ { \sigma _ { d } ^ { * } ( j ) } ( c _ { j } ^ { \mathcal { T } } ) + \mathcal { L } _ { \mathrm { b o x } } ( b _ { j } ^ { \mathcal { T } } , b _ { \sigma _ { d } ^ { * } ( j ) } ^ { S } ) . +$$ + +So the overall loss during KD is $\mathcal { L } = \alpha \mathcal { L } _ { \mathrm { s u p } } + \beta \mathcal { L } _ { \mathrm { d i s t i l l } }$ , where $\alpha$ and $\beta$ balance the supervised loss and distillation loss. In practice, we use $\alpha = \beta = 1$ . + +Table 1: Comparisons to recent works. Our method is robust to whether to use NMS. $^ *$ : implementations with the same PointPillars backbone. $\ddagger$ : implementations with the same SparseConv backbone. + +
MethodNDS↑mAP个mATE√mASE↓mAOE↓mAVE↓mAAE↓NMS
PointPillars [4]53.340.0二二:二二√
SSN[63]54.8341.56二二二二-√
FreeAnchor [64]55.343.7二二-二二√
RegNetX-400MF-SECFPN[65]55.241.2----√
Pillar-OD[5]56.8444.41-----√
CenterPoint (pillar) [6] *59.5647.4831.2725.8133.7832.2520.20√
CenterPoint (pillar) [6] *55.0840.2735.1426.4436.7532.6619.55
CenterPoint (voxel) [6] ‡64.1954.9929.8325.7132.5626.0818.89√
CenterPoint (voxel) [6]‡57.0045.3231.6627.1440.4737.2320.14
Ours (pillar) *62.9753.3134.6226.5631.6126.0218.71√
Ours (pillar) *62.8053.2034.6226.5631.6226.0719.10
Ours (voxel)‡66.1058.7333.3126.3228.8025.1119.08√
Ours (voxel) t66.0458.6233.3326.3428.8025.1119.06
+ +Table 2: Comparisons of different distillation approaches. + +
MethodNDS↑mAP↑mATE↓mASE↓mAOE↓mAVE↓mAAE↓
Baseline (without distillation)62.8053.2034.6226.5631.6226.0219.10
Feature distill (voxel->pillar)62.8453.2934.5526.7831.6126.1119.02
Pseudo labels (voxel-→pillar)63.1853.3134.5826.5531.2925.9918.87
Set-to-set distill (voxel-→pillar)63.3753.8934.3426.2531.0125.5718.77
+ +# 6 Experiments + +We present our experiments in four parts. We introduce the dataset, metrics, implementation, and optimization details in $\ S 6 . 1$ . Then, we demonstrate performance on the nuScenes dataset [66] in $\ S 6 . 2$ . We present knowledge distillation results in $\ S 6 . 3$ . Finally, we provide ablation studies in $\ S 6 . 4$ . + +# 6.1 Training & testing procedures + +Dataset. We experiment on the nuScenes dataset [66]. nuScenes provides rich annotations and diverse scenes. It has 1K short sequences captured in Boston and Singapore with 700, 150, 150 sequences for training, validation, and testing, respectively. Each sequence is ${ \sim } 2 0 \mathrm { s }$ and contains 400 frames. This dataset provides annotation every 0.5s, leading to 28K, 6K, 6K annotated frames for training, validation, and testing. nuScenes uses 32-beam LiDAR, producing 30K points per frame. Following common practice, we use calibrated vehicle pose information to aggregate every 9 non-key frames to key frames, so each annotated frame has $\sim 3 0 0 \mathrm { K }$ points. The annotations include 23 classes with a long-tail distribution, of which 10 classes are included in the benchmark. + +Metrics. The major metrics are mean average precision (mAP) and the nuScenes detection score (NDS). In addition, we use a set of true positive metrics (TP metrics), which include average translation error (ATE), average scale error (ASE), average orientation error (AOE), average velocity error (AVE), and average attribute error (AAE). These metrics are computed in the physical unit. + +Model architecture. Our model consists of three parts: a point-based feature extractor, a DGCNN to encode object queries and to connect the point cloud features to object queries, and a detection head to output the categorical label and bounding box parameters. We experiment with PointPillars [4] and SparseConv [60] as feature extractors. The three blocks of the PointPillars backbone have [3, 5, 5] convolutional layers, with dimensions [64, 128, 256] and strides [2, 2, 2]; the input features are downsampled to 1/2, 1/4, 1/8 of the original feature map. For SparseConv, we use four blocks of [3, 3, 3, 2] 3D sparse convolutional layers, with dimensions [16, 32, 64, 128] and strides [2, 2, 2, 1]; the input features are downsampled to 1/2, 1/4, 1/8, 1/8 of the original feature map. For SparseConv, we transform the features into BEV by collapsing the $z$ -axis. Both backbones use two deformable self-attention [17] layers with dimensions [256, 256] to transform the BEV features. Then, we use two DGCNNs to encode the object queries. Each DGCNN [8] contains two EdgeConv layers with dimensions [256, 256], both with 16 nearest neighbors. For each object query, we predict four points in the BEV to obtain and aggregate the BEV features. The final feature for this object query is the weighted sum of features of these four BEV points. The final detection head takes the features of each object query and predicts class label and bounding box parameters w.r.t. the reference point. + +Table 3: Comparisons of self-distillation versus baselines. + +
MethodNDS ↑mAP↑mATEmASE↓mAOE↓mAVE↓mAAE↓
Baseline (pillar,without distillation)62.8053.2034.6226.5631.6226.0219.10
Self-distillation (pillar->pillar)63.4153.8934.2126.1931.1125.6718.54
Baseline (voxel,without distillation)66.0458.6233.3326.3428.8025.1119.06
Self-distillation (voxel->voxel)66.4559.2531.1725.7730.7325.7218.77
+ +Table 4: Self-distillation with privileged information. + +
MethodNDS ↑mAP↑mATE↓mASE↓mAOE↓mAVE↓mAAE↓
Sparse-→sparse (pillar)42.1238.8944.0127.7864.01144.0139.21
Dense-→sparse (pillar)42.7939.1043.8927.7764.01143.9739.11
Sparse-→sparse (voxel)59.5549.8431.1725.7733.7332.2220.22
Dense->sparse (voxel)59.8950.1231.1125.7633.7032.1920.11
+ +Training & inference. We use AdamW [67] to train the model. The weight decay for AdamW is $1 0 ^ { - 2 }$ . Following a cyclic schedule [68], the learning rate is initially $1 0 ^ { - 4 }$ and gradually increased to $1 0 ^ { - 3 }$ , which is finally decreased to $1 0 ^ { - 8 }$ . The model is initialized with a pre-trained PointPillars network on the same dataset. We train for 20 epochs on 8 RTX 3090 GPUs. During inference, we take the top 100 objects with highest classification scores as the final predictions.We do not use any post-processing such as NMS. For evaluation, we use the toolkit provided with the nuScenes dataset. + +# 6.2 Object DGCNN + +We compare to top-performing methods on the nuScenes dataset in Table 1. PointPillars [4] is an anchor-based method with reasonable trade-off between performance and efficiency. FreeAnchor [64] extends PointPillars by learning how to assign anchors to the ground-truth. RegNetX-400MFSECFPN [65] uses neural architecture search (NAS) to learn a flexible neural network for 3D detection; it is essentially a variant of PointPillars with an enhanced backbone network. Different from anchor-based methods, Pillar-OD [5] makes predictions per pillar, alleviating the class imbalance issue caused by anchors. CenterPoint [6] exploits similar detection heads, with better performance using better training scheduling and data augmentation. For these methods, we use re-implementations in MMDetection3D [69], which match the performances in the original papers. + +We mainly compare to CenterPoint with both PointPillars and SparseConv backbones, denoted as “voxel” and “pillar” respectively. Our method outperforms other methods significantly including CenterPoint with NMS. Without NMS, the performance of CenterPoint drops considerably while our method is unaffected by NMS. This finding verifies the DGCNN implicitly models object relations and removes redundant boxes. + +# 6.3 Set-to-set distillation + +In this section, we present experiments involving our set-to-set distillation pipeline. We conduct three types of distillation. First, we distill a teacher model with a SparseConv backbone to a student model with a PointPillars backbone (denoted as “voxel pillar”). This aligns with the common knowledge distillation setup for classification. We compare to feature-based distillation and pseudo label based methods. The objective of feature-based distillation is to align the middle-level features of the teacher model and the student model while the pseudo label based methods generate pseudo training examples with the pre-trained teacher networks. As Table 2 shows, our set-to-set distillation achieves better performance, confirming that distilling the last stage of the object detection model is more effective than distilling feature maps. + +Second, we perform self-distillation [49] (denoted as “voxel voxel” and “pillar pillar”), where the teacher and the student are identical and take the same point clouds as input. As Table 3 shows, even when the teacher network and the student network have the same capacity, self-distillation still introduces a performance boost. This finding is consistent with the results in [49]. + +Finally, we try distillation with privileged information [56], where the teacher gets access to privileged information but the student does not. Following [10], the teacher takes dense point clouds, and the student takes sparse point clouds (denoted as "dense sparse"). To limit computation time, we train each model over a shorter period of time. The goal is for the student model to learn the same representations as the teacher model without knowing the dense inputs. In Table 4, we compare this setup with self-distillation, where the difference is the teacher model and the student model take the same sparse point clouds in self-distillation. The student achieves better performance when the teacher takes dense point clouds. The result suggests that set-to-set knowledge distillation is an effective approach to transfer insight from privileged information. + +# 6.4 Ablation + +We provide ablation studies on different components of our model to verify assorted design choices. First, we study the improvements of DGCNN over its counterpart, multi-head self-attention [9]. The multi-head self-attention has 8 heads with embedding dimension 256 and LayerNorm [70], following common usage. The DGCNN has two EdgeConv layers with dimensions [256, 256]. The number of neighbors $K$ in EdgeConv is 16. In principle, DGCNN is a sparse version of multi-head self-attention; the sparse structure reduces overhead in back-propagation and leads to sharper “attention maps” as well as faster convergence. + +Table 5: DGCNN versus multi-head self-attention. + +
MethodMetricMulti-head self-attentionDGCNN
NDS39.8941.32
mAP36.3537.81
+ +Table 6: Models with different # DGCNNs. + +
#layersMetric123456
NDS35.9139.7541.1541.2641.0741.32
mAP32.3236.5437.2537.7537.7837.81
+ +Table 7: The number of neighbors in DGCNN. + +
# neigbhorsMetric148163264
NDS40.2140.4540.5141.3240.1739.80
mAP36.8137.1537.4637.8137.1236.70
+ +Table 8: The distribution of the output scores with respect to overlapping boxes. + +
Methodfiltered boxesremainingboxesall boxes
CenterPoint0.07640.18590.0829
Ours0.12220.17110.1604
+ +Table 9: Complexity comparsion between DGCNN and Multi-head self-attention. + +
ModuleComplexity# parameters
DGCNNO(n²d)262144
Multi-head self-attentionO(n²d)263168
+ +Table 5 shows the comparisons: DGCNN consistently outperforms multi-head self-attention. This aligns with our hypothesis: objects are distributed sparsely in the scene, so dense interactions among objects are neither efficient nor effective. Furthermore, we study the effect of number of neighbors in DGCNN. When it is 1, The model reduces to an architecture without object interaction. As we increase the number, it approaches multi-head self-attention. As shown in Table 7, the sweet spot is 16, which appears to balance object interactions and sparsity. + +We also investigate improvements introduced when more DGCNNs are stacked in Table 6. This result suggests it is beneficial to incorporate multiple DGCNNs to model the dynamic object relations. + +Moreover, we hypothesize our method produces different distribution of the output scores with respect to overlapping boxes. To verify this hypothesis (Table 8, we compute average scores for three types of boxes: filtered boxes after NMS, remaining boxes after NMS, and all boxes. Below we show the results. We also compute the percentage of filtered boxes by NMS in our method and Centerpoint. In our method, $2 1 . 9 \%$ + +boxes are removed while in Centerpoint $8 5 . 1 6 \%$ boxes are filtered. Hence, we conclude that our method indeed exhibits a different distribution pattern from Centerpoint. + +Finally, we include a complexity comparison between DGCNN and Multi-head self-attention. Table 9 shows the results; DGCNN layer is on a par with Multi-head self-attention. + +# 7 Conclusion + +Object DGCNN is a highly-efficient 3D object detector for point clouds. It is able to learn object interactions via dynamic graphs and is optimized through a set-to-set loss, leading to NMS-free detection. The success of Object DGCNN indicates that many post-processing operations in 3D object detection are likely unnecessary and can be replaced with suitable neural network modules. Moreover, we introduce a set-to-set knowledge distillation pipeline enabled by the Object DGCNN. This new pipeline significantly simplifies knowledge distillation for 3D object detection and may be applicable to other tasks like 3D model compression. Beyond the direct usage of our model, our experiments suggest several future directions to address current limitations. For example, our method is initialized with a pre-trained backbone network. Training the model from scratch remains elusive due to the sparse set-to-set supervision; solving this issue may yield improved generalization as in [71]. Furthermore, studying 3D-specific feature extractors will improve the speed and generalizability of 3D object detection. Finally, the large amount of unlabeled data available at training time can serve as another type of privileged information to apply self-supervised learning to 3D domains through set-to-set distillation. + +Potential impact. Our method aims to improve the object detection pipeline, which is crucial for the safety of autonomous driving systems. One potential negative impact of our work is that it still lacks theoretical guarantees, similar to many deep learning methods. Future work to improve applicability in this domain might consider challenges of explainability and transparency. + +# 8 Acknowledgement + +The MIT Geometric Data Processing group acknowledges the generous support of Army Research Office grants W911NF2010168 and W911NF2110293, of Air Force Office of Scientific Research award FA9550-19-1-031, of National Science Foundation grants IIS-1838071 and CHS-1955697, from the CSAIL Systems that Learn program, from the MIT–IBM Watson AI Laboratory, from the Toyota–CSAIL Joint Research Center, from a gift from Adobe Systems, from an MIT.nano Immersion Lab/NCSOFT Gaming Program seed grant, and from the Skoltech–MIT Next Generation Program. + +# References + +[1] Shaoshuai Shi, Xiaogang Wang, and Hongsheng Li. Pointrcnn: 3d object proposal generation and detection from point cloud. 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[N/A] \ No newline at end of file diff --git a/parse/train/HwGNkx1WcIs/HwGNkx1WcIs_content_list.json b/parse/train/HwGNkx1WcIs/HwGNkx1WcIs_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..db6248821128f0891f9d5bb13b83d1b78799ee3b --- /dev/null +++ b/parse/train/HwGNkx1WcIs/HwGNkx1WcIs_content_list.json @@ -0,0 +1,1492 @@ +[ + { + "type": "text", + "text": "Object DGCNN: 3D Object Detection using Dynamic Graphs ", + "text_level": 1, + "bbox": [ + 232, + 122, + 764, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Yue Wang Massachusetts Institute of Technology yuewang@csail.mit.edu ", + "bbox": [ + 220, + 226, + 472, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Justin Solomon Massachusetts Institute of Technology jsolomon@mit.edu ", + "bbox": [ + 524, + 226, + 777, + 268 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 304, + 535, + 320 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "3D object detection often involves complicated training and testing pipelines, which require substantial domain knowledge about individual datasets. Inspired by recent non-maximum suppression-free 2D object detection models, we propose a 3D object detection architecture on point clouds. Our method models 3D object detection as message passing on a dynamic graph, generalizing the DGCNN framework to predict a set of objects. In our construction, we remove the necessity of post-processing via object confidence aggregation or non-maximum suppression. To facilitate object detection from sparse point clouds, we also propose a set-to-set distillation approach customized to 3D detection. This approach aligns the outputs of the teacher model and the student model in a permutation-invariant fashion, significantly simplifying knowledge distillation for the 3D detection task. Our method achieves state-of-the-art performance on autonomous driving benchmarks. We also provide abundant analysis of the detection model and distillation framework. ", + "bbox": [ + 233, + 335, + 767, + 515 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Methods for 3D object detection have progressed rapidly, yielding deployable autonomous driving perception systems. Following common practice in 2D vision, 3D object detection often employs complex training and testing pipelines including many post-processing operations to achieve superior performance. These operations are typically non-parallelizable and inefficient even with modern deep learning frameworks, implying a steep trade-off between between efficiency and effectiveness. ", + "bbox": [ + 174, + 535, + 825, + 604 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Modern methods usually employ two stages [1, 2], including a region proposal network [3] that can introduce significant training overhead. Subsequent efforts simplify this pipeline for 3D object detection. PointPillars [4] introduces a one-stage anchor-based design, simplfying training. PillarOD [5] and CenterPoint [6] improve the one-stage model by making per-pillar predictions, that is, one prediction per point on the ground plane. They assign ground-truth bounding boxes to multiple outputs while training to ease optimization. However, they predict redundant boxes, which can overlap in the same positions; extra boxes are eliminated a posteriori using non-maximum suppression (NMS). It remains elusive to remove hand-designed components like NMS in training and testing. ", + "bbox": [ + 174, + 609, + 825, + 720 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We introduce Object DGCNN, a streamlined architecture for 3D object detection from point clouds. Like DETR for 2D object detection [7], we predict a set of bounding boxes from the raw data, enabling an NMS-free pipeline that achieves real-time performance. A critical new component is to treat each object query as a point in a set whose embedding is learned using DGCNN [8]. Compared to the self-attention module [9] in DETR, DGCNN leverages a sparse set of object relations, which reflects the real object distribution in the scene. In contrast to PointPillars [4], PillarOD [5], and CenterPoint [6], our method does not require post-processing. ", + "bbox": [ + 174, + 727, + 825, + 824 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We also provide a knowledge distillation approach customized to 3D object detection. Existing methods typically distill dense feature maps from a teacher model to a student model, whose training objective does not necessarily capture 3D object detection performance [10]. In contrast, we propose set-to-set distillation training that aligns the outputs of the teacher and the student in a permutationinvariant fashion. This process is enabled by the unified Object DGCNN architecture. In addition to obtaining better performance, through this process our model can benefit from privileged information (e.g., dense point clouds) only available at training time. ", + "bbox": [ + 174, + 830, + 825, + 900 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 823, + 119 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions. We summarize our key contributions as follows: ", + "bbox": [ + 173, + 126, + 598, + 140 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose a post-processing-free 3D object detection model achieving state-of-the-art performance. To our knowledge, this is the first NMS-free 3D object detector. \n• We generalize DGCNN to model objects as a point set. The DGCNN module outperforms its self-attention counterpart thanks to its sparse structure. \n• We propose a set-to-set distillation method for 3D object detection. In our construction, knowledge distillation on object detection simply penalizes differences between the outputs of the teacher model and the student model. \n• We show our model can use privileged information (such as dense point clouds) that is naturally available at training time to improve the model performance at inference time. \n• We release our code to promote reproducibility and future research. 1 ", + "bbox": [ + 173, + 151, + 826, + 291 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 309, + 321, + 327 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2D object detection. Object recognition research has been transitioning from models with handcrafted components to models with limited post-processing. One-stage detectors [11–13] remove the complicated region proposal networks in two-stage objectors [3, 14], yielding more efficient training and testing. Anchor-free methods [15, 16] further simplify the one-stage pipeline by shifting from per-anchor prediction to per-pixel prediction. However, these methods still make dense predictions and rely on NMS to reduce redundancy. To alleviate this issue, DETR [7] formulates object detection as a set-to-set prediction problem. It introduces a set-to-set loss that implicitly penalizes redundant boxes, removing the necessity of post-processing. To accelerate convergence, Deformable DETR [17] proposes deformable self-attention and streamlines the optimization process. Our method also formulates 3D object detection as set prediction, but with a customized design for 3D. ", + "bbox": [ + 173, + 340, + 825, + 479 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3D object detection. VoxelNet [18] generalizes one-stage object detection to 3D. It uses 3D dense convolutions to learn representations on voxelized point clouds, which is too inefficient to capture fine-grained features. To address that, PIXOR [19] and PointPillars [4] project points to a birds-eye view (BEV) and operate on 2D feature maps; PointNet [20] aggregates features within each BEV pixel. We use a variant of PointPillars [4] for 3D detection (§3). These methods are efficient but drop information along the vertical axis. To accompany the BEV projection, MVF [21] adds a spherical projection. PillarOd [5] and CenterPoint [6] use pillar-centric object detection, making predictions per BEV pixel (pillar) rather than per anchor. These anchor-free methods simplify 3D object detection while maintaining efficiency. Beyond SSD-style [11] one-stage models, Complex-YOLO [22] extends YOLO to 3D for real-time perception. PointRCNN [23] employs a two-stage architecture for high-quality detection. To improve representations of two-stage models, PVRCNN [2] proposes a point-voxel feature set abstraction layer to leverage the flexible receptive fields of PointNet-based networks. Unlike works on point clouds, LaserNet [24] operates on raw range scans with comparable performance. [25–27] combine point clouds with camera images. Frustum-PointNet [28] leverages 2D object detectors to form a frustum crop of points and then uses PointNet to aggregate features. [29] describes an end-to-end learnable architecture that exploits continuous convolutions to fuse feature maps. VoteNet [30, 31] generalizes Hough voting [32] for 3D object detection in point clouds. DOPS [33] extends VoteNet and predicts 3D object shapes. In addition to visual input, [34] shows that high-definition (HD) maps can boost performance of 3D object detectors. [35] argues that multi-tasking can learn better representations than single-tasking. Beyond supervised learning, [36] learns a perception model for unknown classes. ", + "bbox": [ + 174, + 486, + 825, + 776 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "DGCNN. DGCNN [8] pioneered learning point cloud representations via dynamic graphs. It models point clouds as connected graphs, which are dynamically built using $k$ -nearest neighbors in the latent space. DGCNN learns per-point features through message passing. However, it operates on point clouds for single object recognition and semantic segmentation. One of our key contributions is to generalize DGCNN to model scene-level object relations for 3D detection. ", + "bbox": [ + 174, + 782, + 825, + 852 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Knowledge distillation (KD). KD compresses knowledge from an ensemble of models into a single smaller model [37]. [38] generalizes this idea and combines it with deep learning. KD transfers knowledge from a teacher model to a student model by minimizing a loss, in which the target is the distribution of class probabilities induced by the teacher. [39–50] improve knowledge distillation for classification. Beyond image classification, KD has been extended to improve object detection. [51] leverages FitNets for object detection, addressing obstacles such as class imbalance, loss instability, and feature distribution mismatch. [52] distills between region proposals, accelerating training with added instability. To address this issue, [53] uses fine-grained representation imitation using object masks. [54] uses KD to tackle a continual learning problem. ", + "bbox": [ + 174, + 858, + 821, + 886 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 189 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Privileged information. [55] introduces the framework of learning with privileged information in the context of support vector machines (SVMs), wherein additional information is accessible during training but not testing. [56] unifies KD and learning using privileged information theoretically. [57] identifies practical applications, e.g., transferring knowledge from localized data to non-localized data, from high resolution to low resolution, from color images to edge images, and from regular images to distorted images. To mediate uncertainty and improve training efficiency, [58] makes the variance of Dropout [59] a function of privileged information. We extend these methods to 3D data, in which privileged information consists of dense point clouds aggregated from LiDAR sequences. ", + "bbox": [ + 174, + 194, + 825, + 306 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Overview ", + "text_level": 1, + "bbox": [ + 174, + 324, + 284, + 342 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our target application of object detection differs from the recognition and segmentation tasks considered for DGCNN. Our point clouds typically contain too many points to apply DGCNN and its peers directly to the entire scene. Moreover, the size of our output set, a small set of bounding boxes, differs from the size of our input set, a huge set of points in $\\mathbb { R } ^ { 3 }$ . ", + "bbox": [ + 174, + 356, + 825, + 411 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Following state-of-the-art in large-scale object detection, our pipeline learns a grid-based intermediate representation to capture local features (§3). We test two standard learning-based methods for collecting local point cloud features on a birds-eye view (BEV) grid. While in principle it might be possible to avoid grids entirely in our pipeline, this BEV representation is far more efficient and—as observed in previous work—is sufficient to find objects reliably in autonomous driving, where there is likely only one object above any given grid cell on the ground plane. ", + "bbox": [ + 174, + 417, + 825, + 502 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our main architecture contribution is the Object DGCNN pipeline (§4), which transitions from this BEV grid of features to a set of object bounding boxes. Object DGCNN draws inspiration from the DGCNN architecture; its layers alternate between local feature transformations and $k$ - nearest neighbor aggregation to capture relationships between objects. Unlike conventional DGCNN, however, Object DGCNN incorporates features from the BEV grid in each of its layers; each layer incorporates several queries into the BEV to refine object position estimates. The output of Object DGCNN is a set of objects in the scene. We use a permutation-invariant loss (10) to measure divergence from the ground truth set of objects. ", + "bbox": [ + 174, + 507, + 825, + 618 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The pipeline above does not require hand-designed post-processing like NMS; our output boxes are usable directly for object detection. Beyond simplifying the object detection pipeline, this allows us to propose object detection-specific distillation procedures $( \\ S 6 . 3 )$ that further improve performance. These use one network to train another, e.g., to train a network operating on sparse point clouds to output features that imitate those learned by a network trained on denser, more detailed point clouds. ", + "bbox": [ + 174, + 625, + 825, + 694 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Local Features ", + "text_level": 1, + "bbox": [ + 174, + 713, + 328, + 731 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We begin with a point cloud $\\mathcal { X } \\ = \\ \\{ \\pmb { x } _ { 1 } , \\dots , \\pmb { x } _ { i } , \\dots , \\pmb { x } _ { N } \\} \\ \\subset \\ \\mathbb { R } ^ { 3 }$ with per-point features ${ \\mathcal { F } } =$ $\\{ \\pmb { f } _ { 1 } , \\ldots , \\pmb { f } _ { i } , \\ldots , \\pmb { \\bar { f } } _ { N } \\} \\subset \\mathbb { R } ^ { K }$ , ground-truth bounding boxes $\\boldsymbol { B } = \\{ b _ { 1 } , \\dots , \\bar { b } _ { j } , \\dots , b _ { M } \\} \\subset \\mathbb { R } ^ { 9 }$ , and categorical labels $\\bar { \\mathcal { C } } \\overset { \\cdot } { = } \\{ c _ { j } , \\dotsc , c _ { j } , \\dotsc , c _ { M } \\} \\subset \\mathbb { Z }$ . Each $b _ { j }$ contains position, size, heading angle, and velocity in the birds-eye view (BEV); our architecture aims to predict these boxes and their labels from the point cloud and its features. ", + "bbox": [ + 174, + 744, + 825, + 814 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "As an initial step, modern 3D object detection models scatter points into either BEV pillars or 3D voxels and then use convolutional neural networks to extract features on a grid. This strategy accelerates object detection for large point clouds. We test two neural network architectures for BEV feature extraction, detailed below. ", + "bbox": [ + 174, + 820, + 825, + 876 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "PointPillars [4] maps sparse point clouds onto a dense BEV pillar map on which 2D convolutions can be applied. Suppose $F _ { P } ( i )$ returns the points in pillar $i$ , that is, the set of points in a vertical column above point $i$ on the ground. When collecting features from points to pillars, multiple points can fall into the same pillar. In this case, PointNet [20] (PN) is used to obtain pillar-wise features: ", + "bbox": [ + 173, + 882, + 821, + 911 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/4961b1385646350b25971984d745696f6c00c96db1551f123510f35addb612d1.jpg", + "image_caption": [ + "Figure 1: Overview. Point cloud features are learned in BEV, followed by $L$ DGCNNs to model object relations. We predict a set of bounding boxes and compute loss in a one-to-one manner. " + ], + "image_footnote": [], + "bbox": [ + 174, + 87, + 825, + 218 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 276, + 823, + 305 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/400f1b399e37245a01345e45230268dc050714e77a078e0d1c0512f1a000a2ec.jpg", + "text": "$$\n\\pmb { f } _ { i } ^ { \\mathrm { p i l l a r } } = \\operatorname { P N } ( \\{ \\pmb { f } _ { j } | \\pmb { x } _ { j } \\in F _ { P } ( \\pmb { p } _ { i } ) \\} ) ,\n$$", + "text_format": "latex", + "bbox": [ + 382, + 311, + 614, + 332 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $f _ { i } ^ { \\mathrm { p i l l a r } }$ is the feature of pillar $\\mathbf { \\nabla } _ { \\pmb { p } _ { i } }$ . We set the features for empty pillars to 0. This results in a dense 2D grid $\\mathcal { F } ^ { \\mathrm { p i l l a r } } \\subset \\mathbb { R } ^ { H ^ { \\mathrm { p } } \\times W ^ { \\mathrm { p } } \\times C ^ { \\mathrm { p } } }$ , where $H ^ { \\mathrm { p } } , W ^ { \\mathrm { p } }$ and $C ^ { \\mathrm { p } }$ are the height, width, number of channels of this 2D pillar map, respectively. Multiple stacked convolutional layers further embed the pillar features to the final feature map F d ⊂ RHd×W d×Cd . ", + "bbox": [ + 173, + 338, + 825, + 400 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "An alternative BEV embedding is SparseConv [60]. If $F _ { V } ( i )$ returns the set of points in voxel $i$ SparseConv collects point-wise features into voxel-wise features by ", + "bbox": [ + 171, + 406, + 825, + 435 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/a8ee7f4d6af13bfb1a1f1b0568dbc81c6dba4bd9c7ddf206e0d5634a489a66d5.jpg", + "text": "$$\n\\pmb { f } _ { i } ^ { \\mathrm { v o x e l } } = \\mathrm { P N } ( \\{ \\pmb { f } _ { j } | \\pmb { x } _ { j } \\in F _ { V } ( i ) \\} ) ,\n$$", + "text_format": "latex", + "bbox": [ + 387, + 440, + 609, + 462 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $f _ { i } ^ { \\mathrm { v o x e l } }$ contains the features of voxel $i$ . In contrast to PointPillars, SparseConv conducts 3D sparse convolutions to refine the voxel-wise features. Finally, we compress these sparse voxels to a BEV 2D grid by filling empty voxels with zeros and averaging along the $z$ -axis. For ease of notation, we also denote the resulting 2D grid F d ⊂ RHd×W d×Cd . ", + "bbox": [ + 173, + 468, + 825, + 527 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 Object DGCNN ", + "text_level": 1, + "bbox": [ + 174, + 546, + 339, + 564 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "After obtaining the BEV features ${ \\mathcal { F } } ^ { \\mathrm { d } }$ using one of the architectures above, we predict a set of bounding boxes as well as a label for each box. The key difference between our architecture and most recent 3D object detection methods is that ours produces a set of bounding boxes rather than a box per grid cell followed by NMS, as in [5, 6]. Hence, we need to transition from a grid of per-pillar features to an unordered set of objects; we detail our approach below. We address two key issues: prediction of the bounding boxes and evaluation of the loss. ", + "bbox": [ + 173, + 577, + 825, + 661 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Desiderata. Object DGCNN uses a DGCNN-inspired architecture but incorporates grid-based BEV features, built on the philosophy that local features (§3) are reasonable to store on a dense grid, but object predictions are better modeled using sets. Hence, we require a new architecture and set-to-set loss that encourage bounding box diversity. ", + "bbox": [ + 173, + 666, + 826, + 723 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Object DGCNN uses $L$ layers that follow a series of set-based computations to produce bounding box predictions from the BEV feature maps. Each layer employs the following steps (Figure 1): ", + "bbox": [ + 171, + 729, + 823, + 758 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1. predict a set of query points and attention weights; \n2. collect BEV features from keypoints determined by the queries; and \n3. model object-object interactions via DGCNN. ", + "bbox": [ + 173, + 768, + 642, + 810 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Each layer results in a more refined set of bounding box predictions, one per query. At the end of these layers, we match the prediction set with the ground-truth set in a one-to-one fashion and evaluate a set-to-set object detection loss. ", + "bbox": [ + 173, + 820, + 823, + 863 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Single layer. Inspired by DETR [7], each layer $\\ell \\in \\{ 0 , \\ldots , L - 1 \\}$ of Object DGCNN operates on a set of object queries $\\mathcal { Q } _ { \\ell } = \\{ \\pmb q _ { \\ell 1 } , \\dots , \\pmb q _ { \\ell M ^ { * } } \\} \\subset \\mathbb { R } ^ { Q }$ , producing a new set $\\mathcal { Q } _ { \\ell + 1 }$ . Although queries are fully learnable, our intuition is that they represent progressively refined object positions. ", + "bbox": [ + 174, + 868, + 825, + 911 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The initial set of object queries $\\mathcal { Q } _ { 0 }$ is learned jointly with the neural network weights, yielding a dataset-specific prior. Beyond this fixed initial set, below we detail how to incorporate scene information to obtain $\\mathcal { Q } _ { \\ell + 1 }$ from $\\mathcal { Q } _ { \\ell }$ using an approach inspired by DGCNN [8] and deformable self-attention [17]. For notational convenience, we drop the $\\ell$ subscript. ", + "bbox": [ + 173, + 90, + 826, + 147 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Starting from each query $\\pmb q _ { i }$ (or, without the index dropped, $\\mathbf { \\Delta } q _ { \\ell i } .$ ), we decode a reference point $\\pmb { p } _ { i } \\in \\mathbb { R } ^ { 2 }$ , a set of offsets $\\{ \\delta _ { i 0 } , \\dots , \\delta _ { i K } \\} \\subset \\mathbb { R } ^ { 2 }$ , and a set of attention weights $\\{ w _ { i 0 } , \\dots , w _ { i K } \\} \\subset \\mathbb { R }$ : ", + "bbox": [ + 169, + 152, + 825, + 183 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/cbb852b72cd79ac95bd17fc4d221b448e653bf4c117de74e31bd235ef181fb34.jpg", + "text": "$$\n\\begin{array} { c } { p _ { i } = \\Phi _ { \\mathrm { r e f } } ( { \\boldsymbol { q } } _ { i } ) , \\qquad \\{ \\delta _ { i } ^ { 0 } , \\ldots , \\delta _ { i } ^ { k } , \\ldots \\delta _ { i } ^ { K } \\} = \\Phi _ { \\mathrm { n e i g h b o r } } ( { \\boldsymbol { q } } _ { i } ) , } \\\\ { \\{ w _ { i } ^ { 0 } , \\ldots , w _ { i } ^ { k } , \\ldots w _ { i } ^ { K } \\} = \\Phi _ { \\mathrm { a t t e n } } ( { \\boldsymbol { q } } _ { i } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 188, + 710, + 231 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\Phi _ { \\mathrm { r e f } }$ , $\\Phi _ { \\mathrm { n e i g h b o r } }$ , and $\\Phi _ { \\mathrm { a t t e n } }$ are shared neural networks among the queries. We think of $\\mathbf { \\nabla } _ { \\pmb { p } _ { i } }$ as a hypothesis for the center of the $i$ -th object; the $\\delta$ ’s represent the positions of $K$ informative points relative to the position of the object that determine its geometry. ", + "bbox": [ + 174, + 236, + 825, + 279 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Next, we collect a BEV feature $\\mathbf { \\boldsymbol { f } } _ { i k }$ associated to each neighbor point $\\pmb { p } _ { i k } = \\pmb { p } _ { i } + \\pmb { \\delta } _ { i k }$ ", + "bbox": [ + 173, + 284, + 740, + 300 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/afd1673a654602ffb7ce5456caecc68b3d2ef571dddeb9d6a8e290f64fff6a91.jpg", + "text": "$$\n\\pmb { f } _ { i k } = f _ { \\mathrm { b i l i n e a r } } ( \\mathcal { F } ^ { d } , \\pmb { p } _ { i } + \\delta _ { i k } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 397, + 306, + 599, + 327 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $f _ { \\mathrm { b i l i n e a r } }$ bilinearly interpolates the BEV feature map ${ \\mathcal { F } } ^ { d }$ . Note this step is the interaction between our set-based architecture manipulating query points $\\pmb q _ { i }$ and the grid-based feature map ${ \\mathcal { F } } ^ { d }$ . We then aggregate a single object query feature $\\pmb { f } _ { i } ^ { o }$ from the $\\pmb { f } _ { i k } \\mathbf { s }$ : ", + "bbox": [ + 173, + 334, + 826, + 377 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/5d64a2b1060fc290d7dba28aaf1ae422f5c7690d10dd418742426dbc34a32777.jpg", + "text": "$$\nf _ { i } ^ { o } = \\sum _ { k } { \\frac { e ^ { w _ { i k } } } { \\sum _ { k } { e ^ { w _ { i k } } } } } { f _ { i k } } .\n$$", + "text_format": "latex", + "bbox": [ + 419, + 385, + 578, + 421 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This generates scene-aware features; each object query “attends” to a certain area in the scene. ", + "bbox": [ + 173, + 428, + 792, + 444 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In the current layer $\\ell$ , the queries have not yet interacted with each other. To incorporate neighborhood information in object detection estimates, we use DGCNN-style operations to model a sparse set of relations. We construct a graph between the queries using a nearest neighbor search in feature space. In particular, we connect each query feature $\\pmb { f } _ { i } ^ { o }$ to its 16 nearest neighbors as ablated in Table 7. Identically to DGCNN, we learn a feature per edge $e _ { i j }$ and then aggregate back to the vertices $i$ to produce the new set of object queries. In detail, we write: ", + "bbox": [ + 173, + 449, + 826, + 534 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/986649696473c131b93df5b09569c9f14114b94dca51e40e2bc0229abc52799f.jpg", + "text": "$$\n{ \\pmb q } _ { ( \\ell + 1 ) i } = \\operatorname* { m a x } _ { \\mathrm { e d g e s } ~ e _ { i j } } \\Phi _ { \\mathrm { e d g e } } ( { \\pmb f } _ { i } ^ { o } , { \\pmb f } _ { j } ^ { o } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 388, + 540, + 607, + 565 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where max denotes a channel-wise maximum and $\\Phi _ { \\mathrm { e d g e } }$ is a neural network for computing edge features. This completes our layer for computing $\\mathcal { Q } _ { \\ell + 1 }$ from $\\mathcal { Q } _ { \\ell }$ . Optionally, we repeat this last step multiple times, in effect applying DGCNN to the features $\\pmb { f } _ { i } ^ { o }$ to get the point set $\\mathcal { Q } _ { \\ell + 1 }$ . ", + "bbox": [ + 174, + 571, + 825, + 616 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Set-to-set loss. After $L$ Object DGCNN layers as described above, we are left with a set of $M ^ { * }$ queries $\\mathcal { Q } _ { L }$ used to predict our bounding boxes. For each query $\\mathbf { \\nabla } _ { \\mathbf { q } _ { L i } }$ , we use a classification network to predict a categorical label $\\hat { c _ { i } }$ and a regression network to predict bounding box parameters $\\hat { b _ { i } }$ . Our final task is to assign the predictions to the ground-truth boxes and compute a set-to-set loss. ", + "bbox": [ + 173, + 619, + 826, + 680 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Most object detection models minimize a loss $\\mathcal { L } _ { \\mathrm { o d } }$ given by ", + "bbox": [ + 173, + 685, + 565, + 700 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0bb5ea3ee77868bc6744ec4e167a690db3ba3df23e9fbb318500ba56e3d57031.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { o d } } = \\sum _ { j = 1 } ^ { \\hat { M } } - \\log \\hat { p } _ { \\hat { \\sigma } ( j ) } ( \\hat { c } _ { j } ) + 1 _ { \\{ c _ { \\hat { \\sigma } ( j ) } \\neq \\emptyset \\} } \\mathcal { L } _ { \\mathrm { b o x } } ( \\hat { b } _ { j } , b _ { \\hat { \\sigma } ( j ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 312, + 707, + 684, + 755 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\hat { M } = H ^ { \\mathrm { d } } * W ^ { \\mathrm { d } }$ , $\\hat { \\sigma } ( * )$ returns the corresponding index of the ground-truth bounding box, $\\hat { p } _ { \\hat { \\sigma } ( j ) } ( c _ { j } )$ is the probability of class $c _ { \\hat { \\sigma } ( j ) }$ for the prediction with index $\\sigma ( j )$ , $\\mathcal { D }$ denotes an invalid box, and $\\mathcal { L } _ { \\mathrm { b o x } }$ is typically the $\\mathcal { L } _ { 1 }$ distance. Different matchings $\\hat { \\sigma }$ yield different optimization landscapes and hence different prediction models. Pillar-OD [5] and CenterPoint [6] employ a simple $\\hat { \\sigma }$ to determine the ground-truth box used to evaluate the box predicted at BEV pixel $j$ : ", + "bbox": [ + 173, + 762, + 826, + 835 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d9cf1d8fc868dc8a444cf590156df40e93008f67cd6f2e971255fa9724df0185.jpg", + "text": "$$\n\\begin{array} { r } { \\hat { \\sigma } _ { \\mathrm { o v e r l a p } } ( j ) = \\left\\{ \\begin{array} { l l } { j ^ { ' } , \\quad } & { \\mathrm { i f } \\ b _ { j ^ { ' } } \\ \\mathrm { o v e r l a p s \\ w i t h \\ B E V \\ p i x e l } \\ j ; } \\\\ { \\varnothing , \\quad } & { \\mathrm { o t h e r w i s e } . } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 840, + 694, + 876 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "This strategy can assign a box to multiple nearby BEV pixels. This one-to-many assignment provides dense supervision for the object detector and eases optimization. Since the training objective encourages each BEV pixel to predict the same surrounding box, however, redundant boxes are inevitable. So, NMS is usually required to remove redundant boxes at inference time. ", + "bbox": [ + 171, + 882, + 821, + 912 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/b3a034d74f018210cdbafa53766ce355aeb8c11224c01f00432b81db0fb7d96f.jpg", + "image_caption": [ + "Figure 2: The set-to-set distillation pipeline. The student network is trained with the ground-truth supervision as well as with the supervision from a fixed teacher network. " + ], + "image_footnote": [], + "bbox": [ + 173, + 88, + 844, + 229 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 292, + 823, + 321 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Rather than performing dense predictions in the BEV, we make per-query predictions. Typically, $M ^ { * }$ is much larger than the number of ground-truth boxes $M$ . To account for this difference, we pad the set of ground-truth boxes with $\\mathcal { O } \\mathbf { s }$ (no object) up to $M ^ { * }$ . Following [7], we use an objective built on an optimal matching between these two sets. We define the optimal bipartite matching as ", + "bbox": [ + 174, + 327, + 825, + 383 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/d42ff577c2cee5c4cd49f2cfdce3aae5ff73a0134054f81d344895186a46c112.jpg", + "text": "$$\n\\boldsymbol { \\sigma } ^ { * } = \\underset { \\boldsymbol { \\sigma } \\in \\mathcal { P } } { \\arg \\operatorname* { m i n } } \\sum _ { j = 1 } ^ { M } - 1 _ { \\{ c _ { j } \\neq \\emptyset \\} } \\hat { p } _ { \\boldsymbol { \\sigma } ( j ) } ( c _ { j } ) + 1 _ { \\{ c _ { j } = \\emptyset \\} } \\mathcal { L } _ { \\mathrm { b o x } } ( \\boldsymbol { b } _ { j } , \\hat { \\boldsymbol { b } } _ { \\boldsymbol { \\sigma } ( j ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 282, + 391, + 714, + 436 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\mathcal { P }$ denotes the set of permutations, $\\hat { p } _ { \\sigma ( j ) } ( \\boldsymbol { c } _ { j } )$ is the probability of class $c _ { j }$ for the prediction with index $\\sigma ( j )$ , and $\\mathcal { L } _ { \\mathrm { b o x } }$ is the $\\mathcal { L } _ { 1 }$ loss for bounding box parameters. We use the Hungarian algorithm [61] to solve this assignment problem, as in [62, 7]. Our final set-to-set loss adapts (7): ", + "bbox": [ + 174, + 445, + 826, + 489 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/4aef0d1dcc4e91bd0e7c913e52a80a46ab5984985a532fc61102d3ead74eb2d9.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { s u p } } = \\sum _ { j = 1 } ^ { N } - \\log \\hat { p } _ { \\sigma ^ { * } ( j ) } ( c _ { j } ) + 1 _ { \\{ c _ { j } \\neq \\emptyset \\} } \\mathcal { L } _ { \\mathrm { b o x } } \\bigl ( b _ { j } , \\hat { b } _ { \\sigma ^ { * } ( j ) } \\bigr ) .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 497, + 687, + 542 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "5 Distillation ", + "text_level": 1, + "bbox": [ + 174, + 559, + 297, + 575 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Object DGCNN enables a new set-to-set knowledge distillation (KD) pipeline. KD usually involves a teacher model $\\tau$ and a student model $s$ . The common practice is to align the outputs of the student with those of the teacher using $\\mathcal { L } _ { 2 }$ distance or KL-divergence. In past 3D object detection methods, since final performance heavily relies on NMS and the predictions are post-processed to be a smaller set, distilling the teacher to the student is neither efficient nor effective. Since our set-based detection model is NMS-free, we can easily distill the information between models with homogeneous detection heads (per-query object detection head in our case). First, we train a teacher $\\tau$ using the method above with the loss in (10). Then, we train a student $s$ with supervision given by $\\tau$ and the ground-truth. The class label and box parameters predicted by the teacher for each object query are $c _ { j } ^ { \\mathcal { T } }$ and $b _ { j } ^ { \\mathcal { T } }$ , respectively. The corresponding student outputs are $c _ { j } ^ { S }$ and $b _ { j } ^ { S }$ . We find an optimal matching between the output set of the teacher and that of the student: ", + "bbox": [ + 173, + 590, + 825, + 747 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/7b1a5e8ff8310dcc5566f437c1e3a79813c27c67e4159e02f4f42e3e588b2ee6.jpg", + "text": "$$\n\\sigma _ { d } ^ { * } = \\underset { \\sigma _ { d } \\in \\mathcal { P } } { \\arg \\operatorname* { m i n } } \\sum _ { j } ^ { N } - \\log p _ { \\sigma _ { d } ( j ) } ( c _ { j } ^ { \\mathcal { T } } ) + \\mathcal { L } _ { \\mathrm { b o x } } ( { b } _ { j } ^ { { \\mathcal { T } } } , { b } _ { \\sigma _ { d } ( j ) } ^ { S } ) .\n$$", + "text_format": "latex", + "bbox": [ + 313, + 753, + 684, + 799 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Then, the optimal matching’s KD loss is given by ", + "bbox": [ + 173, + 806, + 500, + 823 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/599748f3f5963f73ef09a7674bb46ac548af48a7eaa4e5d54a0318e6d1d68eb3.jpg", + "text": "$$\n\\mathcal { L } _ { \\mathrm { d i s t i l l } } = \\sum _ { j } ^ { N } - \\log \\hat { p } _ { \\sigma _ { d } ^ { * } ( j ) } ( c _ { j } ^ { \\mathcal { T } } ) + \\mathcal { L } _ { \\mathrm { b o x } } ( b _ { j } ^ { \\mathcal { T } } , b _ { \\sigma _ { d } ^ { * } ( j ) } ^ { S } ) .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 830, + 671, + 875 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "So the overall loss during KD is $\\mathcal { L } = \\alpha \\mathcal { L } _ { \\mathrm { s u p } } + \\beta \\mathcal { L } _ { \\mathrm { d i s t i l l } }$ , where $\\alpha$ and $\\beta$ balance the supervised loss and distillation loss. In practice, we use $\\alpha = \\beta = 1$ . ", + "bbox": [ + 174, + 882, + 821, + 911 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/ceade318ce1706a231d8c7a63f0f72bcd32a0f949e143009e2647b591a31fe87.jpg", + "table_caption": [ + "Table 1: Comparisons to recent works. Our method is robust to whether to use NMS. $^ *$ : implementations with the same PointPillars backbone. $\\ddagger$ : implementations with the same SparseConv backbone. " + ], + "table_footnote": [], + "table_body": "
MethodNDS↑mAP个mATE√mASE↓mAOE↓mAVE↓mAAE↓NMS
PointPillars [4]53.340.0二二:二二√
SSN[63]54.8341.56二二二二-√
FreeAnchor [64]55.343.7二二-二二√
RegNetX-400MF-SECFPN[65]55.241.2----√
Pillar-OD[5]56.8444.41-----√
CenterPoint (pillar) [6] *59.5647.4831.2725.8133.7832.2520.20√
CenterPoint (pillar) [6] *55.0840.2735.1426.4436.7532.6619.55
CenterPoint (voxel) [6] ‡64.1954.9929.8325.7132.5626.0818.89√
CenterPoint (voxel) [6]‡57.0045.3231.6627.1440.4737.2320.14
Ours (pillar) *62.9753.3134.6226.5631.6126.0218.71√
Ours (pillar) *62.8053.2034.6226.5631.6226.0719.10
Ours (voxel)‡66.1058.7333.3126.3228.8025.1119.08√
Ours (voxel) t66.0458.6233.3326.3428.8025.1119.06
", + "bbox": [ + 176, + 137, + 821, + 309 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/0e91e35f79d4d8a7baf54d501e89d21ced7eab35c0219af5a6bd94f4fdc493f6.jpg", + "table_caption": [ + "Table 2: Comparisons of different distillation approaches. " + ], + "table_footnote": [], + "table_body": "
MethodNDS↑mAP↑mATE↓mASE↓mAOE↓mAVE↓mAAE↓
Baseline (without distillation)62.8053.2034.6226.5631.6226.0219.10
Feature distill (voxel->pillar)62.8453.2934.5526.7831.6126.1119.02
Pseudo labels (voxel-→pillar)63.1853.3134.5826.5531.2925.9918.87
Set-to-set distill (voxel-→pillar)63.3753.8934.3426.2531.0125.5718.77
", + "bbox": [ + 178, + 351, + 820, + 419 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 Experiments ", + "text_level": 1, + "bbox": [ + 173, + 450, + 312, + 468 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We present our experiments in four parts. We introduce the dataset, metrics, implementation, and optimization details in $\\ S 6 . 1$ . Then, we demonstrate performance on the nuScenes dataset [66] in $\\ S 6 . 2$ . We present knowledge distillation results in $\\ S 6 . 3$ . Finally, we provide ablation studies in $\\ S 6 . 4$ . ", + "bbox": [ + 174, + 482, + 826, + 523 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6.1 Training & testing procedures ", + "text_level": 1, + "bbox": [ + 176, + 540, + 423, + 556 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Dataset. We experiment on the nuScenes dataset [66]. nuScenes provides rich annotations and diverse scenes. It has 1K short sequences captured in Boston and Singapore with 700, 150, 150 sequences for training, validation, and testing, respectively. Each sequence is ${ \\sim } 2 0 \\mathrm { s }$ and contains 400 frames. This dataset provides annotation every 0.5s, leading to 28K, 6K, 6K annotated frames for training, validation, and testing. nuScenes uses 32-beam LiDAR, producing 30K points per frame. Following common practice, we use calibrated vehicle pose information to aggregate every 9 non-key frames to key frames, so each annotated frame has $\\sim 3 0 0 \\mathrm { K }$ points. The annotations include 23 classes with a long-tail distribution, of which 10 classes are included in the benchmark. ", + "bbox": [ + 173, + 565, + 825, + 678 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Metrics. The major metrics are mean average precision (mAP) and the nuScenes detection score (NDS). In addition, we use a set of true positive metrics (TP metrics), which include average translation error (ATE), average scale error (ASE), average orientation error (AOE), average velocity error (AVE), and average attribute error (AAE). These metrics are computed in the physical unit. ", + "bbox": [ + 174, + 683, + 825, + 739 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Model architecture. Our model consists of three parts: a point-based feature extractor, a DGCNN to encode object queries and to connect the point cloud features to object queries, and a detection head to output the categorical label and bounding box parameters. We experiment with PointPillars [4] and SparseConv [60] as feature extractors. The three blocks of the PointPillars backbone have [3, 5, 5] convolutional layers, with dimensions [64, 128, 256] and strides [2, 2, 2]; the input features are downsampled to 1/2, 1/4, 1/8 of the original feature map. For SparseConv, we use four blocks of [3, 3, 3, 2] 3D sparse convolutional layers, with dimensions [16, 32, 64, 128] and strides [2, 2, 2, 1]; the input features are downsampled to 1/2, 1/4, 1/8, 1/8 of the original feature map. For SparseConv, we transform the features into BEV by collapsing the $z$ -axis. Both backbones use two deformable self-attention [17] layers with dimensions [256, 256] to transform the BEV features. Then, we use two DGCNNs to encode the object queries. Each DGCNN [8] contains two EdgeConv layers with dimensions [256, 256], both with 16 nearest neighbors. For each object query, we predict four points in the BEV to obtain and aggregate the BEV features. The final feature for this object query is the weighted sum of features of these four BEV points. The final detection head takes the features of each object query and predicts class label and bounding box parameters w.r.t. the reference point. ", + "bbox": [ + 174, + 746, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/b55ea507a22f297788549e7ad0c60972bbddf3b051ae3a8ac7b95d72258dc901.jpg", + "table_caption": [ + "Table 3: Comparisons of self-distillation versus baselines. " + ], + "table_footnote": [], + "table_body": "
MethodNDS ↑mAP↑mATEmASE↓mAOE↓mAVE↓mAAE↓
Baseline (pillar,without distillation)62.8053.2034.6226.5631.6226.0219.10
Self-distillation (pillar->pillar)63.4153.8934.2126.1931.1125.6718.54
Baseline (voxel,without distillation)66.0458.6233.3326.3428.8025.1119.06
Self-distillation (voxel->voxel)66.4559.2531.1725.7730.7325.7218.77
", + "bbox": [ + 178, + 104, + 820, + 170 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/3127971340c8ac67568bf68fe9b1c4ec413354755633754383992b5123eda9d0.jpg", + "table_caption": [ + "Table 4: Self-distillation with privileged information. " + ], + "table_footnote": [], + "table_body": "
MethodNDS ↑mAP↑mATE↓mASE↓mAOE↓mAVE↓mAAE↓
Sparse-→sparse (pillar)42.1238.8944.0127.7864.01144.0139.21
Dense-→sparse (pillar)42.7939.1043.8927.7764.01143.9739.11
Sparse-→sparse (voxel)59.5549.8431.1725.7733.7332.2220.22
Dense->sparse (voxel)59.8950.1231.1125.7633.7032.1920.11
", + "bbox": [ + 178, + 191, + 820, + 267 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 275, + 825, + 316 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Training & inference. We use AdamW [67] to train the model. The weight decay for AdamW is $1 0 ^ { - 2 }$ . Following a cyclic schedule [68], the learning rate is initially $1 0 ^ { - 4 }$ and gradually increased to $1 0 ^ { - 3 }$ , which is finally decreased to $1 0 ^ { - 8 }$ . The model is initialized with a pre-trained PointPillars network on the same dataset. We train for 20 epochs on 8 RTX 3090 GPUs. During inference, we take the top 100 objects with highest classification scores as the final predictions.We do not use any post-processing such as NMS. For evaluation, we use the toolkit provided with the nuScenes dataset. ", + "bbox": [ + 173, + 323, + 825, + 406 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.2 Object DGCNN ", + "text_level": 1, + "bbox": [ + 174, + 422, + 323, + 438 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare to top-performing methods on the nuScenes dataset in Table 1. PointPillars [4] is an anchor-based method with reasonable trade-off between performance and efficiency. FreeAnchor [64] extends PointPillars by learning how to assign anchors to the ground-truth. RegNetX-400MFSECFPN [65] uses neural architecture search (NAS) to learn a flexible neural network for 3D detection; it is essentially a variant of PointPillars with an enhanced backbone network. Different from anchor-based methods, Pillar-OD [5] makes predictions per pillar, alleviating the class imbalance issue caused by anchors. CenterPoint [6] exploits similar detection heads, with better performance using better training scheduling and data augmentation. For these methods, we use re-implementations in MMDetection3D [69], which match the performances in the original papers. ", + "bbox": [ + 174, + 448, + 825, + 573 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We mainly compare to CenterPoint with both PointPillars and SparseConv backbones, denoted as “voxel” and “pillar” respectively. Our method outperforms other methods significantly including CenterPoint with NMS. Without NMS, the performance of CenterPoint drops considerably while our method is unaffected by NMS. This finding verifies the DGCNN implicitly models object relations and removes redundant boxes. ", + "bbox": [ + 173, + 579, + 825, + 648 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "6.3 Set-to-set distillation ", + "text_level": 1, + "bbox": [ + 174, + 665, + 356, + 679 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this section, we present experiments involving our set-to-set distillation pipeline. We conduct three types of distillation. First, we distill a teacher model with a SparseConv backbone to a student model with a PointPillars backbone (denoted as “voxel pillar”). This aligns with the common knowledge distillation setup for classification. We compare to feature-based distillation and pseudo label based methods. The objective of feature-based distillation is to align the middle-level features of the teacher model and the student model while the pseudo label based methods generate pseudo training examples with the pre-trained teacher networks. As Table 2 shows, our set-to-set distillation achieves better performance, confirming that distilling the last stage of the object detection model is more effective than distilling feature maps. ", + "bbox": [ + 174, + 689, + 825, + 815 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Second, we perform self-distillation [49] (denoted as “voxel voxel” and “pillar pillar”), where the teacher and the student are identical and take the same point clouds as input. As Table 3 shows, even when the teacher network and the student network have the same capacity, self-distillation still introduces a performance boost. This finding is consistent with the results in [49]. ", + "bbox": [ + 174, + 820, + 825, + 877 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finally, we try distillation with privileged information [56], where the teacher gets access to privileged information but the student does not. Following [10], the teacher takes dense point clouds, and the student takes sparse point clouds (denoted as \"dense sparse\"). To limit computation time, we train each model over a shorter period of time. The goal is for the student model to learn the same representations as the teacher model without knowing the dense inputs. In Table 4, we compare this setup with self-distillation, where the difference is the teacher model and the student model take the same sparse point clouds in self-distillation. The student achieves better performance when the teacher takes dense point clouds. The result suggests that set-to-set knowledge distillation is an effective approach to transfer insight from privileged information. ", + "bbox": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 92, + 825, + 189 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6.4 Ablation ", + "text_level": 1, + "bbox": [ + 173, + 207, + 272, + 222 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We provide ablation studies on different components of our model to verify assorted design choices. First, we study the improvements of DGCNN over its counterpart, multi-head self-attention [9]. The multi-head self-attention has 8 heads with embedding dimension 256 and LayerNorm [70], following common usage. The DGCNN has two EdgeConv layers with dimensions [256, 256]. The number of neighbors $K$ in EdgeConv is 16. In principle, DGCNN is a sparse version of multi-head self-attention; the sparse structure reduces overhead in back-propagation and leads to sharper “attention maps” as well as faster convergence. ", + "bbox": [ + 173, + 234, + 825, + 332 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/d9c467f4bd8a8af5b677bcf0232a76978a3b9e3dd296b0053345336a127ba145.jpg", + "table_caption": [ + "Table 5: DGCNN versus multi-head self-attention. " + ], + "table_footnote": [], + "table_body": "
MethodMetricMulti-head self-attentionDGCNN
NDS39.8941.32
mAP36.3537.81
", + "bbox": [ + 176, + 358, + 498, + 407 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/74c9db5d83550347811f140bf5261a34e29781fe8e0e8af947b524f9b33c7717.jpg", + "table_caption": [ + "Table 6: Models with different # DGCNNs. " + ], + "table_footnote": [], + "table_body": "
#layersMetric123456
NDS35.9139.7541.1541.2641.0741.32
mAP32.3236.5437.2537.7537.7837.81
", + "bbox": [ + 176, + 430, + 496, + 470 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/c7f0743292b06fbc84dfef043abba3482f61b4b4beb5537e850ee5698b56f99d.jpg", + "table_caption": [ + "Table 7: The number of neighbors in DGCNN. " + ], + "table_footnote": [], + "table_body": "
# neigbhorsMetric148163264
NDS40.2140.4540.5141.3240.1739.80
mAP36.8137.1537.4637.8137.1236.70
", + "bbox": [ + 176, + 496, + 496, + 532 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/cf878a8434565d68ea11f6c198653db1951ad771b0d8058103cf87d64c8c7728.jpg", + "table_caption": [ + "Table 8: The distribution of the output scores with respect to overlapping boxes. " + ], + "table_footnote": [], + "table_body": "
Methodfiltered boxesremainingboxesall boxes
CenterPoint0.07640.18590.0829
Ours0.12220.17110.1604
", + "bbox": [ + 176, + 570, + 496, + 607 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/5ae78604979d0cd1e972fe644a43a4036ee6aedd810fe1fe12e313c8ec6a15ed.jpg", + "table_caption": [ + "Table 9: Complexity comparsion between DGCNN and Multi-head self-attention. " + ], + "table_footnote": [], + "table_body": "
ModuleComplexity# parameters
DGCNNO(n²d)262144
Multi-head self-attentionO(n²d)263168
", + "bbox": [ + 176, + 643, + 496, + 685 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 5 shows the comparisons: DGCNN consistently outperforms multi-head self-attention. This aligns with our hypothesis: objects are distributed sparsely in the scene, so dense interactions among objects are neither efficient nor effective. Furthermore, we study the effect of number of neighbors in DGCNN. When it is 1, The model reduces to an architecture without object interaction. As we increase the number, it approaches multi-head self-attention. As shown in Table 7, the sweet spot is 16, which appears to balance object interactions and sparsity. ", + "bbox": [ + 514, + 338, + 825, + 503 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We also investigate improvements introduced when more DGCNNs are stacked in Table 6. This result suggests it is beneficial to incorporate multiple DGCNNs to model the dynamic object relations. ", + "bbox": [ + 514, + 511, + 825, + 579 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Moreover, we hypothesize our method produces different distribution of the output scores with respect to overlapping boxes. To verify this hypothesis (Table 8, we compute average scores for three types of boxes: filtered boxes after NMS, remaining boxes after NMS, and all boxes. Below we show the results. We also compute the percentage of filtered boxes by NMS in our method and Centerpoint. In our method, $2 1 . 9 \\%$ ", + "bbox": [ + 514, + 585, + 825, + 710 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "boxes are removed while in Centerpoint $8 5 . 1 6 \\%$ boxes are filtered. Hence, we conclude that our method indeed exhibits a different distribution pattern from Centerpoint. ", + "bbox": [ + 176, + 710, + 820, + 738 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, we include a complexity comparison between DGCNN and Multi-head self-attention. Table 9 shows the results; DGCNN layer is on a par with Multi-head self-attention. ", + "bbox": [ + 176, + 744, + 823, + 772 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "7 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 795, + 299, + 811 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Object DGCNN is a highly-efficient 3D object detector for point clouds. It is able to learn object interactions via dynamic graphs and is optimized through a set-to-set loss, leading to NMS-free detection. The success of Object DGCNN indicates that many post-processing operations in 3D object detection are likely unnecessary and can be replaced with suitable neural network modules. Moreover, we introduce a set-to-set knowledge distillation pipeline enabled by the Object DGCNN. This new pipeline significantly simplifies knowledge distillation for 3D object detection and may be applicable to other tasks like 3D model compression. Beyond the direct usage of our model, our experiments suggest several future directions to address current limitations. For example, our method is initialized with a pre-trained backbone network. Training the model from scratch remains elusive due to the sparse set-to-set supervision; solving this issue may yield improved generalization as in [71]. Furthermore, studying 3D-specific feature extractors will improve the speed and generalizability of 3D object detection. Finally, the large amount of unlabeled data available at training time can serve as another type of privileged information to apply self-supervised learning to 3D domains through set-to-set distillation. ", + "bbox": [ + 174, + 827, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 825, + 202 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Potential impact. Our method aims to improve the object detection pipeline, which is crucial for the safety of autonomous driving systems. One potential negative impact of our work is that it still lacks theoretical guarantees, similar to many deep learning methods. Future work to improve applicability in this domain might consider challenges of explainability and transparency. ", + "bbox": [ + 174, + 208, + 825, + 263 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "8 Acknowledgement ", + "text_level": 1, + "bbox": [ + 174, + 282, + 359, + 300 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "The MIT Geometric Data Processing group acknowledges the generous support of Army Research Office grants W911NF2010168 and W911NF2110293, of Air Force Office of Scientific Research award FA9550-19-1-031, of National Science Foundation grants IIS-1838071 and CHS-1955697, from the CSAIL Systems that Learn program, from the MIT–IBM Watson AI Laboratory, from the Toyota–CSAIL Joint Research Center, from a gift from Adobe Systems, from an MIT.nano Immersion Lab/NCSOFT Gaming Program seed grant, and from the Skoltech–MIT Next Generation Program. 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MMDetection3D: OpenMMLab next-generation platform for general 3D object detection. https://github.com/open-mmlab/mmdetection3d, 2020. \n[70] Lei Jimmy Ba, Jamie Ryan Kiros, and Geoffrey E. Hinton. Layer normalization. CoRR, 2016. \n[71] Kaiming He, Ross Girshick, and Piotr Dollár. Rethinking imagenet pre-training. arXiv preprint arXiv:1811.08883, 2018. ", + "bbox": [ + 178, + 449, + 826, + 914 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 65, + 828, + 919 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 63, + 828, + 917 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 88, + 828, + 734 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Checklist ", + "text_level": 1, + "bbox": [ + 174, + 755, + 254, + 771 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "1. For all authors... 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Inspired by re-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 288, + 470, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 288, + 470, + 301 + ], + "score": 1.0, + "content": "cent non-maximum suppression-free 2D object detection models, we propose a", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 298, + 470, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 298, + 470, + 312 + ], + "score": 1.0, + "content": "3D object detection architecture on point clouds. Our method models 3D object", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 309, + 471, + 324 + ], + "spans": [ + { + "bbox": [ + 141, + 309, + 471, + 324 + ], + "score": 1.0, + "content": "detection as message passing on a dynamic graph, generalizing the DGCNN frame-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 321, + 470, + 333 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 470, + 333 + ], + "score": 1.0, + "content": "work to predict a set of objects. In our construction, we remove the necessity of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 333, + 471, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 333, + 471, + 344 + ], + "score": 1.0, + "content": "post-processing via object confidence aggregation or non-maximum suppression.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 342, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 342, + 470, + 356 + ], + "score": 1.0, + "content": "To facilitate object detection from sparse point clouds, we also propose a set-to-set", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 354, + 470, + 366 + ], + "spans": [ + { + "bbox": [ + 141, + 354, + 470, + 366 + ], + "score": 1.0, + "content": "distillation approach customized to 3D detection. This approach aligns the outputs", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 365, + 470, + 377 + ], + "spans": [ + { + "bbox": [ + 141, + 365, + 470, + 377 + ], + "score": 1.0, + "content": "of the teacher model and the student model in a permutation-invariant fashion, sig-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 376, + 469, + 387 + ], + "spans": [ + { + "bbox": [ + 142, + 376, + 469, + 387 + ], + "score": 1.0, + "content": "nificantly simplifying knowledge distillation for the 3D detection task. Our method", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 387, + 470, + 399 + ], + "spans": [ + { + "bbox": [ + 141, + 387, + 470, + 399 + ], + "score": 1.0, + "content": "achieves state-of-the-art performance on autonomous driving benchmarks. We also", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 397, + 451, + 410 + ], + "spans": [ + { + "bbox": [ + 141, + 397, + 451, + 410 + ], + "score": 1.0, + "content": "provide abundant analysis of the detection model and distillation framework.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15, + "bbox_fs": [ + 141, + 267, + 471, + 410 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 479 + ], + "lines": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 505, + 437 + ], + "score": 1.0, + "content": "Methods for 3D object detection have progressed rapidly, yielding deployable autonomous driving", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 506, + 448 + ], + "score": 1.0, + "content": "perception systems. Following common practice in 2D vision, 3D object detection often employs", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "complex training and testing pipelines including many post-processing operations to achieve superior", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 455, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 470 + ], + "score": 1.0, + "content": "performance. These operations are typically non-parallelizable and inefficient even with modern deep", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 467, + 485, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 467, + 485, + 480 + ], + "score": 1.0, + "content": "learning frameworks, implying a steep trade-off between between efficiency and effectiveness.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 423, + 506, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 483, + 505, + 571 + ], + "lines": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 497 + ], + "score": 1.0, + "content": "Modern methods usually employ two stages [1, 2], including a region proposal network [3] that can", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "introduce significant training overhead. Subsequent efforts simplify this pipeline for 3D object detec-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 519 + ], + "score": 1.0, + "content": "tion. PointPillars [4] introduces a one-stage anchor-based design, simplfying training. PillarOD [5]", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 507, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 507, + 529 + ], + "score": 1.0, + "content": "and CenterPoint [6] improve the one-stage model by making per-pillar predictions, that is, one pre-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "diction per point on the ground plane. They assign ground-truth bounding boxes to multiple outputs", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 538, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 551 + ], + "score": 1.0, + "content": "while training to ease optimization. However, they predict redundant boxes, which can overlap in the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 562 + ], + "score": 1.0, + "content": "same positions; extra boxes are eliminated a posteriori using non-maximum suppression (NMS). It", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 559, + 458, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 458, + 573 + ], + "score": 1.0, + "content": "remains elusive to remove hand-designed components like NMS in training and testing.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 483, + 507, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 505, + 653 + ], + "lines": [ + { + "bbox": [ + 105, + 576, + 507, + 589 + ], + "spans": [ + { + "bbox": [ + 105, + 576, + 507, + 589 + ], + "score": 1.0, + "content": "We introduce Object DGCNN, a streamlined architecture for 3D object detection from point clouds.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 586, + 507, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 507, + 601 + ], + "score": 1.0, + "content": "Like DETR for 2D object detection [7], we predict a set of bounding boxes from the raw data,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 506, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 506, + 612 + ], + "score": 1.0, + "content": "enabling an NMS-free pipeline that achieves real-time performance. A critical new component is to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 622 + ], + "score": 1.0, + "content": "treat each object query as a point in a set whose embedding is learned using DGCNN [8]. Compared", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 619, + 505, + 633 + ], + "score": 1.0, + "content": "to the self-attention module [9] in DETR, DGCNN leverages a sparse set of object relations, which", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 644 + ], + "score": 1.0, + "content": "reflects the real object distribution in the scene. In contrast to PointPillars [4], PillarOD [5], and", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 642, + 355, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 355, + 655 + ], + "score": 1.0, + "content": "CenterPoint [6], our method does not require post-processing.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 576, + 507, + 655 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 658, + 505, + 713 + ], + "lines": [ + { + "bbox": [ + 105, + 657, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 671 + ], + "score": 1.0, + "content": "We also provide a knowledge distillation approach customized to 3D object detection. Existing", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "score": 1.0, + "content": "methods typically distill dense feature maps from a teacher model to a student model, whose training", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 680, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 505, + 693 + ], + "score": 1.0, + "content": "objective does not necessarily capture 3D object detection performance [10]. In contrast, we propose", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 691, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 506, + 704 + ], + "score": 1.0, + "content": "set-to-set distillation training that aligns the outputs of the teacher and the student in a permutation-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 702, + 505, + 714 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 505, + 714 + ], + "score": 1.0, + "content": "invariant fashion. This process is enabled by the unified Object DGCNN architecture. In addition to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "obtaining better performance, through this process our model can benefit from privileged information", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 333, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 333, + 97 + ], + "score": 1.0, + "content": "(e.g., dense point clouds) only available at training time.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 657, + 506, + 714 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 504, + 95 + ], + "lines": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "spans": [ + { + "bbox": [ + 105, + 72, + 505, + 86 + ], + "score": 1.0, + "content": "obtaining better performance, through this process our model can benefit from privileged information", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 333, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 333, + 97 + ], + "score": 1.0, + "content": "(e.g., dense point clouds) only available at training time.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 366, + 111 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 367, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 367, + 113 + ], + "score": 1.0, + "content": "Contributions. We summarize our key contributions as follows:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 106, + 120, + 506, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 506, + 133 + ], + "score": 1.0, + "content": "• We propose a post-processing-free 3D object detection model achieving state-of-the-art perfor-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 112, + 131, + 404, + 144 + ], + "spans": [ + { + "bbox": [ + 112, + 131, + 404, + 144 + ], + "score": 1.0, + "content": "mance. To our knowledge, this is the first NMS-free 3D object detector.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 141, + 506, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 141, + 506, + 156 + ], + "score": 1.0, + "content": "• We generalize DGCNN to model objects as a point set. The DGCNN module outperforms its", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 113, + 153, + 336, + 166 + ], + "spans": [ + { + "bbox": [ + 113, + 153, + 336, + 166 + ], + "score": 1.0, + "content": "self-attention counterpart thanks to its sparse structure.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 163, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 163, + 505, + 177 + ], + "score": 1.0, + "content": "• We propose a set-to-set distillation method for 3D object detection. In our construction, knowledge", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 113, + 174, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 113, + 174, + 506, + 188 + ], + "score": 1.0, + "content": "distillation on object detection simply penalizes differences between the outputs of the teacher", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 113, + 185, + 235, + 197 + ], + "spans": [ + { + "bbox": [ + 113, + 185, + 235, + 197 + ], + "score": 1.0, + "content": "model and the student model.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 196, + 505, + 210 + ], + "spans": [ + { + "bbox": [ + 106, + 196, + 505, + 210 + ], + "score": 1.0, + "content": "• We show our model can use privileged information (such as dense point clouds) that is naturally", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 113, + 208, + 429, + 220 + ], + "spans": [ + { + "bbox": [ + 113, + 208, + 429, + 220 + ], + "score": 1.0, + "content": "available at training time to improve the model performance at inference time.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 218, + 392, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 218, + 392, + 232 + ], + "score": 1.0, + "content": "• We release our code to promote reproducibility and future research. 1", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 107, + 245, + 197, + 259 + ], + "lines": [ + { + "bbox": [ + 105, + 244, + 198, + 260 + ], + "spans": [ + { + "bbox": [ + 105, + 244, + 198, + 260 + ], + "score": 1.0, + "content": "1 Related Work", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 270, + 505, + 380 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 506, + 283 + ], + "score": 1.0, + "content": "2D object detection. Object recognition research has been transitioning from models with hand-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "crafted components to models with limited post-processing. One-stage detectors [11–13] remove the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 307 + ], + "score": 1.0, + "content": "complicated region proposal networks in two-stage objectors [3, 14], yielding more efficient training", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "and testing. Anchor-free methods [15, 16] further simplify the one-stage pipeline by shifting from", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 314, + 506, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 506, + 328 + ], + "score": 1.0, + "content": "per-anchor prediction to per-pixel prediction. However, these methods still make dense predictions", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 338 + ], + "score": 1.0, + "content": "and rely on NMS to reduce redundancy. To alleviate this issue, DETR [7] formulates object detection", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 506, + 349 + ], + "score": 1.0, + "content": "as a set-to-set prediction problem. It introduces a set-to-set loss that implicitly penalizes redundant", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "boxes, removing the necessity of post-processing. To accelerate convergence, Deformable DETR [17]", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "proposes deformable self-attention and streamlines the optimization process. Our method also", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 369, + 452, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 452, + 381 + ], + "score": 1.0, + "content": "formulates 3D object detection as set prediction, but with a customized design for 3D.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 385, + 505, + 398 + ], + "score": 1.0, + "content": "3D object detection. VoxelNet [18] generalizes one-stage object detection to 3D. 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To address that, PIXOR [19] and PointPillars [4] project points to a birds-eye", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 506, + 432 + ], + "score": 1.0, + "content": "view (BEV) and operate on 2D feature maps; PointNet [20] aggregates features within each BEV", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 505, + 442 + ], + "score": 1.0, + "content": "pixel. We use a variant of PointPillars [4] for 3D detection (§3). These methods are efficient but drop", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 441, + 505, + 452 + ], + "score": 1.0, + "content": "information along the vertical axis. 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Beyond SSD-style [11] one-stage models, Complex-YOLO [22]", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "extends YOLO to 3D for real-time perception. PointRCNN [23] employs a two-stage architecture for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "high-quality detection. 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VoteNet [30, 31] generalizes Hough voting [32] for 3D object detection in point clouds.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "score": 1.0, + "content": "DOPS [33] extends VoteNet and predicts 3D object shapes. In addition to visual input, [34] shows", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "that high-definition (HD) maps can boost performance of 3D object detectors. [35] argues that", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "multi-tasking can learn better representations than single-tasking. Beyond supervised learning, [36]", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 604, + 298, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 298, + 617 + ], + "score": 1.0, + "content": "learns a perception model for unknown classes.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "DGCNN. DGCNN [8] pioneered learning point cloud representations via dynamic graphs. It models", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 376, + 643 + ], + "score": 1.0, + "content": "point clouds as connected graphs, which are dynamically built using", + "type": "text" + }, + { + "bbox": [ + 376, + 631, + 383, + 641 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "-nearest neighbors in the latent", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 641, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 656 + ], + "score": 1.0, + "content": "space. DGCNN learns per-point features through message passing. 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KD compresses knowledge from an ensemble of models into a single", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 690, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 704 + ], + "score": 1.0, + "content": "smaller model [37]. [38] generalizes this idea and combines it with deep learning. KD transfers", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 120, + 712, + 288, + 722 + ], + "lines": [ + { + "bbox": [ + 120, + 710, + 290, + 724 + ], + "spans": [ + { + "bbox": [ + 120, + 710, + 290, + 724 + ], + "score": 1.0, + "content": "1https://github.com/WangYueFt/detr3d", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 310, + 753 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 8 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 504, + 95 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 72, + 505, + 97 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 100, + 366, + 111 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 367, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 367, + 113 + ], + "score": 1.0, + "content": "Contributions. We summarize our key contributions as follows:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 106, + 99, + 367, + 113 + ] + }, + { + "type": "list", + "bbox": [ + 106, + 120, + 506, + 231 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 506, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 506, + 133 + ], + "score": 1.0, + "content": "• We propose a post-processing-free 3D object detection model achieving state-of-the-art perfor-", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 112, + 131, + 404, + 144 + ], + "spans": [ + { + "bbox": [ + 112, + 131, + 404, + 144 + ], + "score": 1.0, + "content": "mance. 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Object recognition research has been transitioning from models with hand-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "crafted components to models with limited post-processing. One-stage detectors [11–13] remove the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 291, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 307 + ], + "score": 1.0, + "content": "complicated region proposal networks in two-stage objectors [3, 14], yielding more efficient training", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "and testing. 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It introduces a set-to-set loss that implicitly penalizes redundant", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "boxes, removing the necessity of post-processing. To accelerate convergence, Deformable DETR [17]", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "proposes deformable self-attention and streamlines the optimization process. 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Beyond SSD-style [11] one-stage models, Complex-YOLO [22]", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "extends YOLO to 3D for real-time perception. PointRCNN [23] employs a two-stage architecture for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 508 + ], + "score": 1.0, + "content": "high-quality detection. To improve representations of two-stage models, PVRCNN [2] proposes a", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 518 + ], + "score": 1.0, + "content": "point-voxel feature set abstraction layer to leverage the flexible receptive fields of PointNet-based", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "networks. Unlike works on point clouds, LaserNet [24] operates on raw range scans with comparable", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 526, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 541 + ], + "score": 1.0, + "content": "performance. [25–27] combine point clouds with camera images. Frustum-PointNet [28] leverages", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 537, + 507, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 507, + 551 + ], + "score": 1.0, + "content": "2D object detectors to form a frustum crop of points and then uses PointNet to aggregate features.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 505, + 561 + ], + "score": 1.0, + "content": "[29] describes an end-to-end learnable architecture that exploits continuous convolutions to fuse", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 560, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 506, + 572 + ], + "score": 1.0, + "content": "feature maps. VoteNet [30, 31] generalizes Hough voting [32] for 3D object detection in point clouds.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 506, + 584 + ], + "score": 1.0, + "content": "DOPS [33] extends VoteNet and predicts 3D object shapes. In addition to visual input, [34] shows", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 506, + 595 + ], + "score": 1.0, + "content": "that high-definition (HD) maps can boost performance of 3D object detectors. [35] argues that", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "multi-tasking can learn better representations than single-tasking. Beyond supervised learning, [36]", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 604, + 298, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 298, + 617 + ], + "score": 1.0, + "content": "learns a perception model for unknown classes.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 34, + "bbox_fs": [ + 104, + 385, + 507, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 620, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 505, + 633 + ], + "score": 1.0, + "content": "DGCNN. DGCNN [8] pioneered learning point cloud representations via dynamic graphs. It models", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 631, + 505, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 376, + 643 + ], + "score": 1.0, + "content": "point clouds as connected graphs, which are dynamically built using", + "type": "text" + }, + { + "bbox": [ + 376, + 631, + 383, + 641 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 631, + 505, + 643 + ], + "score": 1.0, + "content": "-nearest neighbors in the latent", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 641, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 641, + 506, + 656 + ], + "score": 1.0, + "content": "space. DGCNN learns per-point features through message passing. However, it operates on point", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "spans": [ + { + "bbox": [ + 106, + 653, + 505, + 665 + ], + "score": 1.0, + "content": "clouds for single object recognition and semantic segmentation. One of our key contributions is to", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 664, + 406, + 677 + ], + "spans": [ + { + "bbox": [ + 106, + 664, + 406, + 677 + ], + "score": 1.0, + "content": "generalize DGCNN to model scene-level object relations for 3D detection.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 620, + 506, + 677 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 680, + 503, + 702 + ], + "lines": [ + { + "bbox": [ + 105, + 679, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 505, + 693 + ], + "score": 1.0, + "content": "Knowledge distillation (KD). KD compresses knowledge from an ensemble of models into a single", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 690, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 704 + ], + "score": 1.0, + "content": "smaller model [37]. [38] generalizes this idea and combines it with deep learning. KD transfers", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "knowledge from a teacher model to a student model by minimizing a loss, in which the target is the", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "distribution of class probabilities induced by the teacher. [39–50] improve knowledge distillation for", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 107 + ], + "score": 1.0, + "content": "classification. Beyond image classification, KD has been extended to improve object detection. [51]", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "leverages FitNets for object detection, addressing obstacles such as class imbalance, loss instability,", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "and feature distribution mismatch. [52] distills between region proposals, accelerating training with", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "added instability. To address this issue, [53] uses fine-grained representation imitation using object", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 348, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 348, + 150 + ], + "score": 1.0, + "content": "masks. [54] uses KD to tackle a continual learning problem.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 679, + 505, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "knowledge from a teacher model to a student model by minimizing a loss, in which the target is the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "distribution of class probabilities induced by the teacher. [39–50] improve knowledge distillation for", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 95, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 95, + 505, + 107 + ], + "score": 1.0, + "content": "classification. Beyond image classification, KD has been extended to improve object detection. [51]", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 105, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 104, + 105, + 506, + 119 + ], + "score": 1.0, + "content": "leverages FitNets for object detection, addressing obstacles such as class imbalance, loss instability,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 116, + 505, + 129 + ], + "score": 1.0, + "content": "and feature distribution mismatch. [52] distills between region proposals, accelerating training with", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 141 + ], + "score": 1.0, + "content": "added instability. To address this issue, [53] uses fine-grained representation imitation using object", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 348, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 348, + 150 + ], + "score": 1.0, + "content": "masks. [54] uses KD to tackle a continual learning problem.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "Privileged information. [55] introduces the framework of learning with privileged information in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "the context of support vector machines (SVMs), wherein additional information is accessible during", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "training but not testing. [56] unifies KD and learning using privileged information theoretically. [57]", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "identifies practical applications, e.g., transferring knowledge from localized data to non-localized", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "data, from high resolution to low resolution, from color images to edge images, and from regular", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "images to distorted images. To mediate uncertainty and improve training efficiency, [58] makes the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "variance of Dropout [59] a function of privileged information. We extend these methods to 3D data,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 231, + 503, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 503, + 244 + ], + "score": 1.0, + "content": "in which privileged information consists of dense point clouds aggregated from LiDAR sequences.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 107, + 257, + 174, + 271 + ], + "lines": [ + { + "bbox": [ + 104, + 255, + 177, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 255, + 177, + 275 + ], + "score": 1.0, + "content": "2 Overview", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 505, + 294 + ], + "score": 1.0, + "content": "Our target application of object detection differs from the recognition and segmentation tasks", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "considered for DGCNN. Our point clouds typically contain too many points to apply DGCNN and its", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 507, + 317 + ], + "score": 1.0, + "content": "peers directly to the entire scene. Moreover, the size of our output set, a small set of bounding boxes,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 362, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 345, + 328 + ], + "score": 1.0, + "content": "differs from the size of our input set, a huge set of points in", + "type": "text" + }, + { + "bbox": [ + 345, + 315, + 358, + 325 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 315, + 362, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "Following state-of-the-art in large-scale object detection, our pipeline learns a grid-based intermediate", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "representation to capture local features (§3). We test two standard learning-based methods for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 354, + 504, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 504, + 366 + ], + "score": 1.0, + "content": "collecting local point cloud features on a birds-eye view (BEV) grid. While in principle it might be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "possible to avoid grids entirely in our pipeline, this BEV representation is far more efficient and—as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "observed in previous work—is sufficient to find objects reliably in autonomous driving, where there", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 387, + 391, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 391, + 399 + ], + "score": 1.0, + "content": "is likely only one object above any given grid cell on the ground plane.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "Our main architecture contribution is the Object DGCNN pipeline (§4), which transitions from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "this BEV grid of features to a set of object bounding boxes. Object DGCNN draws inspiration", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 507, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 495, + 438 + ], + "score": 1.0, + "content": "from the DGCNN architecture; its layers alternate between local feature transformations and", + "type": "text" + }, + { + "bbox": [ + 495, + 425, + 502, + 434 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 424, + 507, + 438 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 447 + ], + "score": 1.0, + "content": "nearest neighbor aggregation to capture relationships between objects. Unlike conventional DGCNN,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 445, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 460 + ], + "score": 1.0, + "content": "however, Object DGCNN incorporates features from the BEV grid in each of its layers; each layer", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "incorporates several queries into the BEV to refine object position estimates. The output of Object", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 467, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 482 + ], + "score": 1.0, + "content": "DGCNN is a set of objects in the scene. We use a permutation-invariant loss (10) to measure", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 478, + 298, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 298, + 492 + ], + "score": 1.0, + "content": "divergence from the ground truth set of objects.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "The pipeline above does not require hand-designed post-processing like NMS; our output boxes are", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "usable directly for object detection. Beyond simplifying the object detection pipeline, this allows us", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 342, + 531 + ], + "score": 1.0, + "content": "to propose object detection-specific distillation procedures", + "type": "text" + }, + { + "bbox": [ + 343, + 518, + 367, + 528 + ], + "score": 0.28, + "content": "( \\ S 6 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "that further improve performance.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "These use one network to train another, e.g., to train a network operating on sparse point clouds to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "output features that imitate those learned by a network trained on denser, more detailed point clouds.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 565, + 201, + 579 + ], + "lines": [ + { + "bbox": [ + 104, + 563, + 204, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 204, + 582 + ], + "score": 1.0, + "content": "3 Local Features", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 231, + 605 + ], + "score": 1.0, + "content": "We begin with a point cloud", + "type": "text" + }, + { + "bbox": [ + 232, + 590, + 379, + 603 + ], + "score": 0.9, + "content": "\\mathcal { X } \\ = \\ \\{ \\pmb { x } _ { 1 } , \\dots , \\pmb { x } _ { i } , \\dots , \\pmb { x } _ { N } \\} \\ \\subset \\ \\mathbb { R } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 588, + 481, + 605 + ], + "score": 1.0, + "content": "with per-point features", + "type": "text" + }, + { + "bbox": [ + 481, + 591, + 505, + 601 + ], + "score": 0.86, + "content": "{ \\mathcal { F } } =", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 597, + 507, + 616 + ], + "spans": [ + { + "bbox": [ + 107, + 601, + 226, + 614 + ], + "score": 0.89, + "content": "\\{ \\pmb { f } _ { 1 } , \\ldots , \\pmb { f } _ { i } , \\ldots , \\pmb { \\bar { f } } _ { N } \\} \\subset \\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 597, + 351, + 616 + ], + "score": 1.0, + "content": ", ground-truth bounding boxes", + "type": "text" + }, + { + "bbox": [ + 351, + 601, + 484, + 614 + ], + "score": 0.92, + "content": "\\boldsymbol { B } = \\{ b _ { 1 } , \\dots , \\bar { b } _ { j } , \\dots , b _ { M } \\} \\subset \\mathbb { R } ^ { 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 597, + 507, + 616 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 611, + 507, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 180, + 627 + ], + "score": 1.0, + "content": "categorical labels", + "type": "text" + }, + { + "bbox": [ + 181, + 613, + 308, + 624 + ], + "score": 0.91, + "content": "\\bar { \\mathcal { C } } \\overset { \\cdot } { = } \\{ c _ { j } , \\dotsc , c _ { j } , \\dotsc , c _ { M } \\} \\subset \\mathbb { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 611, + 335, + 627 + ], + "score": 1.0, + "content": ". Each", + "type": "text" + }, + { + "bbox": [ + 336, + 613, + 347, + 625 + ], + "score": 0.87, + "content": "b _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 611, + 507, + 627 + ], + "score": 1.0, + "content": "contains position, size, heading angle,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "and velocity in the birds-eye view (BEV); our architecture aims to predict these boxes and their labels", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 633, + 256, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 256, + 647 + ], + "score": 1.0, + "content": "from the point cloud and its features.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 662 + ], + "score": 1.0, + "content": "As an initial step, modern 3D object detection models scatter points into either BEV pillars or", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 660, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 675 + ], + "score": 1.0, + "content": "3D voxels and then use convolutional neural networks to extract features on a grid. This strategy", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "accelerates object detection for large point clouds. We test two neural network architectures for BEV", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 683, + 244, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 244, + 695 + ], + "score": 1.0, + "content": "feature extraction, detailed below.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5 + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "PointPillars [4] maps sparse point clouds onto a dense BEV pillar map on which 2D convolutions can", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 189, + 724 + ], + "score": 1.0, + "content": "be applied. Suppose", + "type": "text" + }, + { + "bbox": [ + 189, + 711, + 214, + 723 + ], + "score": 0.92, + "content": "F _ { P } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 710, + 321, + 724 + ], + "score": 1.0, + "content": "returns the points in pillar", + "type": "text" + }, + { + "bbox": [ + 321, + 712, + 325, + 721 + ], + "score": 0.8, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 710, + 505, + 724 + ], + "score": 1.0, + "content": ", that is, the set of points in a vertical column", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 309, + 752 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 150 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 104, + 73, + 506, + 150 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 154, + 505, + 243 + ], + "lines": [ + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 155, + 505, + 167 + ], + "score": 1.0, + "content": "Privileged information. [55] introduces the framework of learning with privileged information in", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 179 + ], + "score": 1.0, + "content": "the context of support vector machines (SVMs), wherein additional information is accessible during", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 505, + 189 + ], + "score": 1.0, + "content": "training but not testing. [56] unifies KD and learning using privileged information theoretically. [57]", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "identifies practical applications, e.g., transferring knowledge from localized data to non-localized", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "data, from high resolution to low resolution, from color images to edge images, and from regular", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "images to distorted images. To mediate uncertainty and improve training efficiency, [58] makes the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 232 + ], + "score": 1.0, + "content": "variance of Dropout [59] a function of privileged information. We extend these methods to 3D data,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 231, + 503, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 503, + 244 + ], + "score": 1.0, + "content": "in which privileged information consists of dense point clouds aggregated from LiDAR sequences.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 155, + 506, + 244 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 257, + 174, + 271 + ], + "lines": [ + { + "bbox": [ + 104, + 255, + 177, + 275 + ], + "spans": [ + { + "bbox": [ + 104, + 255, + 177, + 275 + ], + "score": 1.0, + "content": "2 Overview", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 107, + 282, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 106, + 283, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 283, + 505, + 294 + ], + "score": 1.0, + "content": "Our target application of object detection differs from the recognition and segmentation tasks", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 306 + ], + "score": 1.0, + "content": "considered for DGCNN. Our point clouds typically contain too many points to apply DGCNN and its", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 304, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 507, + 317 + ], + "score": 1.0, + "content": "peers directly to the entire scene. Moreover, the size of our output set, a small set of bounding boxes,", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 362, + 328 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 345, + 328 + ], + "score": 1.0, + "content": "differs from the size of our input set, a huge set of points in", + "type": "text" + }, + { + "bbox": [ + 345, + 315, + 358, + 325 + ], + "score": 0.88, + "content": "\\mathbb { R } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 315, + 362, + 328 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 283, + 507, + 328 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 331, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 344 + ], + "score": 1.0, + "content": "Following state-of-the-art in large-scale object detection, our pipeline learns a grid-based intermediate", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 506, + 354 + ], + "score": 1.0, + "content": "representation to capture local features (§3). We test two standard learning-based methods for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 354, + 504, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 504, + 366 + ], + "score": 1.0, + "content": "collecting local point cloud features on a birds-eye view (BEV) grid. While in principle it might be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "possible to avoid grids entirely in our pipeline, this BEV representation is far more efficient and—as", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 505, + 388 + ], + "score": 1.0, + "content": "observed in previous work—is sufficient to find objects reliably in autonomous driving, where there", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 387, + 391, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 391, + 399 + ], + "score": 1.0, + "content": "is likely only one object above any given grid cell on the ground plane.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 331, + 506, + 399 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 402, + 505, + 490 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "Our main architecture contribution is the Object DGCNN pipeline (§4), which transitions from", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 505, + 426 + ], + "score": 1.0, + "content": "this BEV grid of features to a set of object bounding boxes. Object DGCNN draws inspiration", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 424, + 507, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 495, + 438 + ], + "score": 1.0, + "content": "from the DGCNN architecture; its layers alternate between local feature transformations and", + "type": "text" + }, + { + "bbox": [ + 495, + 425, + 502, + 434 + ], + "score": 0.75, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 424, + 507, + 438 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 435, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 506, + 447 + ], + "score": 1.0, + "content": "nearest neighbor aggregation to capture relationships between objects. Unlike conventional DGCNN,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 445, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 460 + ], + "score": 1.0, + "content": "however, Object DGCNN incorporates features from the BEV grid in each of its layers; each layer", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "incorporates several queries into the BEV to refine object position estimates. The output of Object", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 467, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 482 + ], + "score": 1.0, + "content": "DGCNN is a set of objects in the scene. We use a permutation-invariant loss (10) to measure", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 478, + 298, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 298, + 492 + ], + "score": 1.0, + "content": "divergence from the ground truth set of objects.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 402, + 507, + 492 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 550 + ], + "lines": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "The pipeline above does not require hand-designed post-processing like NMS; our output boxes are", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 519 + ], + "score": 1.0, + "content": "usable directly for object detection. Beyond simplifying the object detection pipeline, this allows us", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 517, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 342, + 531 + ], + "score": 1.0, + "content": "to propose object detection-specific distillation procedures", + "type": "text" + }, + { + "bbox": [ + 343, + 518, + 367, + 528 + ], + "score": 0.28, + "content": "( \\ S 6 . 3 )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 517, + 506, + 531 + ], + "score": 1.0, + "content": "that further improve performance.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "These use one network to train another, e.g., to train a network operating on sparse point clouds to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "output features that imitate those learned by a network trained on denser, more detailed point clouds.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 496, + 506, + 552 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 565, + 201, + 579 + ], + "lines": [ + { + "bbox": [ + 104, + 563, + 204, + 582 + ], + "spans": [ + { + "bbox": [ + 104, + 563, + 204, + 582 + ], + "score": 1.0, + "content": "3 Local Features", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 107, + 590, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 605 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 231, + 605 + ], + "score": 1.0, + "content": "We begin with a point cloud", + "type": "text" + }, + { + "bbox": [ + 232, + 590, + 379, + 603 + ], + "score": 0.9, + "content": "\\mathcal { X } \\ = \\ \\{ \\pmb { x } _ { 1 } , \\dots , \\pmb { x } _ { i } , \\dots , \\pmb { x } _ { N } \\} \\ \\subset \\ \\mathbb { R } ^ { 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 588, + 481, + 605 + ], + "score": 1.0, + "content": "with per-point features", + "type": "text" + }, + { + "bbox": [ + 481, + 591, + 505, + 601 + ], + "score": 0.86, + "content": "{ \\mathcal { F } } =", + "type": "inline_equation" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 597, + 507, + 616 + ], + "spans": [ + { + "bbox": [ + 107, + 601, + 226, + 614 + ], + "score": 0.89, + "content": "\\{ \\pmb { f } _ { 1 } , \\ldots , \\pmb { f } _ { i } , \\ldots , \\pmb { \\bar { f } } _ { N } \\} \\subset \\mathbb { R } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 597, + 351, + 616 + ], + "score": 1.0, + "content": ", ground-truth bounding boxes", + "type": "text" + }, + { + "bbox": [ + 351, + 601, + 484, + 614 + ], + "score": 0.92, + "content": "\\boldsymbol { B } = \\{ b _ { 1 } , \\dots , \\bar { b } _ { j } , \\dots , b _ { M } \\} \\subset \\mathbb { R } ^ { 9 }", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 597, + 507, + 616 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 611, + 507, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 180, + 627 + ], + "score": 1.0, + "content": "categorical labels", + "type": "text" + }, + { + "bbox": [ + 181, + 613, + 308, + 624 + ], + "score": 0.91, + "content": "\\bar { \\mathcal { C } } \\overset { \\cdot } { = } \\{ c _ { j } , \\dotsc , c _ { j } , \\dotsc , c _ { M } \\} \\subset \\mathbb { Z }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 611, + 335, + 627 + ], + "score": 1.0, + "content": ". Each", + "type": "text" + }, + { + "bbox": [ + 336, + 613, + 347, + 625 + ], + "score": 0.87, + "content": "b _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 611, + 507, + 627 + ], + "score": 1.0, + "content": "contains position, size, heading angle,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "and velocity in the birds-eye view (BEV); our architecture aims to predict these boxes and their labels", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 633, + 256, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 256, + 647 + ], + "score": 1.0, + "content": "from the point cloud and its features.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 588, + 507, + 647 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 505, + 694 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 662 + ], + "score": 1.0, + "content": "As an initial step, modern 3D object detection models scatter points into either BEV pillars or", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 660, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 675 + ], + "score": 1.0, + "content": "3D voxels and then use convolutional neural networks to extract features on a grid. This strategy", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 673, + 505, + 685 + ], + "score": 1.0, + "content": "accelerates object detection for large point clouds. We test two neural network architectures for BEV", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 683, + 244, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 683, + 244, + 695 + ], + "score": 1.0, + "content": "feature extraction, detailed below.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 651, + 505, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "PointPillars [4] maps sparse point clouds onto a dense BEV pillar map on which 2D convolutions can", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 189, + 724 + ], + "score": 1.0, + "content": "be applied. Suppose", + "type": "text" + }, + { + "bbox": [ + 189, + 711, + 214, + 723 + ], + "score": 0.92, + "content": "F _ { P } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 710, + 321, + 724 + ], + "score": 1.0, + "content": "returns the points in pillar", + "type": "text" + }, + { + "bbox": [ + 321, + 712, + 325, + 721 + ], + "score": 0.8, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 710, + 505, + 724 + ], + "score": 1.0, + "content": ", that is, the set of points in a vertical column", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 107, + 220, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 107, + 220, + 155, + 231 + ], + "score": 1.0, + "content": "above point", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 155, + 221, + 160, + 230 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 160, + 220, + 505, + 231 + ], + "score": 1.0, + "content": "on the ground. When collecting features from points to pillars, multiple points can fall", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 231, + 467, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 467, + 243 + ], + "score": 1.0, + "content": "into the same pillar. In this case, PointNet [20] (PN) is used to obtain pillar-wise features:", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 699, + 505, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 107, + 69, + 505, + 173 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 69, + 505, + 173 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 69, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 107, + 69, + 505, + 173 + ], + "score": 0.946, + "type": "image", + "image_path": "4961b1385646350b25971984d745696f6c00c96db1551f123510f35addb612d1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 107, + 69, + 505, + 103.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 107, + 103.66666666666666, + 505, + 138.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 107, + 138.33333333333331, + 505, + 172.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 177, + 504, + 200 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 177, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 413, + 190 + ], + "score": 1.0, + "content": "Figure 1: Overview. Point cloud features are learned in BEV, followed by", + "type": "text" + }, + { + "bbox": [ + 414, + 178, + 422, + 187 + ], + "score": 0.58, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 177, + 505, + 190 + ], + "score": 1.0, + "content": "DGCNNs to model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 188, + 486, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 188, + 486, + 202 + ], + "score": 1.0, + "content": "object relations. We predict a set of bounding boxes and compute loss in a one-to-one manner.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 106, + 219, + 504, + 242 + ], + "lines": [ + { + "bbox": [ + 107, + 220, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 107, + 220, + 155, + 231 + ], + "score": 1.0, + "content": "above point", + "type": "text" + }, + { + "bbox": [ + 155, + 221, + 160, + 230 + ], + "score": 0.66, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 220, + 505, + 231 + ], + "score": 1.0, + "content": "on the ground. When collecting features from points to pillars, multiple points can fall", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 231, + 467, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 467, + 243 + ], + "score": 1.0, + "content": "into the same pillar. In this case, PointNet [20] (PN) is used to obtain pillar-wise features:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "interline_equation", + "bbox": [ + 234, + 247, + 376, + 263 + ], + "lines": [ + { + "bbox": [ + 234, + 247, + 376, + 263 + ], + "spans": [ + { + "bbox": [ + 234, + 247, + 376, + 263 + ], + "score": 0.92, + "content": "\\pmb { f } _ { i } ^ { \\mathrm { p i l l a r } } = \\operatorname { P N } ( \\{ \\pmb { f } _ { j } | \\pmb { x } _ { j } \\in F _ { P } ( \\pmb { p } _ { i } ) \\} ) ,", + "type": "interline_equation", + "image_path": "400f1b399e37245a01345e45230268dc050714e77a078e0d1c0512f1a000a2ec.jpg" + } + ] + } + ], + "index": 7, + "virtual_lines": [ + { + "bbox": [ + 234, + 247, + 376, + 263 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 505, + 317 + ], + "lines": [ + { + "bbox": [ + 104, + 265, + 507, + 284 + ], + "spans": [ + { + "bbox": [ + 104, + 265, + 134, + 284 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 268, + 160, + 282 + ], + "score": 0.92, + "content": "f _ { i } ^ { \\mathrm { p i l l a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 265, + 253, + 284 + ], + "score": 1.0, + "content": "is the feature of pillar", + "type": "text" + }, + { + "bbox": [ + 253, + 271, + 264, + 281 + ], + "score": 0.84, + "content": "\\mathbf { \\nabla } _ { \\pmb { p } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 265, + 507, + 284 + ], + "score": 1.0, + "content": ". We set the features for empty pillars to 0. This results in", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 102, + 277, + 508, + 296 + ], + "spans": [ + { + "bbox": [ + 102, + 277, + 174, + 296 + ], + "score": 1.0, + "content": "a dense 2D grid", + "type": "text" + }, + { + "bbox": [ + 174, + 281, + 270, + 292 + ], + "score": 0.9, + "content": "\\mathcal { F } ^ { \\mathrm { p i l l a r } } \\subset \\mathbb { R } ^ { H ^ { \\mathrm { p } } \\times W ^ { \\mathrm { p } } \\times C ^ { \\mathrm { p } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 277, + 301, + 296 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 302, + 282, + 337, + 293 + ], + "score": 0.38, + "content": "H ^ { \\mathrm { p } } , W ^ { \\mathrm { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 277, + 356, + 296 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 356, + 282, + 370, + 292 + ], + "score": 0.85, + "content": "C ^ { \\mathrm { p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 277, + 508, + 296 + ], + "score": 1.0, + "content": "are the height, width, number of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 306 + ], + "score": 1.0, + "content": "channels of this 2D pillar map, respectively. Multiple stacked convolutional layers further embed the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 303, + 345, + 319 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 345, + 319 + ], + "score": 1.0, + "content": "pillar features to the final feature map F d ⊂ RHd×W d×Cd .", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 105, + 322, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 502, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 334, + 336 + ], + "score": 1.0, + "content": "An alternative BEV embedding is SparseConv [60]. If", + "type": "text" + }, + { + "bbox": [ + 334, + 322, + 360, + 335 + ], + "score": 0.92, + "content": "F _ { V } ( i )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 321, + 497, + 336 + ], + "score": 1.0, + "content": "returns the set of points in voxel", + "type": "text" + }, + { + "bbox": [ + 498, + 324, + 502, + 333 + ], + "score": 0.52, + "content": "i", + "type": "inline_equation" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 332, + 379, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 379, + 347 + ], + "score": 1.0, + "content": "SparseConv collects point-wise features into voxel-wise features by", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "interline_equation", + "bbox": [ + 237, + 349, + 373, + 366 + ], + "lines": [ + { + "bbox": [ + 237, + 349, + 373, + 366 + ], + "spans": [ + { + "bbox": [ + 237, + 349, + 373, + 366 + ], + "score": 0.92, + "content": "\\pmb { f } _ { i } ^ { \\mathrm { v o x e l } } = \\mathrm { P N } ( \\{ \\pmb { f } _ { j } | \\pmb { x } _ { j } \\in F _ { V } ( i ) \\} ) ,", + "type": "interline_equation", + "image_path": "a8ee7f4d6af13bfb1a1f1b0568dbc81c6dba4bd9c7ddf206e0d5634a489a66d5.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 237, + 349, + 373, + 366 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 371, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 102, + 369, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 102, + 369, + 133, + 388 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 371, + 160, + 384 + ], + "score": 0.91, + "content": "f _ { i } ^ { \\mathrm { v o x e l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 369, + 282, + 388 + ], + "score": 1.0, + "content": "contains the features of voxel", + "type": "text" + }, + { + "bbox": [ + 282, + 373, + 287, + 382 + ], + "score": 0.71, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 369, + 505, + 388 + ], + "score": 1.0, + "content": ". In contrast to PointPillars, SparseConv conducts 3D", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "sparse convolutions to refine the voxel-wise features. Finally, we compress these sparse voxels to a", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 394, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 393, + 406 + ], + "score": 1.0, + "content": "BEV 2D grid by filling empty voxels with zeros and averaging along the", + "type": "text" + }, + { + "bbox": [ + 394, + 396, + 400, + 404 + ], + "score": 0.78, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 394, + 506, + 406 + ], + "score": 1.0, + "content": "-axis. For ease of notation,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 403, + 338, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 338, + 420 + ], + "score": 1.0, + "content": "we also denote the resulting 2D grid F d ⊂ RHd×W d×Cd .", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "title", + "bbox": [ + 107, + 433, + 208, + 447 + ], + "lines": [ + { + "bbox": [ + 104, + 432, + 209, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 209, + 451 + ], + "score": 1.0, + "content": "4 Object DGCNN", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 104, + 456, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 249, + 471 + ], + "score": 1.0, + "content": "After obtaining the BEV features", + "type": "text" + }, + { + "bbox": [ + 249, + 457, + 263, + 468 + ], + "score": 0.87, + "content": "{ \\mathcal { F } } ^ { \\mathrm { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 456, + 506, + 471 + ], + "score": 1.0, + "content": "using one of the architectures above, we predict a set of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "bounding boxes as well as a label for each box. The key difference between our architecture and most", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "recent 3D object detection methods is that ours produces a set of bounding boxes rather than a box", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 104, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "per grid cell followed by NMS, as in [5, 6]. Hence, we need to transition from a grid of per-pillar", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 502, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 514 + ], + "score": 1.0, + "content": "features to an unordered set of objects; we detail our approach below. We address two key issues:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 513, + 347, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 347, + 525 + ], + "score": 1.0, + "content": "prediction of the bounding boxes and evaluation of the loss.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 506, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Desiderata. Object DGCNN uses a DGCNN-inspired architecture but incorporates grid-based BEV", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "features, built on the philosophy that local features (§3) are reasonable to store on a dense grid, but", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "object predictions are better modeled using sets. 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Multiple stacked convolutional layers further embed the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 303, + 345, + 319 + ], + "spans": [ + { + "bbox": [ + 104, + 303, + 345, + 319 + ], + "score": 1.0, + "content": "pillar features to the final feature map F d ⊂ RHd×W d×Cd .", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5, + "bbox_fs": [ + 102, + 265, + 508, + 319 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 322, + 505, + 345 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 502, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 334, + 336 + ], + "score": 1.0, + "content": "An alternative BEV embedding is SparseConv [60]. 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For ease of notation,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 403, + 338, + 420 + ], + "spans": [ + { + "bbox": [ + 104, + 403, + 338, + 420 + ], + "score": 1.0, + "content": "we also denote the resulting 2D grid F d ⊂ RHd×W d×Cd .", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 102, + 369, + 506, + 420 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 433, + 208, + 447 + ], + "lines": [ + { + "bbox": [ + 104, + 432, + 209, + 451 + ], + "spans": [ + { + "bbox": [ + 104, + 432, + 209, + 451 + ], + "score": 1.0, + "content": "4 Object DGCNN", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 457, + 505, + 524 + ], + "lines": [ + { + "bbox": [ + 104, + 456, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 104, + 456, + 249, + 471 + ], + "score": 1.0, + "content": "After obtaining the BEV features", + "type": "text" + }, + { + "bbox": [ + 249, + 457, + 263, + 468 + ], + "score": 0.87, + "content": "{ \\mathcal { F } } ^ { \\mathrm { d } }", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 456, + 506, + 471 + ], + "score": 1.0, + "content": "using one of the architectures above, we predict a set of", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 506, + 482 + ], + "score": 1.0, + "content": "bounding boxes as well as a label for each box. The key difference between our architecture and most", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "recent 3D object detection methods is that ours produces a set of bounding boxes rather than a box", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 490, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 104, + 490, + 506, + 504 + ], + "score": 1.0, + "content": "per grid cell followed by NMS, as in [5, 6]. Hence, we need to transition from a grid of per-pillar", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 502, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 506, + 514 + ], + "score": 1.0, + "content": "features to an unordered set of objects; we detail our approach below. We address two key issues:", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 513, + 347, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 513, + 347, + 525 + ], + "score": 1.0, + "content": "prediction of the bounding boxes and evaluation of the loss.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 456, + 506, + 525 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 506, + 573 + ], + "lines": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 541 + ], + "score": 1.0, + "content": "Desiderata. Object DGCNN uses a DGCNN-inspired architecture but incorporates grid-based BEV", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 506, + 552 + ], + "score": 1.0, + "content": "features, built on the philosophy that local features (§3) are reasonable to store on a dense grid, but", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "object predictions are better modeled using sets. Hence, we require a new architecture and set-to-set", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 561, + 282, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 282, + 574 + ], + "score": 1.0, + "content": "loss that encourage bounding box diversity.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 529, + 506, + 574 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 578, + 504, + 601 + ], + "lines": [ + { + "bbox": [ + 106, + 577, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 577, + 194, + 591 + ], + "score": 1.0, + "content": "Object DGCNN uses", + "type": "text" + }, + { + "bbox": [ + 195, + 579, + 203, + 588 + ], + "score": 0.77, + "content": "L", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 577, + 505, + 591 + ], + "score": 1.0, + "content": "layers that follow a series of set-based computations to produce bounding", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 588, + 491, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 491, + 602 + ], + "score": 1.0, + "content": "box predictions from the BEV feature maps. Each layer employs the following steps (Figure 1):", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 577, + 505, + 602 + ] + }, + { + "type": "index", + "bbox": [ + 106, + 609, + 393, + 642 + ], + "lines": [ + { + "bbox": [ + 106, + 608, + 323, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 608, + 323, + 621 + ], + "score": 1.0, + "content": "1. predict a set of query points and attention weights;", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 618, + 393, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 618, + 393, + 633 + ], + "score": 1.0, + "content": "2. collect BEV features from keypoints determined by the queries; and", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 105, + 631, + 305, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 305, + 643 + ], + "score": 1.0, + "content": "3. model object-object interactions via DGCNN.", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + } + ], + "index": 33, + "bbox_fs": [ + 105, + 608, + 393, + 643 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 650, + 504, + 684 + ], + "lines": [ + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 506, + 663 + ], + "score": 1.0, + "content": "Each layer results in a more refined set of bounding box predictions, one per query. At the end", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 505, + 674 + ], + "score": 1.0, + "content": "of these layers, we match the prediction set with the ground-truth set in a one-to-one fashion and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 673, + 274, + 685 + ], + "spans": [ + { + "bbox": [ + 106, + 673, + 274, + 685 + ], + "score": 1.0, + "content": "evaluate a set-to-set object detection loss.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 649, + 506, + 685 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 688, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 294, + 702 + ], + "score": 1.0, + "content": "Single layer. Inspired by DETR [7], each layer", + "type": "text" + }, + { + "bbox": [ + 295, + 689, + 373, + 701 + ], + "score": 0.91, + "content": "\\ell \\in \\{ 0 , \\ldots , L - 1 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 687, + 506, + 702 + ], + "score": 1.0, + "content": "of Object DGCNN operates on a", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 188, + 713 + ], + "score": 1.0, + "content": "set of object queries", + "type": "text" + }, + { + "bbox": [ + 188, + 700, + 309, + 712 + ], + "score": 0.92, + "content": "\\mathcal { Q } _ { \\ell } = \\{ \\pmb q _ { \\ell 1 } , \\dots , \\pmb q _ { \\ell M ^ { * } } \\} \\subset \\mathbb { R } ^ { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 699, + 393, + 713 + ], + "score": 1.0, + "content": ", producing a new set", + "type": "text" + }, + { + "bbox": [ + 394, + 700, + 416, + 712 + ], + "score": 0.91, + "content": "\\mathcal { Q } _ { \\ell + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 699, + 505, + 713 + ], + "score": 1.0, + "content": ". Although queries are", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 710, + 460, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 460, + 723 + ], + "score": 1.0, + "content": "fully learnable, our intuition is that they represent progressively refined object positions.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 39, + "bbox_fs": [ + 105, + 687, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 506, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 71, + 506, + 87 + ], + "spans": [ + { + "bbox": [ + 105, + 71, + 240, + 87 + ], + "score": 1.0, + "content": "The initial set of object queries", + "type": "text" + }, + { + "bbox": [ + 240, + 73, + 254, + 84 + ], + "score": 0.88, + "content": "\\mathcal { Q } _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 71, + 506, + 87 + ], + "score": 1.0, + "content": "is learned jointly with the neural network weights, yielding", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "a dataset-specific prior. 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We think of", + "type": "text" + }, + { + "bbox": [ + 476, + 190, + 486, + 200 + ], + "score": 0.85, + "content": "\\mathbf { \\nabla } _ { \\pmb { p } _ { i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 186, + 507, + 202 + ], + "score": 1.0, + "content": "as a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 235, + 211 + ], + "score": 1.0, + "content": "hypothesis for the center of the", + "type": "text" + }, + { + "bbox": [ + 235, + 199, + 239, + 208 + ], + "score": 0.8, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 240, + 199, + 297, + 211 + ], + "score": 1.0, + "content": "-th object; the", + "type": "text" + }, + { + "bbox": [ + 297, + 199, + 304, + 208 + ], + "score": 0.8, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 199, + 416, + 211 + ], + "score": 1.0, + "content": "’s represent the positions of", + "type": "text" + }, + { + "bbox": [ + 417, + 199, + 427, + 208 + ], + "score": 0.85, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 199, + 505, + 211 + ], + "score": 1.0, + "content": "informative points", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 364, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 364, + 223 + ], + "score": 1.0, + "content": "relative to the position of the object that determine its geometry.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 106, + 225, + 453, + 238 + ], + "lines": [ + { + "bbox": [ + 104, + 223, + 450, + 240 + ], + "spans": [ + { + "bbox": [ + 104, + 223, + 233, + 240 + ], + "score": 1.0, + "content": "Next, we collect a BEV feature", + "type": "text" + }, + { + "bbox": [ + 234, + 226, + 249, + 238 + ], + "score": 0.91, + "content": "\\mathbf { \\boldsymbol { f } } _ { i k }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 223, + 385, + 240 + ], + "score": 1.0, + "content": "associated to each neighbor point", + "type": "text" + }, + { + "bbox": [ + 385, + 226, + 450, + 238 + ], + "score": 0.9, + "content": "\\pmb { p } _ { i k } = \\pmb { p } _ { i } + \\pmb { \\delta } _ { i k }", + "type": "inline_equation" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "interline_equation", + "bbox": [ + 243, + 243, + 367, + 259 + ], + "lines": [ + { + "bbox": [ + 243, + 243, + 367, + 259 + ], + "spans": [ + { + "bbox": [ + 243, + 243, + 367, + 259 + ], + "score": 0.93, + "content": "\\pmb { f } _ { i k } = f _ { \\mathrm { b i l i n e a r } } ( \\mathcal { F } ^ { d } , \\pmb { p } _ { i } + \\delta _ { i k } ) ,", + "type": "interline_equation", + "image_path": "afd1673a654602ffb7ce5456caecc68b3d2ef571dddeb9d6a8e290f64fff6a91.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 243, + 243, + 367, + 259 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 265, + 506, + 299 + ], + "lines": [ + { + "bbox": [ + 104, + 264, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 104, + 264, + 134, + 279 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 266, + 167, + 277 + ], + "score": 0.86, + "content": "f _ { \\mathrm { b i l i n e a r } }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 264, + 353, + 279 + ], + "score": 1.0, + "content": "bilinearly interpolates the BEV feature map", + "type": "text" + }, + { + "bbox": [ + 353, + 265, + 367, + 276 + ], + "score": 0.88, + "content": "{ \\mathcal { F } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 367, + 264, + 506, + 279 + ], + "score": 1.0, + "content": ". Note this step is the interaction", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 275, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 352, + 290 + ], + "score": 1.0, + "content": "between our set-based architecture manipulating query points", + "type": "text" + }, + { + "bbox": [ + 352, + 279, + 362, + 288 + ], + "score": 0.85, + "content": "\\pmb q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 275, + 489, + 290 + ], + "score": 1.0, + "content": "and the grid-based feature map", + "type": "text" + }, + { + "bbox": [ + 489, + 276, + 503, + 286 + ], + "score": 0.88, + "content": "{ \\mathcal { F } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 275, + 506, + 290 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 286, + 372, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 298, + 301 + ], + "score": 1.0, + "content": "We then aggregate a single object query feature", + "type": "text" + }, + { + "bbox": [ + 298, + 288, + 310, + 299 + ], + "score": 0.9, + "content": "\\pmb { f } _ { i } ^ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 286, + 348, + 301 + ], + "score": 1.0, + "content": "from the", + "type": "text" + }, + { + "bbox": [ + 348, + 288, + 367, + 299 + ], + "score": 0.9, + "content": "\\pmb { f } _ { i k } \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 286, + 372, + 301 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 305, + 354, + 334 + ], + "lines": [ + { + "bbox": [ + 257, + 305, + 354, + 334 + ], + "spans": [ + { + "bbox": [ + 257, + 305, + 354, + 334 + ], + "score": 0.95, + "content": "f _ { i } ^ { o } = \\sum _ { k } { \\frac { e ^ { w _ { i k } } } { \\sum _ { k } { e ^ { w _ { i k } } } } } { f _ { i k } } .", + "type": "interline_equation", + "image_path": "5d64a2b1060fc290d7dba28aaf1ae422f5c7690d10dd418742426dbc34a32777.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 257, + 305, + 354, + 319.5 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 257, + 319.5, + 354, + 334.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 485, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 485, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 485, + 353 + ], + "score": 1.0, + "content": "This generates scene-aware features; each object query “attends” to a certain area in the scene.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 506, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 182, + 368 + ], + "score": 1.0, + "content": "In the current layer", + "type": "text" + }, + { + "bbox": [ + 182, + 357, + 188, + 366 + ], + "score": 0.66, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 357, + 505, + 368 + ], + "score": 1.0, + "content": ", the queries have not yet interacted with each other. To incorporate neighborhood", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "information in object detection estimates, we use DGCNN-style operations to model a sparse set of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 377, + 507, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 377, + 507, + 391 + ], + "score": 1.0, + "content": "relations. We construct a graph between the queries using a nearest neighbor search in feature space.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 507, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 290, + 402 + ], + "score": 1.0, + "content": "In particular, we connect each query feature", + "type": "text" + }, + { + "bbox": [ + 291, + 389, + 303, + 401 + ], + "score": 0.9, + "content": "\\pmb { f } _ { i } ^ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 389, + 507, + 402 + ], + "score": 1.0, + "content": "to its 16 nearest neighbors as ablated in Table 7.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 315, + 413 + ], + "score": 1.0, + "content": "Identically to DGCNN, we learn a feature per edge", + "type": "text" + }, + { + "bbox": [ + 315, + 402, + 328, + 412 + ], + "score": 0.88, + "content": "e _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 399, + 488, + 413 + ], + "score": 1.0, + "content": "and then aggregate back to the vertices", + "type": "text" + }, + { + "bbox": [ + 489, + 401, + 493, + 410 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 399, + 506, + 413 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 338, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 338, + 423 + ], + "score": 1.0, + "content": "produce the new set of object queries. 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This completes our layer for computing", + "type": "text" + }, + { + "bbox": [ + 303, + 465, + 326, + 477 + ], + "score": 0.91, + "content": "\\mathcal { Q } _ { \\ell + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 463, + 349, + 479 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 349, + 465, + 362, + 476 + ], + "score": 0.87, + "content": "\\mathcal { Q } _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 463, + 505, + 479 + ], + "score": 1.0, + "content": ". Optionally, we repeat this last step", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 474, + 455, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 338, + 489 + ], + "score": 1.0, + "content": "multiple times, in effect applying DGCNN to the features", + "type": "text" + }, + { + "bbox": [ + 339, + 476, + 351, + 488 + ], + "score": 0.9, + "content": "\\pmb { f } _ { i } ^ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 474, + 428, + 489 + ], + "score": 1.0, + "content": "to get the point set", + "type": "text" + }, + { + "bbox": [ + 428, + 476, + 451, + 488 + ], + "score": 0.91, + "content": "\\mathcal { Q } _ { \\ell + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 474, + 455, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 506, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 504, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 196, + 504 + ], + "score": 1.0, + "content": "Set-to-set loss. 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Note this step is the interaction", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 275, + 506, + 290 + ], + "spans": [ + { + "bbox": [ + 104, + 275, + 352, + 290 + ], + "score": 1.0, + "content": "between our set-based architecture manipulating query points", + "type": "text" + }, + { + "bbox": [ + 352, + 279, + 362, + 288 + ], + "score": 0.85, + "content": "\\pmb q _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 275, + 489, + 290 + ], + "score": 1.0, + "content": "and the grid-based feature map", + "type": "text" + }, + { + "bbox": [ + 489, + 276, + 503, + 286 + ], + "score": 0.88, + "content": "{ \\mathcal { F } } ^ { d }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 275, + 506, + 290 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 286, + 372, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 298, + 301 + ], + "score": 1.0, + "content": "We then aggregate a single object query feature", + "type": "text" + }, + { + "bbox": [ + 298, + 288, + 310, + 299 + ], + "score": 0.9, + "content": "\\pmb { f } _ { i } ^ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 286, + 348, + 301 + ], + "score": 1.0, + "content": "from the", + "type": "text" + }, + { + "bbox": [ + 348, + 288, + 367, + 299 + ], + "score": 0.9, + "content": "\\pmb { f } _ { i k } \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 286, + 372, + 301 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15, + "bbox_fs": [ + 104, + 264, + 506, + 301 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 257, + 305, + 354, + 334 + ], + "lines": [ + { + "bbox": [ + 257, + 305, + 354, + 334 + ], + "spans": [ + { + "bbox": [ + 257, + 305, + 354, + 334 + ], + "score": 0.95, + "content": "f _ { i } ^ { o } = \\sum _ { k } { \\frac { e ^ { w _ { i k } } } { \\sum _ { k } { e ^ { w _ { i k } } } } } { f _ { i k } } .", + "type": "interline_equation", + "image_path": "5d64a2b1060fc290d7dba28aaf1ae422f5c7690d10dd418742426dbc34a32777.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 257, + 305, + 354, + 319.5 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 257, + 319.5, + 354, + 334.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 339, + 485, + 352 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 485, + 353 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 485, + 353 + ], + "score": 1.0, + "content": "This generates scene-aware features; each object query “attends” to a certain area in the scene.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 106, + 339, + 485, + 353 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 356, + 506, + 423 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 182, + 368 + ], + "score": 1.0, + "content": "In the current layer", + "type": "text" + }, + { + "bbox": [ + 182, + 357, + 188, + 366 + ], + "score": 0.66, + "content": "\\ell", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 357, + 505, + 368 + ], + "score": 1.0, + "content": ", the queries have not yet interacted with each other. To incorporate neighborhood", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "information in object detection estimates, we use DGCNN-style operations to model a sparse set of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 377, + 507, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 377, + 507, + 391 + ], + "score": 1.0, + "content": "relations. We construct a graph between the queries using a nearest neighbor search in feature space.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 389, + 507, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 290, + 402 + ], + "score": 1.0, + "content": "In particular, we connect each query feature", + "type": "text" + }, + { + "bbox": [ + 291, + 389, + 303, + 401 + ], + "score": 0.9, + "content": "\\pmb { f } _ { i } ^ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 389, + 507, + 402 + ], + "score": 1.0, + "content": "to its 16 nearest neighbors as ablated in Table 7.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 399, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 315, + 413 + ], + "score": 1.0, + "content": "Identically to DGCNN, we learn a feature per edge", + "type": "text" + }, + { + "bbox": [ + 315, + 402, + 328, + 412 + ], + "score": 0.88, + "content": "e _ { i j }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 399, + 488, + 413 + ], + "score": 1.0, + "content": "and then aggregate back to the vertices", + "type": "text" + }, + { + "bbox": [ + 489, + 401, + 493, + 410 + ], + "score": 0.77, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 399, + 506, + 413 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 411, + 338, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 338, + 423 + ], + "score": 1.0, + "content": "produce the new set of object queries. In detail, we write:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 357, + 507, + 423 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 428, + 372, + 448 + ], + "lines": [ + { + "bbox": [ + 238, + 428, + 372, + 448 + ], + "spans": [ + { + "bbox": [ + 238, + 428, + 372, + 448 + ], + "score": 0.93, + "content": "{ \\pmb q } _ { ( \\ell + 1 ) i } = \\operatorname* { m a x } _ { \\mathrm { e d g e s } ~ e _ { i j } } \\Phi _ { \\mathrm { e d g e } } ( { \\pmb f } _ { i } ^ { o } , { \\pmb f } _ { j } ^ { o } ) ,", + "type": "interline_equation", + "image_path": "986649696473c131b93df5b09569c9f14114b94dca51e40e2bc0229abc52799f.jpg" + } + ] + } + ], + "index": 26, + "virtual_lines": [ + { + "bbox": [ + 238, + 428, + 372, + 448 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 453, + 505, + 488 + ], + "lines": [ + { + "bbox": [ + 105, + 453, + 505, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 315, + 468 + ], + "score": 1.0, + "content": "where max denotes a channel-wise maximum and", + "type": "text" + }, + { + "bbox": [ + 315, + 454, + 339, + 465 + ], + "score": 0.91, + "content": "\\Phi _ { \\mathrm { e d g e } }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 453, + 505, + 468 + ], + "score": 1.0, + "content": "is a neural network for computing edge", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 463, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 303, + 479 + ], + "score": 1.0, + "content": "features. This completes our layer for computing", + "type": "text" + }, + { + "bbox": [ + 303, + 465, + 326, + 477 + ], + "score": 0.91, + "content": "\\mathcal { Q } _ { \\ell + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 463, + 349, + 479 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 349, + 465, + 362, + 476 + ], + "score": 0.87, + "content": "\\mathcal { Q } _ { \\ell }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 463, + 505, + 479 + ], + "score": 1.0, + "content": ". Optionally, we repeat this last step", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 474, + 455, + 489 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 338, + 489 + ], + "score": 1.0, + "content": "multiple times, in effect applying DGCNN to the features", + "type": "text" + }, + { + "bbox": [ + 339, + 476, + 351, + 488 + ], + "score": 0.9, + "content": "\\pmb { f } _ { i } ^ { o }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 474, + 428, + 489 + ], + "score": 1.0, + "content": "to get the point set", + "type": "text" + }, + { + "bbox": [ + 428, + 476, + 451, + 488 + ], + "score": 0.91, + "content": "\\mathcal { Q } _ { \\ell + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 474, + 455, + 489 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 453, + 505, + 489 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 491, + 506, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 491, + 504, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 196, + 504 + ], + "score": 1.0, + "content": "Set-to-set loss. 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We use the Hungarian", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 376, + 496, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 496, + 388 + ], + "score": 1.0, + "content": "algorithm [61] to solve this assignment problem, as in [62, 7]. Our final set-to-set loss adapts (7):", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 15 + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 394, + 421, + 430 + ], + "lines": [ + { + "bbox": [ + 190, + 394, + 421, + 430 + ], + "spans": [ + { + "bbox": [ + 190, + 394, + 421, + 430 + ], + "score": 0.94, + "content": "\\mathcal { L } _ { \\mathrm { s u p } } = \\sum _ { j = 1 } ^ { N } - \\log \\hat { p } _ { \\sigma ^ { * } ( j ) } ( c _ { j } ) + 1 _ { \\{ c _ { j } \\neq \\emptyset \\} } \\mathcal { L } _ { \\mathrm { b o x } } \\bigl ( b _ { j } , \\hat { b } _ { \\sigma ^ { * } ( j ) } \\bigr ) .", + "type": "interline_equation", + "image_path": "4aef0d1dcc4e91bd0e7c913e52a80a46ab5984985a532fc61102d3ead74eb2d9.jpg" + } + ] + } + ], + "index": 17.5, + "virtual_lines": [ + { + "bbox": [ + 190, + 394, + 421, + 412.0 + ], + "spans": [], + "index": 17 + }, + { + "bbox": [ + 190, + 412.0, + 421, + 430.0 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "title", + "bbox": [ + 107, + 443, + 182, + 456 + ], + "lines": [ + { + "bbox": [ + 104, + 441, + 184, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 441, + 184, + 460 + ], + "score": 1.0, + "content": "5 Distillation", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 106, + 468, + 505, + 592 + ], + "lines": [ + { + "bbox": [ + 106, + 468, + 505, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 482 + ], + "score": 1.0, + "content": "Object DGCNN enables a new set-to-set knowledge distillation (KD) pipeline. KD usually involves", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 479, + 176, + 493 + ], + "score": 1.0, + "content": "a teacher model", + "type": "text" + }, + { + "bbox": [ + 176, + 480, + 186, + 490 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 479, + 275, + 493 + ], + "score": 1.0, + "content": "and a student model", + "type": "text" + }, + { + "bbox": [ + 275, + 480, + 283, + 490 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 479, + 505, + 493 + ], + "score": 1.0, + "content": ". The common practice is to align the outputs of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 265, + 504 + ], + "score": 1.0, + "content": "student with those of the teacher using", + "type": "text" + }, + { + "bbox": [ + 266, + 492, + 278, + 502 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 490, + 505, + 504 + ], + "score": 1.0, + "content": "distance or KL-divergence. In past 3D object detection", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "methods, since final performance heavily relies on NMS and the predictions are post-processed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "to be a smaller set, distilling the teacher to the student is neither efficient nor effective. Since our", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "set-based detection model is NMS-free, we can easily distill the information between models with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "homogeneous detection heads (per-query object detection head in our case). First, we train a teacher", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 544, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 107, + 546, + 116, + 556 + ], + "score": 0.81, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 544, + 393, + 559 + ], + "score": 1.0, + "content": "using the method above with the loss in (10). 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The class label and box parameters predicted by the teacher for each object", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 567, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 147, + 583 + ], + "score": 1.0, + "content": "query are", + "type": "text" + }, + { + "bbox": [ + 148, + 568, + 160, + 582 + ], + "score": 0.89, + "content": "c _ { j } ^ { \\mathcal { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 567, + 178, + 583 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 567, + 192, + 582 + ], + "score": 0.89, + "content": "b _ { j } ^ { \\mathcal { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 567, + 410, + 583 + ], + "score": 1.0, + "content": ", respectively. The corresponding student outputs are", + "type": "text" + }, + { + "bbox": [ + 410, + 568, + 421, + 582 + ], + "score": 0.88, + "content": "c _ { j } ^ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 567, + 441, + 583 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 441, + 568, + 453, + 582 + ], + "score": 0.9, + "content": "b _ { j } ^ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 567, + 506, + 583 + ], + "score": 1.0, + "content": ". 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The student network is trained with the ground-truth", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 199, + 399, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 399, + 211 + ], + "score": 1.0, + "content": "supervision as well as with the supervision from a fixed teacher network.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 105, + 232, + 504, + 255 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 105, + 232, + 505, + 256 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 259, + 505, + 304 + ], + "lines": [ + { + "bbox": [ + 104, + 258, + 504, + 272 + ], + "spans": [ + { + "bbox": [ + 104, + 258, + 487, + 272 + ], + "score": 1.0, + "content": "Rather than performing dense predictions in the BEV, we make per-query predictions. 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To account for this difference, we pad the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 229, + 294 + ], + "score": 1.0, + "content": "set of ground-truth boxes with", + "type": "text" + }, + { + "bbox": [ + 229, + 282, + 242, + 292 + ], + "score": 0.77, + "content": "\\mathcal { O } \\mathbf { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 281, + 311, + 294 + ], + "score": 1.0, + "content": "(no object) up to", + "type": "text" + }, + { + "bbox": [ + 312, + 282, + 327, + 292 + ], + "score": 0.87, + "content": "M ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 281, + 506, + 294 + ], + "score": 1.0, + "content": ". Following [7], we use an objective built on", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 293, + 464, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 464, + 306 + ], + "score": 1.0, + "content": "an optimal matching between these two sets. We define the optimal bipartite matching as", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 258, + 506, + 306 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 310, + 437, + 346 + ], + "lines": [ + { + "bbox": [ + 173, + 310, + 437, + 346 + ], + "spans": [ + { + "bbox": [ + 173, + 310, + 437, + 346 + ], + "score": 0.94, + "content": "\\boldsymbol { \\sigma } ^ { * } = \\underset { \\boldsymbol { \\sigma } \\in \\mathcal { P } } { \\arg \\operatorname* { m i n } } \\sum _ { j = 1 } ^ { M } - 1 _ { \\{ c _ { j } \\neq \\emptyset \\} } \\hat { p } _ { \\boldsymbol { \\sigma } ( j ) } ( c _ { j } ) + 1 _ { \\{ c _ { j } = \\emptyset \\} } \\mathcal { L } _ { \\mathrm { b o x } } ( \\boldsymbol { b } _ { j } , \\hat { \\boldsymbol { b } } _ { \\boldsymbol { \\sigma } ( j ) } ) ,", + "type": "interline_equation", + "image_path": "d42ff577c2cee5c4cd49f2cfdce3aae5ff73a0134054f81d344895186a46c112.jpg" + } + ] + } + ], + "index": 12, + "virtual_lines": [ + { + "bbox": [ + 173, + 310, + 437, + 322.0 + ], + "spans": [], + "index": 11 + }, + { + "bbox": [ + 173, + 322.0, + 437, + 334.0 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 173, + 334.0, + 437, + 346.0 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 506, + 388 + ], + "lines": [ + { + "bbox": [ + 105, + 352, + 506, + 368 + ], + "spans": [ + { + "bbox": [ + 105, + 352, + 133, + 368 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 354, + 142, + 363 + ], + "score": 0.83, + "content": "\\mathcal { P }", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 352, + 276, + 368 + ], + "score": 1.0, + "content": "denotes the set of permutations,", + "type": "text" + }, + { + "bbox": [ + 276, + 353, + 313, + 366 + ], + "score": 0.93, + "content": "\\hat { p } _ { \\sigma ( j ) } ( \\boldsymbol { c } _ { j } )", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 352, + 420, + 368 + ], + "score": 1.0, + "content": "is the probability of class", + "type": "text" + }, + { + "bbox": [ + 421, + 355, + 430, + 366 + ], + "score": 0.86, + "content": "c _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 431, + 352, + 506, + 368 + ], + "score": 1.0, + "content": "for the prediction", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 364, + 506, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 153, + 378 + ], + "score": 1.0, + "content": "with index", + "type": "text" + }, + { + "bbox": [ + 154, + 365, + 173, + 377 + ], + "score": 0.91, + "content": "\\sigma ( j )", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 364, + 196, + 378 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 197, + 365, + 218, + 376 + ], + "score": 0.91, + "content": "\\mathcal { L } _ { \\mathrm { b o x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 364, + 246, + 378 + ], + "score": 1.0, + "content": "is the", + "type": "text" + }, + { + "bbox": [ + 246, + 365, + 259, + 376 + ], + "score": 0.87, + "content": "\\mathcal { L } _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 364, + 506, + 378 + ], + "score": 1.0, + "content": "loss for bounding box parameters. We use the Hungarian", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 376, + 496, + 388 + ], + "spans": [ + { + "bbox": [ + 106, + 376, + 496, + 388 + ], + "score": 1.0, + "content": "algorithm [61] to solve this assignment problem, as in [62, 7]. 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KD usually involves", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 479, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 104, + 479, + 176, + 493 + ], + "score": 1.0, + "content": "a teacher model", + "type": "text" + }, + { + "bbox": [ + 176, + 480, + 186, + 490 + ], + "score": 0.8, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 479, + 275, + 493 + ], + "score": 1.0, + "content": "and a student model", + "type": "text" + }, + { + "bbox": [ + 275, + 480, + 283, + 490 + ], + "score": 0.79, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 283, + 479, + 505, + 493 + ], + "score": 1.0, + "content": ". The common practice is to align the outputs of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 490, + 505, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 265, + 504 + ], + "score": 1.0, + "content": "student with those of the teacher using", + "type": "text" + }, + { + "bbox": [ + 266, + 492, + 278, + 502 + ], + "score": 0.88, + "content": "\\mathcal { L } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 490, + 505, + 504 + ], + "score": 1.0, + "content": "distance or KL-divergence. In past 3D object detection", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 515 + ], + "score": 1.0, + "content": "methods, since final performance heavily relies on NMS and the predictions are post-processed", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "to be a smaller set, distilling the teacher to the student is neither efficient nor effective. Since our", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "set-based detection model is NMS-free, we can easily distill the information between models with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 547 + ], + "score": 1.0, + "content": "homogeneous detection heads (per-query object detection head in our case). First, we train a teacher", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 544, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 107, + 546, + 116, + 556 + ], + "score": 0.81, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 544, + 393, + 559 + ], + "score": 1.0, + "content": "using the method above with the loss in (10). 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The class label and box parameters predicted by the teacher for each object", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 567, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 104, + 567, + 147, + 583 + ], + "score": 1.0, + "content": "query are", + "type": "text" + }, + { + "bbox": [ + 148, + 568, + 160, + 582 + ], + "score": 0.89, + "content": "c _ { j } ^ { \\mathcal { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 567, + 178, + 583 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 179, + 567, + 192, + 582 + ], + "score": 0.89, + "content": "b _ { j } ^ { \\mathcal { T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 567, + 410, + 583 + ], + "score": 1.0, + "content": ", respectively. The corresponding student outputs are", + "type": "text" + }, + { + "bbox": [ + 410, + 568, + 421, + 582 + ], + "score": 0.88, + "content": "c _ { j } ^ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 567, + 441, + 583 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 441, + 568, + 453, + 582 + ], + "score": 0.9, + "content": "b _ { j } ^ { S }", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 567, + 506, + 583 + ], + "score": 1.0, + "content": ". We find an", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 580, + 422, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 422, + 591 + ], + "score": 1.0, + "content": "optimal matching between the output set of the teacher and that of the student:", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25, + "bbox_fs": [ + 104, + 468, + 506, + 591 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 192, + 597, + 419, + 633 + ], + "lines": [ + { + "bbox": [ + 192, + 597, + 419, + 633 + ], + "spans": [ + { + "bbox": [ + 192, + 597, + 419, + 633 + ], + "score": 0.94, + "content": "\\sigma _ { d } ^ { * } = \\underset { \\sigma _ { d } \\in \\mathcal { P } } { \\arg \\operatorname* { m i n } } \\sum _ { j } ^ { N } - \\log p _ { \\sigma _ { d } ( j ) } ( c _ { j } ^ { \\mathcal { T } } ) + \\mathcal { L } _ { \\mathrm { b o x } } ( { b } _ { j } ^ { { \\mathcal { T } } } , { b } _ { \\sigma _ { d } ( j ) } ^ { S } ) .", + "type": "interline_equation", + "image_path": "7b1a5e8ff8310dcc5566f437c1e3a79813c27c67e4159e02f4f42e3e588b2ee6.jpg" + } + ] + } + ], + "index": 31.5, + "virtual_lines": [ + { + "bbox": [ + 192, + 597, + 419, + 615.0 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 192, + 615.0, + 419, + 633.0 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 639, + 306, + 652 + ], + "lines": [ + { + "bbox": [ + 105, + 637, + 306, + 654 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 306, + 654 + ], + "score": 1.0, + "content": "Then, the optimal matching’s KD loss is given by", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33, + "bbox_fs": [ + 105, + 637, + 306, + 654 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 200, + 658, + 411, + 693 + ], + "lines": [ + { + "bbox": [ + 200, + 658, + 411, + 693 + ], + "spans": [ + { + "bbox": [ + 200, + 658, + 411, + 693 + ], + "score": 0.93, + "content": "\\mathcal { L } _ { \\mathrm { d i s t i l l } } = \\sum _ { j } ^ { N } - \\log \\hat { p } _ { \\sigma _ { d } ^ { * } ( j ) } ( c _ { j } ^ { \\mathcal { T } } ) + \\mathcal { L } _ { \\mathrm { b o x } } ( b _ { j } ^ { \\mathcal { T } } , b _ { \\sigma _ { d } ^ { * } ( j ) } ^ { S } ) .", + "type": "interline_equation", + "image_path": "599748f3f5963f73ef09a7674bb46ac548af48a7eaa4e5d54a0318e6d1d68eb3.jpg" + } + ] + } + ], + "index": 34.5, + "virtual_lines": [ + { + "bbox": [ + 200, + 658, + 411, + 675.5 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 200, + 675.5, + 411, + 693.0 + ], + "spans": [], + "index": 35 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 699, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 237, + 712 + ], + "score": 1.0, + "content": "So the overall loss during KD is", + "type": "text" + }, + { + "bbox": [ + 237, + 701, + 329, + 712 + ], + "score": 0.88, + "content": "\\mathcal { L } = \\alpha \\mathcal { L } _ { \\mathrm { s u p } } + \\beta \\mathcal { L } _ { \\mathrm { d i s t i l l } }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 699, + 360, + 712 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 360, + 703, + 367, + 710 + ], + "score": 0.76, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 699, + 385, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 386, + 702, + 393, + 712 + ], + "score": 0.86, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "balance the supervised loss", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 710, + 311, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 267, + 723 + ], + "score": 1.0, + "content": "and distillation loss. 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MethodNDS↑mAP个mATE√mASE↓mAOE↓mAVE↓mAAE↓NMS
PointPillars [4]53.340.0二二:二二√
SSN[63]54.8341.56二二二二-√
FreeAnchor [64]55.343.7二二-二二√
RegNetX-400MF-SECFPN[65]55.241.2----√
Pillar-OD[5]56.8444.41-----√
CenterPoint (pillar) [6] *59.5647.4831.2725.8133.7832.2520.20√
CenterPoint (pillar) [6] *55.0840.2735.1426.4436.7532.6619.55
CenterPoint (voxel) [6] ‡64.1954.9929.8325.7132.5626.0818.89√
CenterPoint (voxel) [6]‡57.0045.3231.6627.1440.4737.2320.14
Ours (pillar) *62.9753.3134.6226.5631.6126.0218.71√
Ours (pillar) *62.8053.2034.6226.5631.6226.0719.10
Ours (voxel)‡66.1058.7333.3126.3228.8025.1119.08√
Ours (voxel) t66.0458.6233.3326.3428.8025.1119.06
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MethodNDS↑mAP↑mATE↓mASE↓mAOE↓mAVE↓mAAE↓
Baseline (without distillation)62.8053.2034.6226.5631.6226.0219.10
Feature distill (voxel->pillar)62.8453.2934.5526.7831.6126.1119.02
Pseudo labels (voxel-→pillar)63.1853.3134.5826.5531.2925.9918.87
Set-to-set distill (voxel-→pillar)63.3753.8934.3426.2531.0125.5718.77
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Then, we demonstrate performance on the nuScenes dataset [66] in", + "type": "text" + }, + { + "bbox": [ + 485, + 393, + 503, + 404 + ], + "score": 0.85, + "content": "\\ S 6 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 394, + 506, + 405 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 404, + 483, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 282, + 416 + ], + "score": 1.0, + "content": "We present knowledge distillation results in", + "type": "text" + }, + { + "bbox": [ + 283, + 404, + 302, + 415 + ], + "score": 0.85, + "content": "\\ S 6 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 404, + 460, + 416 + ], + "score": 1.0, + "content": ". Finally, we provide ablation studies in", + "type": "text" + }, + { + "bbox": [ + 460, + 405, + 479, + 415 + ], + "score": 0.82, + "content": "\\ S 6 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 404, + 483, + 416 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 428, + 259, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 260, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 260, + 443 + ], + "score": 1.0, + "content": "6.1 Training & testing procedures", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "Dataset. 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Our model consists of three parts: a point-based feature extractor, a DGCNN to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "encode object queries and to connect the point cloud features to object queries, and a detection head", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "to output the categorical label and bounding box parameters. We experiment with PointPillars [4]", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "and SparseConv [60] as feature extractors. The three blocks of the PointPillars backbone have", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 648 + ], + "score": 1.0, + "content": "[3, 5, 5] convolutional layers, with dimensions [64, 128, 256] and strides [2, 2, 2]; the input features", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 645, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 657 + ], + "score": 1.0, + "content": "are downsampled to 1/2, 1/4, 1/8 of the original feature map. 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MethodNDS↑mAP个mATE√mASE↓mAOE↓mAVE↓mAAE↓NMS
PointPillars [4]53.340.0二二:二二√
SSN[63]54.8341.56二二二二-√
FreeAnchor [64]55.343.7二二-二二√
RegNetX-400MF-SECFPN[65]55.241.2----√
Pillar-OD[5]56.8444.41-----√
CenterPoint (pillar) [6] *59.5647.4831.2725.8133.7832.2520.20√
CenterPoint (pillar) [6] *55.0840.2735.1426.4436.7532.6619.55
CenterPoint (voxel) [6] ‡64.1954.9929.8325.7132.5626.0818.89√
CenterPoint (voxel) [6]‡57.0045.3231.6627.1440.4737.2320.14
Ours (pillar) *62.9753.3134.6226.5631.6126.0218.71√
Ours (pillar) *62.8053.2034.6226.5631.6226.0719.10
Ours (voxel)‡66.1058.7333.3126.3228.8025.1119.08√
Ours (voxel) t66.0458.6233.3326.3428.8025.1119.06
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MethodNDS↑mAP↑mATE↓mASE↓mAOE↓mAVE↓mAAE↓
Baseline (without distillation)62.8053.2034.6226.5631.6226.0219.10
Feature distill (voxel->pillar)62.8453.2934.5526.7831.6126.1119.02
Pseudo labels (voxel-→pillar)63.1853.3134.5826.5531.2925.9918.87
Set-to-set distill (voxel-→pillar)63.3753.8934.3426.2531.0125.5718.77
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We introduce the dataset, metrics, implementation, and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 393, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 196, + 405 + ], + "score": 1.0, + "content": "optimization details in", + "type": "text" + }, + { + "bbox": [ + 196, + 393, + 214, + 404 + ], + "score": 0.84, + "content": "\\ S 6 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 394, + 484, + 405 + ], + "score": 1.0, + "content": ". Then, we demonstrate performance on the nuScenes dataset [66] in", + "type": "text" + }, + { + "bbox": [ + 485, + 393, + 503, + 404 + ], + "score": 0.85, + "content": "\\ S 6 . 2", + "type": "inline_equation" + }, + { + "bbox": [ + 504, + 394, + 506, + 405 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 404, + 483, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 282, + 416 + ], + "score": 1.0, + "content": "We present knowledge distillation results in", + "type": "text" + }, + { + "bbox": [ + 283, + 404, + 302, + 415 + ], + "score": 0.85, + "content": "\\ S 6 . 3", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 404, + 460, + 416 + ], + "score": 1.0, + "content": ". Finally, we provide ablation studies in", + "type": "text" + }, + { + "bbox": [ + 460, + 405, + 479, + 415 + ], + "score": 0.82, + "content": "\\ S 6 . 4", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 404, + 483, + 416 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 382, + 506, + 416 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 428, + 259, + 441 + ], + "lines": [ + { + "bbox": [ + 105, + 426, + 260, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 260, + 443 + ], + "score": 1.0, + "content": "6.1 Training & testing procedures", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 537 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "Dataset. We experiment on the nuScenes dataset [66]. nuScenes provides rich annotations and", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 506, + 472 + ], + "score": 1.0, + "content": "diverse scenes. It has 1K short sequences captured in Boston and Singapore with 700, 150, 150", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 470, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 410, + 483 + ], + "score": 1.0, + "content": "sequences for training, validation, and testing, respectively. Each sequence is", + "type": "text" + }, + { + "bbox": [ + 411, + 471, + 435, + 481 + ], + "score": 0.86, + "content": "{ \\sim } 2 0 \\mathrm { s }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 470, + 505, + 483 + ], + "score": 1.0, + "content": "and contains 400", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 482, + 506, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 506, + 494 + ], + "score": 1.0, + "content": "frames. This dataset provides annotation every 0.5s, leading to 28K, 6K, 6K annotated frames for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 506 + ], + "score": 1.0, + "content": "training, validation, and testing. nuScenes uses 32-beam LiDAR, producing 30K points per frame.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 502, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 104, + 502, + 506, + 517 + ], + "score": 1.0, + "content": "Following common practice, we use calibrated vehicle pose information to aggregate every 9 non-key", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 514, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 106, + 515, + 304, + 527 + ], + "score": 1.0, + "content": "frames to key frames, so each annotated frame has", + "type": "text" + }, + { + "bbox": [ + 305, + 514, + 337, + 525 + ], + "score": 0.88, + "content": "\\sim 3 0 0 \\mathrm { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "points. The annotations include 23 classes", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 525, + 426, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 426, + 537 + ], + "score": 1.0, + "content": "with a long-tail distribution, of which 10 classes are included in the benchmark.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 448, + 506, + 537 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 541, + 505, + 586 + ], + "lines": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 554 + ], + "score": 1.0, + "content": "Metrics. The major metrics are mean average precision (mAP) and the nuScenes detection score", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 565 + ], + "score": 1.0, + "content": "(NDS). 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These metrics are computed in the physical unit.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 541, + 505, + 587 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 506, + 603 + ], + "score": 1.0, + "content": "Model architecture. Our model consists of three parts: a point-based feature extractor, a DGCNN to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 602, + 506, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 506, + 614 + ], + "score": 1.0, + "content": "encode object queries and to connect the point cloud features to object queries, and a detection head", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 625 + ], + "score": 1.0, + "content": "to output the categorical label and bounding box parameters. We experiment with PointPillars [4]", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "and SparseConv [60] as feature extractors. The three blocks of the PointPillars backbone have", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 648 + ], + "score": 1.0, + "content": "[3, 5, 5] convolutional layers, with dimensions [64, 128, 256] and strides [2, 2, 2]; the input features", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 645, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 506, + 657 + ], + "score": 1.0, + "content": "are downsampled to 1/2, 1/4, 1/8 of the original feature map. For SparseConv, we use four blocks of", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 506, + 669 + ], + "score": 1.0, + "content": "[3, 3, 3, 2] 3D sparse convolutional layers, with dimensions [16, 32, 64, 128] and strides [2, 2, 2, 1];", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "the input features are downsampled to 1/2, 1/4, 1/8, 1/8 of the original feature map. For SparseConv,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 678, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 325, + 690 + ], + "score": 1.0, + "content": "we transform the features into BEV by collapsing the", + "type": "text" + }, + { + "bbox": [ + 325, + 680, + 331, + 688 + ], + "score": 0.77, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 678, + 505, + 690 + ], + "score": 1.0, + "content": "-axis. Both backbones use two deformable", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "self-attention [17] layers with dimensions [256, 256] to transform the BEV features. Then, we use", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "two DGCNNs to encode the object queries. Each DGCNN [8] contains two EdgeConv layers with", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "dimensions [256, 256], both with 16 nearest neighbors. For each object query, we predict four points", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "in the BEV to obtain and aggregate the BEV features. The final feature for this object query is the", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 229, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 506, + 241 + ], + "score": 1.0, + "content": "weighted sum of features of these four BEV points. The final detection head takes the features of", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 240, + 496, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 496, + 252 + ], + "score": 1.0, + "content": "each object query and predicts class label and bounding box parameters w.r.t. the reference point.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 591, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 109, + 83, + 502, + 135 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 188, + 70, + 421, + 81 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 188, + 68, + 423, + 82 + ], + "spans": [ + { + "bbox": [ + 188, + 68, + 423, + 82 + ], + "score": 1.0, + "content": "Table 3: Comparisons of self-distillation versus baselines.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 109, + 83, + 502, + 135 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 83, + 502, + 135 + ], + "spans": [ + { + "bbox": [ + 109, + 83, + 502, + 135 + ], + "score": 0.769, + "html": "
MethodNDS ↑mAP↑mATEmASE↓mAOE↓mAVE↓mAAE↓
Baseline (pillar,without distillation)62.8053.2034.6226.5631.6226.0219.10
Self-distillation (pillar->pillar)63.4153.8934.2126.1931.1125.6718.54
Baseline (voxel,without distillation)66.0458.6233.3326.3428.8025.1119.06
Self-distillation (voxel->voxel)66.4559.2531.1725.7730.7325.7218.77
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MethodNDS ↑mAP↑mATE↓mASE↓mAOE↓mAVE↓mAAE↓
Sparse-→sparse (pillar)42.1238.8944.0127.7864.01144.0139.21
Dense-→sparse (pillar)42.7939.1043.8927.7764.01143.9739.11
Sparse-→sparse (voxel)59.5549.8431.1725.7733.7332.2220.22
Dense->sparse (voxel)59.8950.1231.1125.7633.7032.1920.11
", + "type": "table", + "image_path": "3127971340c8ac67568bf68fe9b1c4ec413354755633754383992b5123eda9d0.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 109, + 152, + 502, + 172.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 109, + 172.0, + 502, + 192.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 109, + 192.0, + 502, + 212.0 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 5.0 + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 505, + 251 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 505, + 231 + ], + "score": 1.0, + "content": "in the BEV to obtain and aggregate the BEV features. The final feature for this object query is the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 229, + 506, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 506, + 241 + ], + "score": 1.0, + "content": "weighted sum of features of these four BEV points. The final detection head takes the features of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 240, + 496, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 496, + 252 + ], + "score": 1.0, + "content": "each object query and predicts class label and bounding box parameters w.r.t. the reference point.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 268 + ], + "score": 1.0, + "content": "Training & inference. We use AdamW [67] to train the model. The weight decay for AdamW is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 265, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 128, + 277 + ], + "score": 0.89, + "content": "1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 265, + 383, + 280 + ], + "score": 1.0, + "content": ". Following a cyclic schedule [68], the learning rate is initially", + "type": "text" + }, + { + "bbox": [ + 383, + 267, + 405, + 277 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 265, + 506, + 280 + ], + "score": 1.0, + "content": "and gradually increased", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 276, + 507, + 291 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 117, + 291 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 277, + 138, + 289 + ], + "score": 0.89, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 276, + 258, + 291 + ], + "score": 1.0, + "content": ", which is finally decreased to", + "type": "text" + }, + { + "bbox": [ + 259, + 277, + 280, + 288 + ], + "score": 0.9, + "content": "1 0 ^ { - 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 276, + 507, + 291 + ], + "score": 1.0, + "content": ". The model is initialized with a pre-trained PointPillars", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "network on the same dataset. We train for 20 epochs on 8 RTX 3090 GPUs. During inference, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "take the top 100 objects with highest classification scores as the final predictions.We do not use any", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 310, + 507, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 507, + 324 + ], + "score": 1.0, + "content": "post-processing such as NMS. For evaluation, we use the toolkit provided with the nuScenes dataset.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5 + }, + { + "type": "title", + "bbox": [ + 107, + 335, + 198, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 200, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 200, + 349 + ], + "score": 1.0, + "content": "6.2 Object DGCNN", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 367 + ], + "score": 1.0, + "content": "We compare to top-performing methods on the nuScenes dataset in Table 1. PointPillars [4] is an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "anchor-based method with reasonable trade-off between performance and efficiency. FreeAnchor [64]", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 377, + 507, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 507, + 389 + ], + "score": 1.0, + "content": "extends PointPillars by learning how to assign anchors to the ground-truth. RegNetX-400MF-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "SECFPN [65] uses neural architecture search (NAS) to learn a flexible neural network for 3D", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "detection; it is essentially a variant of PointPillars with an enhanced backbone network. Different", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "from anchor-based methods, Pillar-OD [5] makes predictions per pillar, alleviating the class imbalance", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "issue caused by anchors. CenterPoint [6] exploits similar detection heads, with better performance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "using better training scheduling and data augmentation. For these methods, we use re-implementations", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 424, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 424, + 455 + ], + "score": 1.0, + "content": "in MMDetection3D [69], which match the performances in the original papers.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 472 + ], + "score": 1.0, + "content": "We mainly compare to CenterPoint with both PointPillars and SparseConv backbones, denoted as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 467, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 467, + 505, + 484 + ], + "score": 1.0, + "content": "“voxel” and “pillar” respectively. Our method outperforms other methods significantly including", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 479, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 495 + ], + "score": 1.0, + "content": "CenterPoint with NMS. Without NMS, the performance of CenterPoint drops considerably while our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 491, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 491, + 505, + 506 + ], + "score": 1.0, + "content": "method is unaffected by NMS. This finding verifies the DGCNN implicitly models object relations", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 502, + 230, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 230, + 516 + ], + "score": 1.0, + "content": "and removes redundant boxes.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29 + }, + { + "type": "title", + "bbox": [ + 107, + 527, + 218, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 526, + 218, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 218, + 540 + ], + "score": 1.0, + "content": "6.3 Set-to-set distillation", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "In this section, we present experiments involving our set-to-set distillation pipeline. We conduct", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "three types of distillation. First, we distill a teacher model with a SparseConv backbone to a student", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 335, + 582 + ], + "score": 1.0, + "content": "model with a PointPillars backbone (denoted as “voxel", + "type": "text" + }, + { + "bbox": [ + 335, + 570, + 345, + 579 + ], + "score": 0.77, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "pillar”). This aligns with the common", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "knowledge distillation setup for classification. We compare to feature-based distillation and pseudo", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "label based methods. The objective of feature-based distillation is to align the middle-level features", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 600, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 615 + ], + "score": 1.0, + "content": "of the teacher model and the student model while the pseudo label based methods generate pseudo", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "training examples with the pre-trained teacher networks. As Table 2 shows, our set-to-set distillation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "achieves better performance, confirming that distilling the last stage of the object detection model is", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 634, + 280, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 280, + 648 + ], + "score": 1.0, + "content": "more effective than distilling feature maps.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 351, + 663 + ], + "score": 1.0, + "content": "Second, we perform self-distillation [49] (denoted as “voxel", + "type": "text" + }, + { + "bbox": [ + 352, + 652, + 362, + 660 + ], + "score": 0.8, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 650, + 434, + 663 + ], + "score": 1.0, + "content": "voxel” and “pillar", + "type": "text" + }, + { + "bbox": [ + 434, + 652, + 444, + 660 + ], + "score": 0.82, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "pillar”), where", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 661, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 674 + ], + "score": 1.0, + "content": "the teacher and the student are identical and take the same point clouds as input. As Table 3 shows,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "even when the teacher network and the student network have the same capacity, self-distillation still", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 684, + 435, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 435, + 695 + ], + "score": 1.0, + "content": "introduces a performance boost. This finding is consistent with the results in [49].", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Finally, we try distillation with privileged information [56], where the teacher gets access to privileged", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "information but the student does not. Following [10], the teacher takes dense point clouds, and the", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 752 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 109, + 83, + 502, + 135 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 188, + 70, + 421, + 81 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 188, + 68, + 423, + 82 + ], + "spans": [ + { + "bbox": [ + 188, + 68, + 423, + 82 + ], + "score": 1.0, + "content": "Table 3: Comparisons of self-distillation versus baselines.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table_body", + "bbox": [ + 109, + 83, + 502, + 135 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 83, + 502, + 135 + ], + "spans": [ + { + "bbox": [ + 109, + 83, + 502, + 135 + ], + "score": 0.769, + "html": "
MethodNDS ↑mAP↑mATEmASE↓mAOE↓mAVE↓mAAE↓
Baseline (pillar,without distillation)62.8053.2034.6226.5631.6226.0219.10
Self-distillation (pillar->pillar)63.4153.8934.2126.1931.1125.6718.54
Baseline (voxel,without distillation)66.0458.6233.3326.3428.8025.1119.06
Self-distillation (voxel->voxel)66.4559.2531.1725.7730.7325.7218.77
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MethodNDS ↑mAP↑mATE↓mASE↓mAOE↓mAVE↓mAAE↓
Sparse-→sparse (pillar)42.1238.8944.0127.7864.01144.0139.21
Dense-→sparse (pillar)42.7939.1043.8927.7764.01143.9739.11
Sparse-→sparse (voxel)59.5549.8431.1725.7733.7332.2220.22
Dense->sparse (voxel)59.8950.1231.1125.7633.7032.1920.11
", + "type": "table", + "image_path": "3127971340c8ac67568bf68fe9b1c4ec413354755633754383992b5123eda9d0.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 109, + 152, + 502, + 172.0 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 109, + 172.0, + 502, + 192.0 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 109, + 192.0, + 502, + 212.0 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 5.0 + }, + { + "type": "text", + "bbox": [ + 107, + 218, + 505, + 251 + ], + "lines": [], + "index": 9, + "bbox_fs": [ + 105, + 217, + 506, + 252 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 256, + 505, + 322 + ], + "lines": [ + { + "bbox": [ + 106, + 257, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 257, + 505, + 268 + ], + "score": 1.0, + "content": "Training & inference. We use AdamW [67] to train the model. The weight decay for AdamW is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 265, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 106, + 267, + 128, + 277 + ], + "score": 0.89, + "content": "1 0 ^ { - 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 265, + 383, + 280 + ], + "score": 1.0, + "content": ". Following a cyclic schedule [68], the learning rate is initially", + "type": "text" + }, + { + "bbox": [ + 383, + 267, + 405, + 277 + ], + "score": 0.9, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 265, + 506, + 280 + ], + "score": 1.0, + "content": "and gradually increased", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 276, + 507, + 291 + ], + "spans": [ + { + "bbox": [ + 104, + 276, + 117, + 291 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 117, + 277, + 138, + 289 + ], + "score": 0.89, + "content": "1 0 ^ { - 3 }", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 276, + 258, + 291 + ], + "score": 1.0, + "content": ", which is finally decreased to", + "type": "text" + }, + { + "bbox": [ + 259, + 277, + 280, + 288 + ], + "score": 0.9, + "content": "1 0 ^ { - 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 276, + 507, + 291 + ], + "score": 1.0, + "content": ". The model is initialized with a pre-trained PointPillars", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 302 + ], + "score": 1.0, + "content": "network on the same dataset. We train for 20 epochs on 8 RTX 3090 GPUs. During inference, we", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 506, + 313 + ], + "score": 1.0, + "content": "take the top 100 objects with highest classification scores as the final predictions.We do not use any", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 310, + 507, + 324 + ], + "spans": [ + { + "bbox": [ + 105, + 310, + 507, + 324 + ], + "score": 1.0, + "content": "post-processing such as NMS. For evaluation, we use the toolkit provided with the nuScenes dataset.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13.5, + "bbox_fs": [ + 104, + 257, + 507, + 324 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 335, + 198, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 334, + 200, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 200, + 349 + ], + "score": 1.0, + "content": "6.2 Object DGCNN", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 355, + 505, + 454 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 367 + ], + "score": 1.0, + "content": "We compare to top-performing methods on the nuScenes dataset in Table 1. PointPillars [4] is an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "anchor-based method with reasonable trade-off between performance and efficiency. FreeAnchor [64]", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 377, + 507, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 377, + 507, + 389 + ], + "score": 1.0, + "content": "extends PointPillars by learning how to assign anchors to the ground-truth. RegNetX-400MF-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 505, + 400 + ], + "score": 1.0, + "content": "SECFPN [65] uses neural architecture search (NAS) to learn a flexible neural network for 3D", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 106, + 399, + 505, + 411 + ], + "score": 1.0, + "content": "detection; it is essentially a variant of PointPillars with an enhanced backbone network. Different", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 410, + 505, + 423 + ], + "score": 1.0, + "content": "from anchor-based methods, Pillar-OD [5] makes predictions per pillar, alleviating the class imbalance", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 434 + ], + "score": 1.0, + "content": "issue caused by anchors. CenterPoint [6] exploits similar detection heads, with better performance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 505, + 444 + ], + "score": 1.0, + "content": "using better training scheduling and data augmentation. For these methods, we use re-implementations", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 441, + 424, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 424, + 455 + ], + "score": 1.0, + "content": "in MMDetection3D [69], which match the performances in the original papers.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 356, + 507, + 455 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 459, + 505, + 514 + ], + "lines": [ + { + "bbox": [ + 105, + 458, + 506, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 506, + 472 + ], + "score": 1.0, + "content": "We mainly compare to CenterPoint with both PointPillars and SparseConv backbones, denoted as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 104, + 467, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 104, + 467, + 505, + 484 + ], + "score": 1.0, + "content": "“voxel” and “pillar” respectively. Our method outperforms other methods significantly including", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 479, + 506, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 495 + ], + "score": 1.0, + "content": "CenterPoint with NMS. Without NMS, the performance of CenterPoint drops considerably while our", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 104, + 491, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 104, + 491, + 505, + 506 + ], + "score": 1.0, + "content": "method is unaffected by NMS. This finding verifies the DGCNN implicitly models object relations", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 502, + 230, + 516 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 230, + 516 + ], + "score": 1.0, + "content": "and removes redundant boxes.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29, + "bbox_fs": [ + 104, + 458, + 506, + 516 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 527, + 218, + 538 + ], + "lines": [ + { + "bbox": [ + 106, + 526, + 218, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 218, + 540 + ], + "score": 1.0, + "content": "6.3 Set-to-set distillation", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 546, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "spans": [ + { + "bbox": [ + 106, + 547, + 505, + 559 + ], + "score": 1.0, + "content": "In this section, we present experiments involving our set-to-set distillation pipeline. We conduct", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 106, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "three types of distillation. First, we distill a teacher model with a SparseConv backbone to a student", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 568, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 335, + 582 + ], + "score": 1.0, + "content": "model with a PointPillars backbone (denoted as “voxel", + "type": "text" + }, + { + "bbox": [ + 335, + 570, + 345, + 579 + ], + "score": 0.77, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 568, + 505, + 582 + ], + "score": 1.0, + "content": "pillar”). This aligns with the common", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 505, + 591 + ], + "score": 1.0, + "content": "knowledge distillation setup for classification. We compare to feature-based distillation and pseudo", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "label based methods. The objective of feature-based distillation is to align the middle-level features", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 600, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 615 + ], + "score": 1.0, + "content": "of the teacher model and the student model while the pseudo label based methods generate pseudo", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 505, + 625 + ], + "score": 1.0, + "content": "training examples with the pre-trained teacher networks. As Table 2 shows, our set-to-set distillation", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 623, + 505, + 635 + ], + "score": 1.0, + "content": "achieves better performance, confirming that distilling the last stage of the object detection model is", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 634, + 280, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 280, + 648 + ], + "score": 1.0, + "content": "more effective than distilling feature maps.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 547, + 505, + 648 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 650, + 505, + 695 + ], + "lines": [ + { + "bbox": [ + 105, + 650, + 505, + 663 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 351, + 663 + ], + "score": 1.0, + "content": "Second, we perform self-distillation [49] (denoted as “voxel", + "type": "text" + }, + { + "bbox": [ + 352, + 652, + 362, + 660 + ], + "score": 0.8, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 650, + 434, + 663 + ], + "score": 1.0, + "content": "voxel” and “pillar", + "type": "text" + }, + { + "bbox": [ + 434, + 652, + 444, + 660 + ], + "score": 0.82, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 650, + 505, + 663 + ], + "score": 1.0, + "content": "pillar”), where", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 661, + 506, + 674 + ], + "spans": [ + { + "bbox": [ + 105, + 661, + 506, + 674 + ], + "score": 1.0, + "content": "the teacher and the student are identical and take the same point clouds as input. As Table 3 shows,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "spans": [ + { + "bbox": [ + 105, + 672, + 505, + 685 + ], + "score": 1.0, + "content": "even when the teacher network and the student network have the same capacity, self-distillation still", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 684, + 435, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 684, + 435, + 695 + ], + "score": 1.0, + "content": "introduces a performance boost. This finding is consistent with the results in [49].", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 43.5, + "bbox_fs": [ + 105, + 650, + 506, + 695 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "Finally, we try distillation with privileged information [56], where the teacher gets access to privileged", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "information but the student does not. Following [10], the teacher takes dense point clouds, and the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 325, + 85 + ], + "score": 1.0, + "content": "student takes sparse point clouds (denoted as \"dense", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 325, + 74, + 335, + 83 + ], + "score": 0.81, + "content": "", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 336, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "sparse\"). To limit computation time, we", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "train each model over a shorter period of time. The goal is for the student model to learn the same", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "representations as the teacher model without knowing the dense inputs. In Table 4, we compare", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 504, + 117 + ], + "score": 1.0, + "content": "this setup with self-distillation, where the difference is the teacher model and the student model", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "take the same sparse point clouds in self-distillation. The student achieves better performance when", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "the teacher takes dense point clouds. The result suggests that set-to-set knowledge distillation is an", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 371, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 371, + 150 + ], + "score": 1.0, + "content": "effective approach to transfer insight from privileged information.", + "type": "text", + "cross_page": true + } + ], + "index": 6 + } + ], + "index": 46.5, + "bbox_fs": [ + 105, + 699, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 150 + ], + "lines": [ + { + "bbox": [ + 105, + 73, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 105, + 73, + 325, + 85 + ], + "score": 1.0, + "content": "student takes sparse point clouds (denoted as \"dense", + "type": "text" + }, + { + "bbox": [ + 325, + 74, + 335, + 83 + ], + "score": 0.81, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 73, + 505, + 85 + ], + "score": 1.0, + "content": "sparse\"). To limit computation time, we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "train each model over a shorter period of time. The goal is for the student model to learn the same", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "representations as the teacher model without knowing the dense inputs. In Table 4, we compare", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 106, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 106, + 504, + 117 + ], + "score": 1.0, + "content": "this setup with self-distillation, where the difference is the teacher model and the student model", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 129 + ], + "score": 1.0, + "content": "take the same sparse point clouds in self-distillation. The student achieves better performance when", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "the teacher takes dense point clouds. The result suggests that set-to-set knowledge distillation is an", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 138, + 371, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 371, + 150 + ], + "score": 1.0, + "content": "effective approach to transfer insight from privileged information.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "title", + "bbox": [ + 106, + 164, + 167, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 169, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 169, + 177 + ], + "score": 1.0, + "content": "6.4 Ablation", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "score": 1.0, + "content": "We provide ablation studies on different components of our model to verify assorted design choices.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "First, we study the improvements of DGCNN over its counterpart, multi-head self-attention [9]. The", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 206, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 506, + 222 + ], + "score": 1.0, + "content": "multi-head self-attention has 8 heads with embedding dimension 256 and LayerNorm [70], following", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "common usage. The DGCNN has two EdgeConv layers with dimensions [256, 256]. The number of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 147, + 243 + ], + "score": 1.0, + "content": "neighbors", + "type": "text" + }, + { + "bbox": [ + 147, + 230, + 158, + 240 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 229, + 506, + 243 + ], + "score": 1.0, + "content": "in EdgeConv is 16. In principle, DGCNN is a sparse version of multi-head self-attention;", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "the sparse structure reduces overhead in back-propagation and leads to sharper “attention maps” as", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 251, + 216, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 216, + 264 + ], + "score": 1.0, + "content": "well as faster convergence.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11 + }, + { + "type": "table", + "bbox": [ + 108, + 284, + 305, + 323 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 109, + 272, + 306, + 283 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 271, + 307, + 284 + ], + "spans": [ + { + "bbox": [ + 107, + 271, + 307, + 284 + ], + "score": 1.0, + "content": "Table 5: DGCNN versus multi-head self-attention.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "table_body", + "bbox": [ + 108, + 284, + 305, + 323 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 284, + 305, + 323 + ], + "spans": [ + { + "bbox": [ + 108, + 284, + 305, + 323 + ], + "score": 0.969, + "html": "
MethodMetricMulti-head self-attentionDGCNN
NDS39.8941.32
mAP36.3537.81
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#layersMetric123456
NDS35.9139.7541.1541.2641.0741.32
mAP32.3236.5437.2537.7537.7837.81
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# neigbhorsMetric148163264
NDS40.2140.4540.5141.3240.1739.80
mAP36.8137.1537.4637.8137.1236.70
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Methodfiltered boxesremainingboxesall boxes
CenterPoint0.07640.18590.0829
Ours0.12220.17110.1604
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ModuleComplexity# parameters
DGCNNO(n²d)262144
Multi-head self-attentionO(n²d)263168
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Furthermore, we study the effect of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 314, + 334, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 314, + 334, + 506, + 345 + ], + "score": 1.0, + "content": "number of neighbors in DGCNN. When it is 1,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 314, + 344, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 314, + 344, + 505, + 355 + ], + "score": 1.0, + "content": "The model reduces to an architecture without", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 315, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 315, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "object interaction. As we increase the number, it", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 313, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 313, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "approaches multi-head self-attention. As shown", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 313, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 313, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "in Table 7, the sweet spot is 16, which appears", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 313, + 387, + 486, + 401 + ], + "spans": [ + { + "bbox": [ + 313, + 387, + 486, + 401 + ], + "score": 1.0, + "content": "to balance object interactions and sparsity.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38.5 + }, + { + "type": "text", + "bbox": [ + 315, + 405, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 314, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 314, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "We also investigate improvements introduced", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 314, + 415, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 314, + 415, + 506, + 426 + ], + "score": 1.0, + "content": "when more DGCNNs are stacked in Table 6.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 313, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 313, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "This result suggests it is beneficial to incorporate", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 313, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 313, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "multiple DGCNNs to model the dynamic object", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 313, + 447, + 354, + 460 + ], + "spans": [ + { + "bbox": [ + 313, + 447, + 354, + 460 + ], + "score": 1.0, + "content": "relations.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 315, + 464, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 314, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 314, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "Moreover, we hypothesize our method produces", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 314, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 314, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "different distribution of the output scores with", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 313, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 313, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "respect to overlapping boxes. To verify this hy-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 313, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 313, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "pothesis (Table 8, we compute average scores", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 313, + 508, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 313, + 508, + 506, + 520 + ], + "score": 1.0, + "content": "for three types of boxes: filtered boxes after", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 313, + 519, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 313, + 519, + 506, + 531 + ], + "score": 1.0, + "content": "NMS, remaining boxes after NMS, and all boxes.", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 313, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 313, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "Below we show the results. We also compute", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 313, + 541, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 313, + 541, + 506, + 552 + ], + "score": 1.0, + "content": "the percentage of filtered boxes by NMS in our", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 313, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 313, + 551, + 477, + 563 + ], + "score": 1.0, + "content": "method and Centerpoint. In our method,", + "type": "text" + }, + { + "bbox": [ + 478, + 552, + 505, + 562 + ], + "score": 0.88, + "content": "2 1 . 9 \\%", + "type": "inline_equation" + } + ], + "index": 58 + } + ], + "index": 54 + }, + { + "type": "text", + "bbox": [ + 108, + 563, + 502, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 504, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 273, + 575 + ], + "score": 1.0, + "content": "boxes are removed while in Centerpoint", + "type": "text" + }, + { + "bbox": [ + 273, + 563, + 306, + 573 + ], + "score": 0.87, + "content": "8 5 . 1 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 561, + 504, + 575 + ], + "score": 1.0, + "content": "boxes are filtered. Hence, we conclude that our", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 573, + 397, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 397, + 586 + ], + "score": 1.0, + "content": "method indeed exhibits a different distribution pattern from Centerpoint.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 59.5 + }, + { + "type": "text", + "bbox": [ + 108, + 590, + 504, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 504, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 504, + 601 + ], + "score": 1.0, + "content": "Finally, we include a complexity comparison between DGCNN and Multi-head self-attention. Table 9", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 600, + 408, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 408, + 613 + ], + "score": 1.0, + "content": "shows the results; DGCNN layer is on a par with Multi-head self-attention.", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 61.5 + }, + { + "type": "title", + "bbox": [ + 107, + 630, + 183, + 643 + ], + "lines": [ + { + "bbox": [ + 104, + 627, + 185, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 185, + 646 + ], + "score": 1.0, + "content": "7 Conclusion", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 63 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Object DGCNN is a highly-efficient 3D object detector for point clouds. It is able to learn object", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "interactions via dynamic graphs and is optimized through a set-to-set loss, leading to NMS-free", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "detection. The success of Object DGCNN indicates that many post-processing operations in 3D", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "object detection are likely unnecessary and can be replaced with suitable neural network modules.", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 105, + 699, + 507, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 507, + 712 + ], + "score": 1.0, + "content": "Moreover, we introduce a set-to-set knowledge distillation pipeline enabled by the Object DGCNN.", + "type": "text" + } + ], + "index": 68 + }, + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "This new pipeline significantly simplifies knowledge distillation for 3D object detection and may", + "type": "text" + } + ], + "index": 69 + } + ], + "index": 66.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 742, + 308, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 73, + 505, + 150 + ], + "lines": [], + "index": 3, + "bbox_fs": [ + 105, + 73, + 506, + 150 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 106, + 164, + 167, + 176 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 169, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 169, + 177 + ], + "score": 1.0, + "content": "6.4 Ablation", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 263 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 506, + 199 + ], + "score": 1.0, + "content": "We provide ablation studies on different components of our model to verify assorted design choices.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 505, + 209 + ], + "score": 1.0, + "content": "First, we study the improvements of DGCNN over its counterpart, multi-head self-attention [9]. The", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 206, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 206, + 506, + 222 + ], + "score": 1.0, + "content": "multi-head self-attention has 8 heads with embedding dimension 256 and LayerNorm [70], following", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 506, + 231 + ], + "score": 1.0, + "content": "common usage. The DGCNN has two EdgeConv layers with dimensions [256, 256]. The number of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 229, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 229, + 147, + 243 + ], + "score": 1.0, + "content": "neighbors", + "type": "text" + }, + { + "bbox": [ + 147, + 230, + 158, + 240 + ], + "score": 0.8, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 229, + 506, + 243 + ], + "score": 1.0, + "content": "in EdgeConv is 16. In principle, DGCNN is a sparse version of multi-head self-attention;", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 241, + 505, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 253 + ], + "score": 1.0, + "content": "the sparse structure reduces overhead in back-propagation and leads to sharper “attention maps” as", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 251, + 216, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 251, + 216, + 264 + ], + "score": 1.0, + "content": "well as faster convergence.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 186, + 506, + 264 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 284, + 305, + 323 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 109, + 272, + 306, + 283 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 271, + 307, + 284 + ], + "spans": [ + { + "bbox": [ + 107, + 271, + 307, + 284 + ], + "score": 1.0, + "content": "Table 5: DGCNN versus multi-head self-attention.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "table_body", + "bbox": [ + 108, + 284, + 305, + 323 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 284, + 305, + 323 + ], + "spans": [ + { + "bbox": [ + 108, + 284, + 305, + 323 + ], + "score": 0.969, + "html": "
MethodMetricMulti-head self-attentionDGCNN
NDS39.8941.32
mAP36.3537.81
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#layersMetric123456
NDS35.9139.7541.1541.2641.0741.32
mAP32.3236.5437.2537.7537.7837.81
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# neigbhorsMetric148163264
NDS40.2140.4540.5141.3240.1739.80
mAP36.8137.1537.4637.8137.1236.70
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Methodfiltered boxesremainingboxesall boxes
CenterPoint0.07640.18590.0829
Ours0.12220.17110.1604
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ModuleComplexity# parameters
DGCNNO(n²d)262144
Multi-head self-attentionO(n²d)263168
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Furthermore, we study the effect of", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 314, + 334, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 314, + 334, + 506, + 345 + ], + "score": 1.0, + "content": "number of neighbors in DGCNN. When it is 1,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 314, + 344, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 314, + 344, + 505, + 355 + ], + "score": 1.0, + "content": "The model reduces to an architecture without", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 315, + 355, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 315, + 355, + 505, + 367 + ], + "score": 1.0, + "content": "object interaction. As we increase the number, it", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 313, + 366, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 313, + 366, + 505, + 378 + ], + "score": 1.0, + "content": "approaches multi-head self-attention. As shown", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 313, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 313, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "in Table 7, the sweet spot is 16, which appears", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 313, + 387, + 486, + 401 + ], + "spans": [ + { + "bbox": [ + 313, + 387, + 486, + 401 + ], + "score": 1.0, + "content": "to balance object interactions and sparsity.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 38.5, + "bbox_fs": [ + 313, + 268, + 506, + 401 + ] + }, + { + "type": "text", + "bbox": [ + 315, + 405, + 505, + 459 + ], + "lines": [ + { + "bbox": [ + 314, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 314, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "We also investigate improvements introduced", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 314, + 415, + 506, + 426 + ], + "spans": [ + { + "bbox": [ + 314, + 415, + 506, + 426 + ], + "score": 1.0, + "content": "when more DGCNNs are stacked in Table 6.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 313, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 313, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "This result suggests it is beneficial to incorporate", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 313, + 437, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 313, + 437, + 506, + 450 + ], + "score": 1.0, + "content": "multiple DGCNNs to model the dynamic object", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 313, + 447, + 354, + 460 + ], + "spans": [ + { + "bbox": [ + 313, + 447, + 354, + 460 + ], + "score": 1.0, + "content": "relations.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 47, + "bbox_fs": [ + 313, + 404, + 506, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 315, + 464, + 505, + 563 + ], + "lines": [ + { + "bbox": [ + 314, + 465, + 505, + 477 + ], + "spans": [ + { + "bbox": [ + 314, + 465, + 505, + 477 + ], + "score": 1.0, + "content": "Moreover, we hypothesize our method produces", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 314, + 475, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 314, + 475, + 505, + 487 + ], + "score": 1.0, + "content": "different distribution of the output scores with", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 313, + 486, + 506, + 499 + ], + "spans": [ + { + "bbox": [ + 313, + 486, + 506, + 499 + ], + "score": 1.0, + "content": "respect to overlapping boxes. To verify this hy-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 313, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 313, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "pothesis (Table 8, we compute average scores", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 313, + 508, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 313, + 508, + 506, + 520 + ], + "score": 1.0, + "content": "for three types of boxes: filtered boxes after", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 313, + 519, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 313, + 519, + 506, + 531 + ], + "score": 1.0, + "content": "NMS, remaining boxes after NMS, and all boxes.", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 313, + 529, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 313, + 529, + 505, + 542 + ], + "score": 1.0, + "content": "Below we show the results. We also compute", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 313, + 541, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 313, + 541, + 506, + 552 + ], + "score": 1.0, + "content": "the percentage of filtered boxes by NMS in our", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 313, + 551, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 313, + 551, + 477, + 563 + ], + "score": 1.0, + "content": "method and Centerpoint. In our method,", + "type": "text" + }, + { + "bbox": [ + 478, + 552, + 505, + 562 + ], + "score": 0.88, + "content": "2 1 . 9 \\%", + "type": "inline_equation" + } + ], + "index": 58 + } + ], + "index": 54, + "bbox_fs": [ + 313, + 465, + 506, + 563 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 563, + 502, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 504, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 273, + 575 + ], + "score": 1.0, + "content": "boxes are removed while in Centerpoint", + "type": "text" + }, + { + "bbox": [ + 273, + 563, + 306, + 573 + ], + "score": 0.87, + "content": "8 5 . 1 6 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 561, + 504, + 575 + ], + "score": 1.0, + "content": "boxes are filtered. Hence, we conclude that our", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 573, + 397, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 397, + 586 + ], + "score": 1.0, + "content": "method indeed exhibits a different distribution pattern from Centerpoint.", + "type": "text" + } + ], + "index": 60 + } + ], + "index": 59.5, + "bbox_fs": [ + 105, + 561, + 504, + 586 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 590, + 504, + 612 + ], + "lines": [ + { + "bbox": [ + 106, + 589, + 504, + 601 + ], + "spans": [ + { + "bbox": [ + 106, + 589, + 504, + 601 + ], + "score": 1.0, + "content": "Finally, we include a complexity comparison between DGCNN and Multi-head self-attention. Table 9", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 106, + 600, + 408, + 613 + ], + "spans": [ + { + "bbox": [ + 106, + 600, + 408, + 613 + ], + "score": 1.0, + "content": "shows the results; DGCNN layer is on a par with Multi-head self-attention.", + "type": "text" + } + ], + "index": 62 + } + ], + "index": 61.5, + "bbox_fs": [ + 106, + 589, + 504, + 613 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 630, + 183, + 643 + ], + "lines": [ + { + "bbox": [ + 104, + 627, + 185, + 646 + ], + "spans": [ + { + "bbox": [ + 104, + 627, + 185, + 646 + ], + "score": 1.0, + "content": "7 Conclusion", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 63 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "Object DGCNN is a highly-efficient 3D object detector for point clouds. It is able to learn object", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "interactions via dynamic graphs and is optimized through a set-to-set loss, leading to NMS-free", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 505, + 691 + ], + "score": 1.0, + "content": "detection. 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Beyond the direct usage of our model, our", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "experiments suggest several future directions to address current limitations. For example, our method", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "is initialized with a pre-trained backbone network. 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Finally, the large amount of unlabeled data available at training time can serve", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "as another type of privileged information to apply self-supervised learning to 3D domains through", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 150, + 192, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 192, + 160 + ], + "score": 1.0, + "content": "set-to-set distillation.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + } + ], + "index": 66.5, + "bbox_fs": [ + 105, + 655, + 507, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 72, + 505, + 160 + ], + "lines": [ + { + "bbox": [ + 106, + 73, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 73, + 506, + 85 + ], + "score": 1.0, + "content": "be applicable to other tasks like 3D model compression. Beyond the direct usage of our model, our", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "experiments suggest several future directions to address current limitations. For example, our method", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 108 + ], + "score": 1.0, + "content": "is initialized with a pre-trained backbone network. 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Finally, the large amount of unlabeled data available at training time can serve", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 151 + ], + "score": 1.0, + "content": "as another type of privileged information to apply self-supervised learning to 3D domains through", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 150, + 192, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 192, + 160 + ], + "score": 1.0, + "content": "set-to-set distillation.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "Potential impact. Our method aims to improve the object detection pipeline, which is crucial for the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "safety of autonomous driving systems. One potential negative impact of our work is that it still lacks", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 505, + 200 + ], + "score": 1.0, + "content": "theoretical guarantees, similar to many deep learning methods. Future work to improve applicability", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 197, + 415, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 415, + 212 + ], + "score": 1.0, + "content": "in this domain might consider challenges of explainability and transparency.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 107, + 224, + 220, + 238 + ], + "lines": [ + { + "bbox": [ + 104, + 222, + 222, + 241 + ], + "spans": [ + { + "bbox": [ + 104, + 222, + 222, + 241 + ], + "score": 1.0, + "content": "8 Acknowledgement", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 249, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 262 + ], + "score": 1.0, + "content": "The MIT Geometric Data Processing group acknowledges the generous support of Army Research", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 260, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 505, + 272 + ], + "score": 1.0, + "content": "Office grants W911NF2010168 and W911NF2110293, of Air Force Office of Scientific Research", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 270, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 284 + ], + "score": 1.0, + "content": "award FA9550-19-1-031, of National Science Foundation grants IIS-1838071 and CHS-1955697,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "from the CSAIL Systems that Learn program, from the MIT–IBM Watson AI Laboratory, from", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 305 + ], + "score": 1.0, + "content": "the Toyota–CSAIL Joint Research Center, from a gift from Adobe Systems, from an MIT.nano", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "Immersion Lab/NCSOFT Gaming Program seed grant, and from the Skoltech–MIT Next Generation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 313, + 147, + 329 + ], + "spans": [ + { + "bbox": [ + 104, + 313, + 147, + 329 + ], + "score": 1.0, + "content": "Program.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 340, + 163, + 353 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 165, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 165, + 355 + ], + "score": 1.0, + "content": "References", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 109, + 356, + 506, + 724 + ], + "lines": [ + { + "bbox": [ + 109, + 357, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 109, + 357, + 506, + 371 + ], + "score": 1.0, + "content": "[1] Shaoshuai Shi, Xiaogang Wang, and Hongsheng Li. 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MethodNDS↑mAP个mATE√mASE↓mAOE↓mAVE↓mAAE↓NMS
PointPillars [4]53.340.0二二:二二√
SSN[63]54.8341.56二二二二-√
FreeAnchor [64]55.343.7二二-二二√
RegNetX-400MF-SECFPN[65]55.241.2----√
Pillar-OD[5]56.8444.41-----√
CenterPoint (pillar) [6] *59.5647.4831.2725.8133.7832.2520.20√
CenterPoint (pillar) [6] *55.0840.2735.1426.4436.7532.6619.55
CenterPoint (voxel) [6] ‡64.1954.9929.8325.7132.5626.0818.89√
CenterPoint (voxel) [6]‡57.0045.3231.6627.1440.4737.2320.14
Ours (pillar) *62.9753.3134.6226.5631.6126.0218.71√
Ours (pillar) *62.8053.2034.6226.5631.6226.0719.10
Ours (voxel)‡66.1058.7333.3126.3228.8025.1119.08√
Ours (voxel) t66.0458.6233.3326.3428.8025.1119.06
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MethodNDS↑mAP↑mATE↓mASE↓mAOE↓mAVE↓mAAE↓
Baseline (without distillation)62.8053.2034.6226.5631.6226.0219.10
Feature distill (voxel->pillar)62.8453.2934.5526.7831.6126.1119.02
Pseudo labels (voxel-→pillar)63.1853.3134.5826.5531.2925.9918.87
Set-to-set distill (voxel-→pillar)63.3753.8934.3426.2531.0125.5718.77
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MethodNDS ↑mAP↑mATE↓mASE↓mAOE↓mAVE↓mAAE↓
Sparse-→sparse (pillar)42.1238.8944.0127.7864.01144.0139.21
Dense-→sparse (pillar)42.7939.1043.8927.7764.01143.9739.11
Sparse-→sparse (voxel)59.5549.8431.1725.7733.7332.2220.22
Dense->sparse (voxel)59.8950.1231.1125.7633.7032.1920.11
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MethodNDS ↑mAP↑mATEmASE↓mAOE↓mAVE↓mAAE↓
Baseline (pillar,without distillation)62.8053.2034.6226.5631.6226.0219.10
Self-distillation (pillar->pillar)63.4153.8934.2126.1931.1125.6718.54
Baseline (voxel,without distillation)66.0458.6233.3326.3428.8025.1119.06
Self-distillation (voxel->voxel)66.4559.2531.1725.7730.7325.7218.77
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MethodMetricMulti-head self-attentionDGCNN
NDS39.8941.32
mAP36.3537.81
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ModuleComplexity# parameters
DGCNNO(n²d)262144
Multi-head self-attentionO(n²d)263168
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# neigbhorsMetric148163264
NDS40.2140.4540.5141.3240.1739.80
mAP36.8137.1537.4637.8137.1236.70
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Methodfiltered boxesremainingboxesall boxes
CenterPoint0.07640.18590.0829
Ours0.12220.17110.1604
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#layersMetric123456
NDS35.9139.7541.1541.2641.0741.32
mAP32.3236.5437.2537.7537.7837.81
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/dev/null +++ b/parse/train/SklVEnR5K7/images/fc3fc005ad0c6c35b42051c1829110a12875808e7553f2a75e56d966788cb20a.jpg @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4c934551e05bc2fe909f9e50f95cbc3a8942c663fd157d68f8ce6c2a7029a5df +size 61806 diff --git a/parse/train/SyMDXnCcF7/SyMDXnCcF7.md b/parse/train/SyMDXnCcF7/SyMDXnCcF7.md new file mode 100644 index 0000000000000000000000000000000000000000..6e6e31344449ac3122ed64a3a4aa55b0e3185f67 --- /dev/null +++ b/parse/train/SyMDXnCcF7/SyMDXnCcF7.md @@ -0,0 +1,3278 @@ +# A MEAN FIELD THEORY OF BATCH NORMALIZATION + +Greg Yang†∗, Jeffrey Pennington◦, Vinay Rao◦, Jascha Sohl-Dickstein◦, & Samuel S. Schoenholz◦ + +Microsoft Research AI, $\circ$ Google Brain gregyang@microsoft.com, {jpennin,vinaysrao,jaschasd,schsam}@google.com + +# ABSTRACT + +We develop a mean field theory for batch normalization in fully-connected feedforward neural networks. In so doing, we provide a precise characterization of signal propagation and gradient backpropagation in wide batch-normalized networks at initialization. Our theory shows that gradient signals grow exponentially in depth and that these exploding gradients cannot be eliminated by tuning the initial weight variances or by adjusting the nonlinear activation function. Indeed, batch normalization itself is the cause of gradient explosion. As a result, vanilla batch-normalized networks without skip connections are not trainable at large depths for common initialization schemes, a prediction that we verify with a variety of empirical simulations. While gradient explosion cannot be eliminated, it can be reduced by tuning the network close to the linear regime, which improves the trainability of deep batch-normalized networks without residual connections. Finally, we investigate the learning dynamics of batch-normalized networks and observe that after a single step of optimization the networks achieve a relatively stable equilibrium in which gradients have dramatically smaller dynamic range. Our theory leverages Laplace, Fourier, and Gegenbauer transforms and we derive new identities that may be of independent interest. + +# 1 INTRODUCTION + +Deep neural networks have been enormously successful across a broad range of disciplines. These successes are often driven by architectural innovations. For example, the combination of convolutions (LeCun et al., 1990), residual connections (He et al., 2015), and batch normalization (Ioffe & Szegedy, 2015) has allowed for the training of very deep networks and these components have become essential parts of models in vision (Zoph et al.), language (Chen & Wu, 2017), and reinforcement learning (Silver et al., 2017). However, a fundamental problem that has accompanied this rapid progress is a lack of theoretical clarity. An important consequence of this gap between theory and experiment is that two important issues become conflated. In particular, it is generally unclear whether novel neural network components improve generalization or whether they merely increase the fraction of hyperparameter configurations where good generalization can be achieved. Resolving this confusion has the promise of allowing researchers to more effectively and deliberately design neural networks. + +Recently, progress has been made (Poole et al., 2016; Schoenholz et al., 2016; Daniely et al., 2016; Pennington et al., 2017; Hanin & Rolnick, 2018; Yang, 2019) in this direction by considering neural networks at initialization, before any training has occurred. In this case, the parameters of the network are random variables which induces a distribution of the activations of the network as well as the gradients. Studying these distributions is equivalent to understanding the prior over functions that these random neural networks compute. Picking hyperparameters that correspond to well-conditioned priors ensures that the neural network will be trainable and this fact has been extensively verified experimentally. However, to fulfill its promise of making neural network design less of a black box, these techniques must be applied to neural network architectures that are used in practice. Over the past year, this gap has closed significantly and theory for networks with skip connections (Yang & Schoenholz, 2017; 2018), convolutional networks (Xiao et al., 2018), and gated recurrent networks (Chen et al., 2018; Gilboa et al., 2019) have been developed. More recently, Yang (2019) devised a formalism that extends this approach to include an even wider range of architectures. + +Before state-of-the-art models can be analyzed in this framework, a slowly-decreasing number of architectural innovations must be studied. One particularly important component that has thus-far remained elusive is batch normalization. + +Our Contributions. In this paper, we develop a theory of fully-connected networks with batch normalization whose weights and biases are randomly distributed. A significant complication in the case of batch normalization (compared to e.g. layer normalization or weight normalization) is that the statistics of the network depend non-locally on the entire batch. Thus, our first main result is to recast the theory for random fully-connected networks so that it can be applied to batches of data. We then extend the theory to include batch normalization explicitly and validate this theory against Monte-Carlo simulations. We show that as in previous cases we can leverage our theory to predict valid hyperparameter configurations. + +In the process of our investigation, we identify a number of previously unknown properties of batch normalization that make training unstable. In particular, for most nonlinearities used in practice, batchnorm in a deep, randomly initialized network induces high degree of symmetry in the embeddings of the samples in the batch $( \mathrm { T h m } \ 3 . 4 )$ . Whenever this symmetry takes hold, we show that for any choice of nonlinearity, gradients of fully-connected networks with batch normalization explode exponentially in the depth of the network $\mathrm { T h m } \ 3 . 9 $ ). This imposes strong limits on the maximum trainable depth of batch normalized networks. This limit can be lifted partially but not completely by pushing activation functions to be more linear at initialization. It might seem that such gradient explosion ought to lead to learning dynamics that are unfavorable. However, we show that networks with batch normalization causes the scale of the gradients to naturally equilibrate after a single step of gradient descent (provided the initial gradients are not so large as to cause numerical instabilities). For shallower networks, this equilibrating effect is sufficient to allow adequate training. + +Finally, we note that there is a related vein of research that has emerged that leverages the prior over functions induced by random networks to perform exact Bayesian inference (Lee et al., 2017; de G. Matthews et al., 2018; Novak et al., 2019; Garriga-Alonso et al., 2019; Yang, 2019). One of the natural consequences of this work is that the prior for networks with batch normalization can be computed exactly in the wide network limit. As such, it is now possible to perform exact Bayesian inference in the case of wide neural networks with batch normalization. + +# 2 RELATED WORK + +Batch normalization has rapidly become an essential part of the deep learning toolkit. Since then, a number of similar modifications have been proposed including layer normalization (Ba et al., 2016) and weight normalization (Salimans & Kingma, 2016). Comparisons of performance between these different schemes have been challenging and inconclusive (Gitman & Ginsburg, 2017). The original introduction of batchnorm in Ioffe & Szegedy (2015) proposed that batchnorm prevents “internal covariate shift” as an explanation for its effectiveness. Since then, several papers have approached batchnorm from a theoretical angle, especially following Ali Rahimi’s catalyzing call to action at NIPS 2017. Balduzzi et al. (2017) found that batchnorm in resnets allow deep gradient signal propagation in contrast to the case without batchnorm. Santurkar et al. (2018) found that batchnorm does not help covariate shift but helps by smoothing loss landscape. Bjorck et al. (2018) reached the opposite conclusion as our paper for residual networks with batchnorm, that batchnorm works in this setting because it induces beneficial gradient dynamics and thus allows a much bigger learning rate. Luo et al. (2018) explores similar ideas that batchnorm allows large learning rates and likewise uses random matrix theory to support their claims. Kohler et al. (2018) identified situations in which batchnorm can provably induce acceleration in training. Of the above that mathematically analyze batchnorm, all but Santurkar et al. (2018) make simplifying assumptions on the form of batchnorm and typically do not have gradients flowing through the batch variance. Even Santurkar et al. (2018) only analyzes a vanilla network which gets added a single batchnorm at a single moment in training. Our analysis here on the other hand works for networks with arbitrarily many batchnorm layers with very general activation functions, and indeed, this deep stacking of batchnorm is precisely what leads to gradient explosion. It is an initialization time analysis for networks of infinite width, but we show experimentally that the resulting insights predict both training and test time behavior. + +We remark that Philipp & Carbonell (2018) has also empirically noted gradient explosion happens in deep batchnorm networks with various nonlinearities. In this work, in contrast, we develop a precise theoretical characterization of gradient statistics and, as a result, we are able to make significantly stronger conclusions. + +# 3 THEORY + +We begin with a brief recapitulation of mean field theory in the fully-connected setting. In addition to recounting earlier results, we rephrase the formalism developed previously to compute statistics of neural networks over a batch of data. Later, we will extend the theory to include batch normalization. We consider a fully-connected network of depth $L$ whose layers have width $N _ { l }$ , activation function1 $\phi$ , weights $W ^ { l } \bar { \mathbf { \Xi } } \in \mathbb { R } ^ { N _ { l - 1 } \times N _ { l } }$ , and biases $\hat { b } ^ { l } \in \mathbb { R } ^ { N _ { l } }$ . Given a batch of $B$ inputs2 $\{ x _ { i } : x _ { i } \in \mathbb { R } ^ { N _ { 0 } } \} _ { i = 1 , \cdots , B }$ , the pre-activations of the network are defined by the recurrence relation, + +$$ +\pmb { h } _ { i } ^ { 1 } = W ^ { 1 } \pmb { x } _ { i } + b ^ { 1 } \qquad \mathrm { a n d } \qquad \pmb { h } _ { i } ^ { l } = W ^ { l } \phi ( \pmb { h } _ { i } ^ { l - 1 } ) + b ^ { l } \quad \forall l > 1 . +$$ + +At initialization, we choose the weights and biases to be i.i.d. as $W _ { \alpha \beta } ^ { l } \sim \mathcal { N } ( 0 , \sigma _ { w } ^ { 2 } / N _ { l - 1 } )$ and $b _ { \alpha } ^ { l } \sim \mathcal { N } ( 0 , \sigma _ { b } ^ { 2 } )$ . In the following, we use $\alpha , \beta , \ldots$ for the neuron index, and $i , j , \dots$ for the batch index. We will be concerned with understanding the statistics of the pre-activations and the gradients induced by the randomness in the weights and biases. For ease of exposition we will typically take the network to have constant width $N _ { l } = N$ . + +In the mean field approximation, we iteratively replace the pre-activations in Eq. (2) by Gaussian random variables with matching first and second moments. In the infinite width limit this approximation becomes exact (Lee et al., 2017; de G. Matthews et al., 2018; Yang, 2019). Since the weights are i.i.d. with zero mean it follows that the mean of each pre-activation is zero and the covariance between distinct neurons are zero. The pre-activation statistics are therefore given $( h _ { \alpha _ { 1 } 1 } ^ { l } , \cdot \cdot \cdot , h _ { \alpha _ { B } B } ^ { l } ) \xrightarrow { N _ { l - 1 } \infty } \mathcal { N } ( 0 , \Sigma ^ { l } \delta _ { \alpha _ { 1 } \cdots \alpha _ { B } } )$ Nl−1→∞ −−−−−−→ N (0, Σlδα1···αB ) where Σl are B × B covariance matrices and $\bar { \partial } _ { \alpha _ { 1 } , \cdots \alpha _ { B } }$ $\delta$ $\alpha _ { 1 } = \alpha _ { 2 } = \cdot \cdot \cdot = \alpha _ { B }$ + +Definition 3.1. Let $\mathrm { V }$ be the operator on functions $\phi$ such that $\mathrm { V } _ { \phi } ( \Sigma ) = \mathbb { E } [ \phi ( h ) \phi ( h ) ^ { T } : h \sim$ $\mathcal { N } ( 0 , \Sigma ) ]$ computes the matrix of uncentered second moments of $\phi ( h )$ for $h \sim { \mathcal { N } } ( 0 , \Sigma )$ . + +Using the above notation, we can express the covariance matrices by the recurrence relation, + +$$ +\Sigma ^ { l } = \sigma _ { w } ^ { 2 } \mathrm { V } _ { \phi } ( \Sigma ^ { l - 1 } ) + \sigma _ { b } ^ { 2 } \mathbf { 1 } \mathbf { 1 } ^ { T } +$$ + +At first Eq. (2) may seem challenging since the expectation involves a Gaussian integral in $\mathbb { R } ^ { B }$ . However, each term in the expectation of $\mathrm { V } _ { \phi }$ involves at most a pair of pre-activations and so the expectation may be reduced to the evaluation of $\mathcal { O } ( B ^ { 2 } )$ two-dimensional integrals. These integrals can either be performed analytically (Cho & Saul, 2009; Williams, 1997) or efficiently approximated numerically (Lee et al., 2017), and so Eq. (2) defines a computationally efficient method for computing the statistics of neural networks after random initialization. This theme of dimensionality reduction will play a prominent role in the forthcoming discussion on batch normalization. + +Eq. (2) defines a dynamical system over the space of covariance matrices. Studying the statistics of random feed-forward networks therefore amounts to investigating this dynamical system and is an enormous simplification compared with studying the pre-activations of the network directly. As is common in the dynamical systems literature, a significant amount of insight can be gained by investigating the behavior of Eq. (2) in the vicinity of its fixed points. For most common activation functions, Eq. (2) has a fixed point at some $\Sigma ^ { * }$ . Moreover, when the inputs are non-degenerate, this fixed point generally has a simple structure with $\Sigma ^ { * } = q ^ { * } [ ( 1 - c ^ { * } ) I + \dot { c } ^ { * } \mathbf { 1 } \mathbf { 1 } ^ { T } ]$ owing to permutation symmetry among elements of the batch. We refer to fixed points with such symmetry as BSB1 (Batch Symmetry Breaking $^ { l }$ or $^ { l }$ Block Symmetry Breaking) fixed points. As we will discuss later, in the context of batch normalization other fixed points with fewer symmetries (“BSBk fixed points”) + +may become preferred. In the fully-connected setting fixed points may efficiently be computed by solving the fixed point equation induced by Eq. (2) in the special case $B = 2$ . The structure of this fixed point implies that in asymptotically deep feed-forward neural networks all inputs yield pre-activations of identical norm with identical angle between them. Neural networks that are deep enough so that their pre-activation statistics lie in this regime have been shown to be untrainable (Schoenholz et al., 2016). + +Notation As we often talk about matrices and also linear operators over matrices, we write $\mathcal { T } \{ \Sigma \}$ for an operator $\tau$ applied to a matrix $\Sigma$ , and matrix multiplication is still written as juxtaposition. Composition of matrix operators are denoted with $\mathcal { T } _ { 1 } \circ \mathcal { T } _ { 2 }$ . + +Local Convergence to BSB1 Fixed Point. To understand the behavior of Eq. (2) near its BSB1 fixed point we can consider the Taylor series in the deviation from the fixed point, $\Delta \Sigma ^ { l } = \Sigma ^ { l } - \Sigma ^ { * }$ . To lowest order we generically find, + +$$ +\Delta \Sigma ^ { l } = J \{ \Delta \Sigma ^ { l - 1 } \} +$$ + +where $\begin{array} { r } { J = \frac { \mathrm { d } \mathrm { V } _ { \phi } } { \mathrm { d } \Sigma } \big | _ { \Sigma = \Sigma ^ { * } } } \end{array}$ is the $B ^ { 2 } \times B ^ { 2 }$ Jacobian of $\mathrm { V } _ { \phi }$ . In most prior work where $\phi$ was a pointwise non-linearity, Eq. (3) reduces to the case $B = 2$ which naturally gave rise to linearized dynamics in $q ^ { l } = \mathbb { E } [ ( h _ { i } ^ { l } ) ^ { \overline { { 2 } } } ]$ and $c ^ { l } = \mathbb { E } [ h _ { i } ^ { l } z _ { j } ^ { l } ] / q ^ { l }$ . However, in the case of batch normalization we will see that one must consider the evolution of Eq. (3) as a whole. This is qualitatively reminiscent of the case of convolutional networks studied in Xiao et al. (2018) where the evolution of the entire pixel $\times$ pixel covariance matrix had to be evaluated. + +The dynamics induced by Eq. (3) will be controlled by the eigenvalues of $J$ : Suppose $J$ has eigenvalues $\lambda _ { i }$ — ordered such that $\lambda _ { 1 } \geq \lambda _ { 2 } \geq \cdot \cdot \cdot \geq \lambda _ { B ^ { 2 } }$ — with associated eigen“vectors” $e _ { i }$ (note that the $e _ { i }$ will themselves be $B \times B$ matrices). It follows that if $\Delta \Sigma ^ { 0 } = \textstyle \sum _ { i } c _ { i } e _ { i }$ for some choice of constants $c _ { i }$ then $\begin{array} { r } { \Delta \Sigma ^ { l } = \sum _ { i } c _ { i } \lambda _ { i } ^ { l } e _ { i } } \end{array}$ . Thus, if $\lambda _ { i } < 1$ for all $i$ , $\Delta \Sigma ^ { l }$ will approach zero exponentially and the fixed-point will be stable. The number of layers over which $\Sigma$ will approach $\Sigma ^ { * }$ will be given by $- 1 / \log ( \lambda _ { 1 } )$ . By contrast if $\lambda _ { i } > 1$ for any $i$ then the fixed point will be unstable. In this case, there is typically a different, stable, fixed point that must be identified. It follows that if the eigenvalues of $J$ can be computed then the dynamics will follow immediately. + +While $J$ may appear to be a complicated object at first, a moment’s thought shows that each diagonal element $J \{ \Delta \Sigma \} _ { i i }$ is only affected by the corresponding diagonal element $\Delta \Sigma _ { i i }$ , and each off-diagonal element $\dot { J } \{ \Delta \Sigma \} _ { i j }$ is only affected by $\Delta \Sigma _ { i i } , \Delta \Sigma _ { j j }$ , and $\Delta \Sigma _ { i j }$ . Such a Diagonal-Offdiagonal Semidirect $( D O S )$ operator has a simple eigendecomposition with two eigenspaces corresponding to changes in the off-diagonal and changes in the diagonal $\big ( \begin{array} { l l l l l l } \end{array} \begin{array} { r l } \end{array} \begin{array} { r l } \end{array} \begin{array} { r l } \end{array} \big )$ . The associated eigenvalues are precisely those calculated by Schoenholz et al. (2016) in a simplified analysis. DOS operators are a particularly simple form of a more general operator possessing an abundance of symmetries called ultrasymmetric operators, which will play a prominent role in the analysis of batchnorm below. + +Gradient Dynamics. Similar arguments allow us to develop a theory for the statistics of gradients. The backpropogation algorithm gives an efficient method of propagating gradients from the end of the network to the earlier layers as, + +$$ +\frac { \mathrm { d } \mathcal { L } } { \mathrm { d } W ^ { l } } = \sum _ { i } \delta _ { i } ^ { l } \phi ( \pmb { h } _ { i } ^ { l - 1 } ) ^ { T } \qquad \delta _ { \alpha i } ^ { l } = \phi ^ { \prime } ( h _ { \alpha i } ^ { l } ) \sum _ { \beta } W _ { \beta \alpha } ^ { l + 1 } \delta _ { \beta i } ^ { l + 1 } . +$$ + +Here $\mathcal { L }$ is the loss function and $\delta _ { i } ^ { l } \ = \ \nabla _ { h _ { i } ^ { l } } \mathcal { L }$ are $N _ { l }$ -dimensional vectors that describe the error signal from neurons in the $\mathbf { \xi } _ { l } \mathbf { \cdot } \mathbf { \xi } _ { l } ,$ th layer due to the $\because$ ’th element of the batch. The preceding discussion gave a precise characterization of the statistics of the $h _ { i } ^ { l }$ that we can leverage to understand the statistics of $\delta _ { i } ^ { l }$ . Assuming gradient independence, that is, that an iid set of weights are used during backpropagation (see Appendix B for more discussions), it is easy to see that $\mathbb { E } [ \delta _ { \alpha i } ^ { l } ] = 0$ and $\mathbb { E } [ \delta _ { \alpha i } ^ { l } \delta _ { \beta j } ^ { l } ] = \Pi _ { i j } ^ { l }$ if $\alpha = \beta$ and 0 otherwise, where $\Pi ^ { l }$ is a covariance matrix and we may once again drop the neuron index. We can construct a recurrence relation to compute $\Pi ^ { l }$ , + +$$ +\Pi ^ { l } = \sigma _ { w } ^ { 2 } \mathrm { V } _ { \phi ^ { \prime } } ( \Sigma ^ { l } ) \odot \Pi ^ { l + 1 } . +$$ + +Typically, we will be interested in understanding the dynamics of $\Pi ^ { l }$ when $\Sigma ^ { l }$ has converged exponentially towards its fixed point. Thus, we study the approximation, + +$$ +\Pi ^ { l } \approx \sigma _ { w } ^ { 2 } \mathsf { V } _ { \phi ^ { \prime } } ( \Sigma ^ { * } ) \odot \Pi ^ { l + 1 } . +$$ + +Since these dynamics are linear (and in fact componentwise), explosion and vanishing of gradients will be controlled by $\mathrm { V } _ { \phi ^ { \prime } } ( \Sigma ^ { * } )$ . + +# 3.1 BATCH NORMALIZATION + +We now extend the mean field formalism to include batch normalization. Here, the definition for the neural network is modified to be the coupled equations, + +$$ +\pmb { h } _ { i } ^ { l } = W ^ { l } \phi ( \widetilde { \pmb { h } } _ { i } ^ { l - 1 } ) + b ^ { l } \qquad \widetilde { \pmb { h } } _ { \alpha i } ^ { l } = \gamma _ { \alpha } \frac { h _ { \alpha i } ^ { l } - \mu _ { \alpha } } { \sigma _ { \alpha } } + \beta _ { \alpha } +$$ + +where $\gamma _ { \alpha }$ and $\beta _ { \alpha }$ are parameters, and $\begin{array} { r } { \mu _ { \alpha } = \frac { 1 } { N _ { l } } \sum _ { i } h _ { \alpha i } } \end{array}$ and $\begin{array} { r } { \sigma _ { \alpha } ^ { 2 } = \sqrt { \frac { 1 } { N _ { l } } \sum _ { i } ( h _ { \alpha i } - \mu _ { \alpha } ) ^ { 2 } + \epsilon } } \end{array}$ are the per-neuron batch statistics. In practice $\epsilon \approx 1 0 ^ { - 5 }$ or so to prevent division by zero, but in this paper, unless stated otherwise (in the last few sections), $\epsilon$ is assumed to be 0. Unlike in the case of vanilla fully-connected networks, here the pre-activations are invariant to $\sigma _ { w } ^ { 2 }$ and $\sigma _ { b } ^ { 2 }$ . Without a loss of generality, we therefore set $\sigma _ { w } ^ { 2 } = 1$ and $\sigma _ { b } ^ { 2 } = 0$ for the remainder of the text. In principal, batch normalization additionally yields a pair of hyperparameters $\gamma$ and $\beta$ which are set to be constants. However, these may be incorporated into the nonlinearity and so without a loss of generality we set $\gamma = 1$ and $\beta = 0$ . In order to avoid degenerate results, we assume $B \geq 4$ unless stated otherwise; we shall discuss the small $B$ regime in Appendix J. + +If one treats batchnorm as a “batchwise nonlinearity”, then the arguments from the previous section can proceed identically and we conclude that as the width of the network grows, the pre-activations will be jointly Gaussian with identically distributed neurons. Thus, we arrive at an analogous expression to Eq. (2), + +$$ +\Sigma ^ { l } = \mathsf { V } _ { \mathcal { B } _ { \phi } } ( \Sigma ^ { l - 1 } ) \qquad \mathrm { w h e r e } \qquad \mathcal { B } _ { \phi } : \mathbb { R } ^ { B } \to \mathbb { R } ^ { B } , \mathcal { B } _ { \phi } ( h ) = \phi \left( \frac { \sqrt { B } G h } { \vert \vert G h \vert \vert } \right) . +$$ + +Here we have introduced the projection operator $\begin{array} { r } { G = I - \frac { 1 } { B } \mathbf { 1 } \mathbf { 1 } ^ { T } } \end{array}$ which is defined such that $G x =$ $x - \mu \mathbf { 1 }$ with $\mu = \textstyle \sum _ { i } x _ { i } / B$ . Unlike $\phi$ , $B _ { \phi }$ does not act component-wise on $h$ . It is therefore not obvious whether $\mathrm { V } _ { B _ { \phi } }$ can be evaluated without performing a $B$ -dimensional Gaussian integral. + +Theoretical tools. In this paper, we present several ways to analyze high dimensional integrals like the above: 1. the Laplace method 2. the Fourier method 3. spherical integration 4. and the Gegenbauer method. The former two use the Laplace and Fourier transforms to simplify expressions like the above, where the Laplace method requires that $\phi$ be positive homogeneous. Often the Laplace method will give clean, closed form answers for such $\phi$ . Because batchnorm can be thought of as a linear projection $( G )$ followed by projection to the sphere of radius $\sqrt { B }$ , spherical integration techniques are often very useful and in fact is typically the most straightforward way of numerically evaluating quantities. Lastly, the Gegenbauer method expresses objects in terms of the Gegenbauer coefficients of $\phi$ . Briefly, Gegenbauer polynomials $\{ C _ { l } ^ { ( \bar { \alpha } ) } ( x ) \} _ { l = 0 } ^ { \infty }$ are orthogonal polynomials with respect to the measure $( 1 - x ^ { 2 } ) ^ { \alpha - \frac { 1 } { 2 } }$ on $[ - 1 , 1 ]$ . They are intimately related to spherical harmonics (a natural basis for functions on a sphere), which explains their appearance in this context. The Gegenbauer method is the most illuminating amongst them all, and is what allows us to conclude that gradient explosion happens regardless of nonlinearity under general conditions. See Appendix D for a more in-depth discussion of these techniques. In what follows, we will mostly present results by the Laplace method for $\phi =$ relu and by the Gegenbauer method for general $\phi$ , but give pointers to the appendix for others. + +Back to the topic of Eq. (8), we present a pair of results that expresses Eq. (8) in a more manageable form. From previous work (Poole et al., 2016), $\mathrm { V } _ { \phi }$ can be expressed in terms of a two-dimensional Gaussian integrals independent of $B$ . When $\phi$ is degree- $\alpha$ positive homogeneous (e.g. rectified linear activations) we can relate $\mathrm { V } _ { \phi }$ and $\mathrm { V } _ { B _ { \phi } }$ by the Laplace transform. (see Thm E.5). + +Theorem 3.2. Suppose $\phi : \mathbb { R } \mathbb { R }$ is degree-α positive homogeneous. For any positive semi-definite matrix $\Sigma$ define the projection $\Sigma ^ { G } = G \bar { \Sigma } G$ . Then + +$$ +\mathrm { V } _ { \mathcal { B } _ { \phi } } ( \Sigma ) = \frac { B ^ { \alpha } } { \Gamma ( \alpha ) } \int _ { 0 } ^ { \infty } \mathrm { d } s ~ s ^ { \alpha - 1 } \frac { \mathrm { V } _ { \phi } ( \Sigma ^ { G } ( I + 2 s \Sigma ^ { G } ) ^ { - 1 } ) } { \sqrt { \operatorname* { d e t } ( I + 2 s \Sigma ^ { G } ) } } . +$$ + +whenever the integral exists. + +Using this parameterization, when $\mathrm { V } _ { \phi }$ has a closed form solution (like ReLU), $\mathrm { V } _ { B _ { \phi } }$ involves only a single integral. A similar expression can be derived for general $\phi$ by applying Fourier transform; see Appendix E.2. Next we express Eq. (8) as a spherical integral (see Proposition E.34) + +Theorem 3.3. Let $\Phi \in \mathbb { R } ^ { B \times B - 1 }$ have as columns a set of orthonormal basis vectors of $\{ \boldsymbol { x } \in \mathbb { R } ^ { B }$ $G x = x \}$ . Then with $S ^ { B - 2 } \subseteq \mathbb { R } ^ { B - 1 }$ denoting the $( B - 2 )$ -dimensional sphere, + +$$ +\mathrm { V } _ { \mathcal { B } _ { \phi } } ( \Sigma ) = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \frac { \phi ( \sqrt { B } \mathrm { e } v ) ^ { \otimes 2 } } { \sqrt { \operatorname* { d e t } \Phi ^ { T } \Sigma \mathrm { e } } ( v ^ { T } ( \mathrm { e } ^ { T } \Sigma \mathrm { e } ) ^ { - 1 } v ) ^ { \frac { B - 1 } { 2 } } } . +$$ + +Together these theorems provide analytic recurrence relations for random neural networks with batch normalization over a wide range of activation functions. By analogy to the fully-connected case we would like to study the dynamical system over covariance matrices induced by these equations. + +We begin by investigating the fixed point structure of Eq. (8). As in the case of feed-forward networks, permutation symmetry implies that there exist BSB1 fixed points $\Sigma ^ { * } = q ^ { * } [ ( 1 - c ^ { * } ) I + c ^ { * } \mathbf { 1 } \mathbf { 1 } ^ { T } ]$ . We will see that this fixed point is in fact unique, and a clean expression of $q ^ { * }$ and $c ^ { * }$ can be obtained in terms of Gegenbauer basis (see Thm F.13). + +Theorem 3.4 (Gegenbauer expansions of BSB1 fixed point). If $\phi ( \sqrt { B - 1 } x )$ has Gegenbauer expansion P∞l=0 al 1cB−1,l $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ where $\begin{array} { r } { c _ { B - 1 , l } = \frac { B - 3 } { B - 3 + 2 l } } \end{array}$ , then + +$$ +q ^ { * } = \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) , \qquad q ^ { * } c ^ { * } = \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) . +$$ + +Thus the entries of the BSB1 fixed point are diagonal quadratic forms of the Gegenbauer coefficients of $\phi ( \sqrt { B - 1 } x )$ . Even more concise closed forms are available when the activation functions are degree $\alpha$ positive homogeneous (see Thm F.8). In particular, for ReLU we arrive at the following + +Theorem 3.5 (BSB1 fixed point for ReLU). When $\phi =$ relu, then + +$$ +q ^ { * } = { \frac { 1 } { 2 } } , \qquad c ^ { * } = \operatorname { J } _ { 1 } \left( { \frac { - 1 } { B - 1 } } \right) = { \frac { 1 } { \pi } } - { \frac { 1 } { 2 ( B - 1 ) } } + O \left( { \frac { 1 } { ( B - 1 ) ^ { 2 } } } \right) +$$ + +where $\begin{array} { r } { \mathrm { J _ { 1 } } ( c ) = \frac { 1 } { \pi } ( \sqrt { 1 - c ^ { 2 } } + ( \pi - \operatorname { a r c c o s } ( c ) ) c ) } \end{array}$ is the arccosine kernel (Cho & Saul, 2009). + +In the appendix, $\cdot$ also describes a trigonometric integral formula for general nonlinearities. + +In the presence of batch normalization, when the activation function grows quickly, a winner-takeall phenomenon can occur where a subset of samples in the batch have much bigger activations than others. This causes the covariance matrix to form blocks of differing magnitude, breaking the BSB1 symmetry. One notices this, for example, as the degree $\alpha$ of $\alpha$ -ReLU (i.e. αth power of ReLU) increases past a point $\alpha _ { \mathrm { t r a n s i t i o n } } ( B )$ depending on the batch size $B$ (see Fig. 1). However, we observe, through simulations, that by far most of the nonlinearities used in practice, like ReLU, leaky ReLU, tanh, sigmoid, etc, all lead to BSB1 fixed points, and we can prove this rigorously for $\phi = \mathrm { i d }$ (see Corollary F.3). We discuss symmetry-broken fixed points (BSB2 fixed points) in Appendices J and K, but in the main text, from here on, + +Assumption 1. Unless stated otherwise, we assume that any nonlinearity $\phi$ mentioned induces $\Sigma ^ { l }$ to converge to a BSB1 fixed point under the dynamics of Eq. (8). + +# 3.1.1 LINEARIZED DYNAMICS + +With the fixed point structure for batch normalized networks having been described, we now investigate the linearized dynamics of Eq. (8) in the vicinity of these fixed points. + +To determine the eigenvalues of $\frac { \mathrm { d } \mathrm { V } _ { B _ { \phi } } } { \mathrm { d } \Sigma } \Big | _ { \Sigma = \Sigma ^ { * } }$ it is helpful to consider the action of batch normalization in more detail. In particular, we notice that $B _ { \phi }$ can be decomposed into the composition of three separate operations, $B _ { \phi } = \phi \circ \mathsf { n o } G$ . As discussed above, $G h$ subtracts the mean from $h$ and we introduce the new function $\mathsf { n } ( h ) = \sqrt { B } h / | | h | |$ which normalizes a centered $h$ by its standard deviation. Applying the chain rule, we can rewrite the Jacobian as, + +$$ +\frac { \mathrm { d } \mathrm { V } _ { \mathcal { B } _ { \phi } } ( \Sigma ) } { \mathrm { d } \Sigma } = \frac { \mathrm { d } \mathrm { V } _ { [ \phi \circ \boldsymbol { \mathsf { n } } ] } ( \Sigma ^ { G } ) } { \mathrm { d } \Sigma ^ { G } } \circ G ^ { \otimes 2 } +$$ + +where $\circ$ denotes composition and $G ^ { \otimes 2 }$ is the natural extension of $G$ to act on matrices as ${ \cal G } ^ { \otimes 2 } \{ \Sigma \} =$ $G \Sigma G = \Sigma ^ { G }$ . It ends up being advantageous to study $\begin{array} { r } { G ^ { \otimes 2 } \circ \left. \frac { \mathrm { d } \mathrm { V } _ { [ \phi \circ \mathrm { n } ] } } { \mathrm { d } \Sigma } \right| _ { \Sigma = \Sigma ^ { * } } \circ G ^ { \otimes 2 } = : G ^ { \otimes 2 } \circ \hat { J } \circ G ^ { \otimes 2 } } \end{array}$ and to note that the nonzero eigenvalues of this object are identical to the nonzero eigenvalues of the Jacobian (see Lemma F.17). + +At face value, this is a complicated object since it simultaneously has large dimension and possesses an intricate block structure. However, the permutation symmetry of the BSB1 $\Sigma ^ { * }$ induces strong symmetries in $\hat { J }$ that significantly simplify the analysis (see Appendix F.3). In particular while $\hat { J } _ { i j k l }$ is a four-index object, we have $\hat { \hat { J } } _ { i j k l } = \hat { J } _ { \pi ( i ) \pi ( j ) \pi ( k ) \pi ( l ) }$ for all permutations $\pi$ on $B$ and $\hat { J } _ { i j k l } = \hat { J } _ { j i l k }$ . We call linear operators possessing such symmetries ultrasymmetric (Defn E.53) and show that all ultrasymmetric operators conjugated by $G ^ { \otimes 2 }$ admit an eigendecomposition that contains three distinct eigenspaces with associated eigenvalues (see Thm E.62). + +Theorem 3.6. Let $\tau$ be an ultrasymmetric matrix operator. Then on the space of symmetric matrices, $G ^ { \otimes 2 } \circ \mathcal { T } \circ G ^ { \otimes 2 }$ has the following orthogonal (under trace inner product) eigendecomposition, + +1. an eigenspace $\{ \Sigma : \Sigma ^ { G } = 0 \}$ with eigenvalue $O$ . + +2. a $^ { l }$ -dimensional eigenspace $\mathbb { R } G$ with eigenvalue $\lambda _ { G } ^ { G , T }$ . + +3. a $( B - 1 )$ -dimensional eigenspace $\mathbb { L } : = \{ D ^ { G } : D$ diagonal, tr $D = 0 \}$ , with eigenvalue $\lambda _ { \mathbb { L } } ^ { G , T }$ . + +4. B(B−3)2 -dimensional eigenspace M := {Σ : ΣG = Σ, DiagΣ = 0}, with eigenvalue $\lambda _ { \mathbb { M } } ^ { G , T }$ . + +The specific forms of the eigenvalues can be obtained as linear functions of the entries of $\tau$ ; see $\mathrm { T h m } \mathrm { E } . 6 2 $ for details. Note that, as the eigenspaces are orthogonal, this implies that $G ^ { \otimes 2 } \circ T \circ G ^ { \otimes 2 }$ is self-adjoint (even when $\tau$ is not). + +In our context with $\mathcal { T } = \hat { J }$ , the eigenspaces can be roughly interpreted as follows: The deviation $\Delta \Sigma = \Sigma - \Sigma ^ { * }$ from the fixed point decomposes as a linear combination of components in each of the eigenspaces. The $\mathbb { R } G$ -component captures the average norm of elements of the batch (the trace of $\Delta \Sigma$ ), the $\mathbb { L }$ -component captures the fluctuation of such norms, and the $\mathbb { M }$ -component captures the covariances between elements of the batch. + +Because of the explicit normalization of batchnorm, one sees immediately that the $\mathbb { R } G$ -component goes to 0 after 1 step. For positive homogeneous $\phi$ , we can use the Laplace method to obtain closed form expressions for the other eigenvalues (see Thm F.33). The below theorem shows that, as the batch size becomes larger, a deep ReLU-batchnorm network takes more layers to converge to a BSB1 fixed point. + +Theorem 3.7. Let $\phi =$ relu and $B > 3$ . The eigenvalues of $G ^ { \otimes 2 } \circ \hat { J } \circ G ^ { \otimes 2 }$ for L and M are + +$$ +\begin{array} { l } { { \displaystyle \lambda _ { \mathrm { L } } ^ { \uparrow } = \frac { 1 } { 2 ( B + 1 ) \mu ^ { * } } \left( ( B - 2 ) \left[ 1 - { \bf J } _ { 1 } \left( \frac { - 1 } { B - 1 } \right) \right] + \frac { B } { B - 1 } { \bf J } _ { 1 } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) \right) ~ \nearrow ~ 1 } } \\ { { \displaystyle \lambda _ { \mathrm { M } } ^ { \uparrow } = \frac { B } { 2 ( B + 1 ) \mu ^ { * } } { \bf J } _ { 1 } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) ~ \nearrow ~ \frac { \pi } { 2 ( \pi - 1 ) } \approx 0 . 7 3 3 } } \end{array} +$$ + +where $\begin{array} { r } { \mu ^ { * } = q ^ { * } ( 1 - c ^ { * } ) = \frac { 1 } { 2 } \left( 1 - \operatorname { J } _ { 1 } \left( \frac { - 1 } { B - 1 } \right) \right) } \end{array}$ , and $\nearrow$ denotes increasing limit as $B \to \infty$ + +More generally, we can evaluate them for general nonlinearity using spherical integration (Appendix F.3.1) and, more enlightening, using the Gegenbauer method, is the following + +Theorem 3.8. If φ(√B − 1x) has Gegenbauer expansion P∞l=0 al 1cB−1,l C $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) , } \end{array}$ , then + +$$ +\lambda _ { \mathbb { L } } ^ { \uparrow } = \frac { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } w _ { B - 1 , l } + a _ { l } a _ { l + 2 } u _ { B - 1 , l } } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } v _ { B - 1 , l } } , \qquad \lambda _ { \mathbb { M } } ^ { \uparrow } = \frac { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \tilde { w } _ { B - 1 , l } + a _ { l } a _ { l + 2 } \tilde { u } _ { B - 1 , l } } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } v _ { B - 1 , l } } +$$ + +where the coefficients $w _ { B - 1 , l } , u _ { B - 1 , l } , \tilde { w } _ { B - 1 , l } , \tilde { u } _ { B - 1 , l } , v _ { B - 1 , l }$ are given in Thms $\cdot$ and F.24. + +A BSB1 fixed point is not locally attracting if $\lambda _ { \mathbb { L } } ^ { \uparrow } > 1$ or $\lambda _ { \mathbb { M } } ^ { \uparrow } > 1$ . Thus $\mathrm { T h m } 3 . 8$ yields insight on the stability of the BSB1 fixed point, which we can interpret heuristically as follows. The specific forms of the coefficients $w _ { B - 1 , l } , u _ { B - 1 , l } , \tilde { w } _ { B - 1 , l } , \tilde { u } _ { B - 1 , l } , v _ { B - 1 , l }$ show that $\lambda _ { \mathbb { M } } ^ { \uparrow }$ is typically much smaller than $\lambda _ { \mathbb { L } } ^ { \uparrow }$ (but there are exceptions like $\phi = \sin )$ , and $w _ { B - 1 , 1 } < v _ { B - 1 , 1 }$ but $w _ { B - 1 , l } \geq v _ { B - 1 , l }$ for all $l \geq 2$ . Thus one expects that, the larger $a _ { 1 }$ is, i.e. the “more linear” and less explosive $\phi$ is, the smaller $\lambda _ { \mathbb { L } } ^ { \uparrow }$ is and the more likely that Eq. (8) converges to a BSB1 fixed point. This is consistent with the “winner-take-all” intuition for the emergence of BSB2 fixed point explained above. See Appendix F.3.2 for more discussion. + +# 3.1.2 GRADIENT BACKPROPAGATION + +With a mean field theory of the pre-activations of feed-forward networks with batch normalization having been developed, we turn our attention to the backpropagation of gradients. In contrast to the case of networks without batch normalization, we will see that exploding gradients at initialization are a severe problem here. To this end, one of the main results from this section will be to show that fully-connected networks with batch normalization feature exploding gradients for any choice of nonlinearity such that $\Sigma ^ { l }$ converges to a BSB1 fixed point. Below, by rate of gradient explosion we mean the $\beta$ such that the gradient norm squared grows as $\beta ^ { L + o ( L ) }$ with depth $L$ . As before, all computations below assumes gradient independence (see Appendix B for a discussion). + +As a starting point we seek an analog of Eq. (6) in the case of batch normalization. However, because the activation functions no longer act point-wise on the pre-activations, the backpropagation equation becomes, + +$$ +\delta _ { \alpha i } ^ { l } = \sum _ { \beta j } \frac { \partial B _ { \phi } ( \pmb { h } _ { \alpha } ^ { l } ) _ { j } } { \partial h _ { \alpha i } ^ { l } } W _ { \beta \alpha } ^ { l + 1 } \delta _ { \beta j } ^ { l + 1 } +$$ + +where $\pmb { h } _ { \alpha } ^ { l } = ( h _ { \alpha 1 } ^ { l } , \dots , h _ { \alpha B } ^ { l } )$ and we observe the additional sum over the batch. Computing the resulting covariance matrix $\Pi ^ { l }$ , we arrive at the recurrence relation, + +$$ +\Pi ^ { l } = \mathbb { E } \left[ \left( \frac { \mathrm { d } \mathcal { B } _ { \phi } ( h ) } { \mathrm { d } h } \right) ^ { T } \Pi ^ { l + 1 } \frac { \mathrm { d } \mathcal { B _ { \phi } } ( h ) } { \mathrm { d } h } : h \sim \mathcal { N } ( 0 , \Sigma ^ { l } ) \right] = : \mathrm { V } _ { \mathcal { B _ { \phi } ^ { \prime } } } ( \Sigma ^ { l } ) ^ { \dagger } \{ \Pi ^ { l + 1 } \} +$$ + +where we have defined the linear operator $\mathrm { V } _ { F } ( \Sigma ) ^ { \dagger } \{ \cdot \}$ such that $\mathrm { V } _ { F } ( \Sigma ) ^ { \dagger } \{ \Pi \} = \mathbb { E } [ F _ { h } ^ { T } \Pi F _ { h } : h \sim$ $\mathcal { N } ( 0 , \Sigma ) ]$ for any vector-indexed linear operator $F _ { h }$ . As in the case of vanilla feed-forward networks, here we will be concerned with the behavior of gradients when $\Sigma ^ { l }$ is close to its fixed point. We therefore study the asymptotic approximation to Eq. (17) given by $\Pi ^ { l } = \mathrm { V } _ { B _ { \phi } ^ { \prime } } ( \Sigma ^ { * } ) ^ { \dagger } \{ \hat { \Pi } ^ { l + 1 } \}$ . In this case the dynamics of $\Pi$ are linear and are therefore naturally determined by the eigenvalues of $\mathrm { V } _ { B _ { \phi } ^ { \prime } } ( \Sigma ^ { * } ) ^ { \dag }$ . + +As in the forward case, batch normalization is the composition of three operations $B _ { \phi } = \phi \circ \mathsf { n o G }$ . Applying the chain rule, Eq. (17) can be rewritten as, + +$$ +\nabla _ { B _ { \phi } ^ { \prime } } ( \Sigma ) ^ { \dagger } = G ^ { \otimes 2 } \circ \mathbb { E } \left[ \left( \frac { \mathrm { d } ( \phi \circ \boldsymbol { \mathsf { n } } ) ( z ) } { \mathrm { d } z } \bigg | _ { z = G h } ^ { r } \right) ^ { \otimes 2 } : h \sim \mathcal { N } ( 0 , \Sigma ) \right] = : G ^ { \otimes 2 } \circ F ( \Sigma ) +$$ + +with $F ( \Sigma )$ appropriately defined. Note that since $G ^ { \otimes 2 }$ is an idempotent operator, $( \mathrm { V } _ { B _ { \phi } ^ { \prime } } ( \Sigma ) ^ { \dag } ) ^ { n } =$ $( G ^ { \otimes 2 } \circ F ( \Sigma ) ) ^ { n } = ( G ^ { \otimes 2 } \circ F ( \Sigma ) \circ G ^ { \otimes 2 } ) ^ { n - 1 } \circ G ^ { \otimes 2 } \circ F ( \Sigma ) .$ , so that it suffices to study the eigendecomposition of $G ^ { \otimes 2 } \circ F ( \Sigma ^ { * } ) \circ G ^ { \otimes 2 }$ . Due to the symmetry of $\Sigma ^ { * }$ , $F ( \Sigma ^ { * } )$ is ultrasymmetric, so that + +$G ^ { \otimes 2 } \circ F ( \Sigma ^ { * } ) \circ G ^ { \otimes 2 }$ has eigenspaces $\mathbb { R } G , \mathbb { L } , \mathbb { M }$ and we can compute its eigenvalues via $\cdot$ . More illuminating, however, is the Gegenbauer expansion (see Thm G.5). It requires a new identity $\cdot$ involving Gegenbauer polynomials integrated over a sphere, which may be of independent interest. + +Theorem 3.9 (Batchnorm causes gradient explosion). Suppose $\phi ( \sqrt { B - 1 } x ) , \phi ^ { \prime } ( \sqrt { B - 1 } x ) \in$ L2((1 − x2) B−32 ). If φ(√B − 1x) has Gegenbauer expansion P∞l=0 al 1cB−1,l $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ , then gradients explode at the rate of + +$$ +\lambda _ { G } ^ { \downarrow } = \frac { \sum _ { l = 0 } ^ { \infty } \left( \frac { l + B - 3 } { B - 3 } \cdot l \right) a _ { l } ^ { 2 } r _ { l } } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } r _ { l } } +$$ + +where $r _ { l } = c _ { B - 1 , l } ^ { - 1 } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( 1 \right) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) > 0$ . Consequently, for any non-constant $\phi$ 6 G in which case gradients explode at the rate of (i.e. there is a $j > 0$ such that $a _ { j } \neq 0 ,$ ), $\lambda _ { G } ^ { \downarrow } > 1$ $\frac { B - 2 } { B - 3 }$ ; $\phi$ minimizes . $\lambda _ { G } ^ { \downarrow }$ iff it is linear (i.e. $a _ { i } = 0 , \forall i \geq 2$ ), + +This contrasts starkly with the case of non-normalized fully-connected networks, which can use the weight and bias variances to control its mean field network dynamics (Poole et al., 2016; Schoenholz et al., 2016). As a corollary, we disprove the conjecture of the original batchnorm paper (Ioffe & Szegedy, 2015) that “Batch Normalization may lead the layer Jacobians to have singular values close to $1 ^ { \circ }$ in the initialization setting, and in fact prove the exact opposite, that batchnorm forces the layer Jacobian singular values away from $^ { l }$ . + +Appendix G.1 discusses the numerical evaluation of all eigenvalues, and as usual, the Laplace method yields closed forms for positive homogeneous $\phi$ $\begin{array} { c c c c c c c c c c c c c c c c } { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { } & { { } } & { } & { { } } & { } & { { } } & { } & { } { } & { } & { } & { } { } & { } & { } & { } { } & { } & { } & { } & \end{array}$ . We highlight the result for ReLU. + +Theorem 3.10. In a ReLU-batchnorm network, the gradient norm explodes exponentially at the rate of + +$$ +{ \frac { 1 } { B - 3 } } \left( { \frac { B - 1 + \mathrm { J } _ { 1 } ^ { \prime } \left( { \frac { - 1 } { B - 1 } } \right) } { 1 - \mathrm { J } _ { 1 } \left( { \frac { - 1 } { B - 1 } } \right) } } - 1 \right) +$$ + +which decreases to $\textstyle { \frac { \pi } { \pi - 1 } } \approx 1 . 4 6 7$ as $B \infty$ . In contrast, for a linear batchnorm network, the gradient norm explodes exponentially at the rate of $\frac { B - 2 } { B - 3 }$ , which goes to $^ { l }$ as $B \infty$ . + +Fig. 1 shows theory and simulation for ReLU gradient dynamics. + +Weight Gradient While all of the above only study the gradient with respect to the hidden preactivations, Appendix L shows that the weight gradient norms at layer $l$ is just $\langle \Pi ^ { l } , \mu ^ { \ast } G \rangle = \mu ^ { \ast } \mathrm { t r } \Pi ^ { l }$ , and thus by $\mathrm { T h m } 3 . 9$ , the weight gradients explode as well at the same rate $\lambda _ { G } ^ { \downarrow }$ . + +Effect of $\epsilon$ as a hyperparameter In practice, $\epsilon$ is usually treated as small constant and is not regarded as a hyperparameter to be tuned. Nevertheless, we can investigate its effect on gradient explosion. A straightforward generalization of the analysis presented above to the case of $\epsilon > 0$ suggests somewhat larger $\epsilon$ values than typically used can ameliorate (but not eliminate) gradient explosion problems. See Fig. 5(c,d). + +# 3.2 CROSS-BATCH DYNAMICS + +Forward In addition to analyzing the correlation between preactivations of samples in a batch, we also study the correlation between those of different batches. The dynamics Eq. (8) can be generalized to simultaneous propagation of $k$ batches (see Eq. (54) and Appendix H). + +$$ +\widetilde { \Sigma } ^ { l } = \mathrm { V } _ { B _ { \phi } ^ { \oplus k } } ( \widetilde { \Sigma } ^ { l - 1 } ) = \underset { ( h _ { 1 } , \ldots , h _ { k } ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { l - 1 } ) } { \mathbb { E } } ( \mathcal { B } _ { \phi } ( h _ { 1 } ) , \ldots , \mathcal { B } _ { \phi } ( h _ { k } ) ) ^ { \otimes 2 } , \quad \mathrm { w i t h ~ } \widetilde { \Sigma } ^ { l } \in \mathbb { R } ^ { k B \times k B } . +$$ + +Here the domain of the dynamics is the space of block matrices, with diagonal blocks and offdiagonal blocks resp. representing within-batch and cross-batch covariance. We observe empirically + +![](images/efb897665ab6775072dfc47255054a708efc5abdb3c4da6c9fe4aac78d8cc35b.jpg) +Figure 1: Numerical confirmation of theoretical predictions. (a,b) Comparison between theoretical prediction (dashed lines) and Monte Carlo simulations (solid lines) for the eigenvalues of the backwards Jacobian (see Thms 3.10 and G.12) as a function of batch size and the magnitude of gradients as a function of depth respectively for rectified linear networks. In each case Monte Carlo simulations are averaged over 200 sample networks of width 1000 and shaded regions denote 1 standard deviation. Dashed lines are shifted slightly for easier comparison. (c,d) Demonstration of the existence of a BSB1 to BSB2 symmetry breaking transition as a function of $\alpha$ for $\alpha$ -ReLU (i.e. the αth power of ReLU) activations. In (c) we plot the empirical variance of the diagonal and off-diagonal entries of the covariance matrix which clearly shows a jump at the transition. In (d) we conference paper at ICLR 2019plot representative covariance matrices for the two phases (BSB1 bottom, BSB2 top). + +![](images/6c04ca6ffc5dda281f31ee174cfcc17477c0b86d7ecdb3211affcd060b8c7d38.jpg) +Figure A.1: Batch norm leads to a chaotic input-output map with increasingFigure 2: Batch norm leads to a chaotic input-output map with increasing depth. A linear network with batch norm is shown acting on two minibatches of size 64 after ranetwork with batch norm is shown acting on two minibatches of size 64 after random orthogonal initialization. The datapoints in the minibatch are chosen to form a 2d circle in ininitialization. The datapoints in the minibatch are chosen to form a 2d circle in input space, except for one datapoint that is perturbed separately in each minibatch (leftmost datapointfor one datapoint that is perturbed separately in each minibatch (leftmost datapoint at input layer 0). c input-output map with increasing depth. A linearBecause the network is linear, for a given minibatch it performs an affine transformaBecause the network is linear, for a given minibatch it performs an affine transformation on its inputs n two minibatches of size 64 after random orthogonal– a circle in input space remains an ellipse throughout the network. However, d– a circle in input space remains an ellipse throughout the network. However, due to batch norm h are chosen to form a 2d circle in input space, exceptthe coefficients of that affine transformation change nonlinearly as the datapointsthe coefficients of that affine transformation change nonlinearly as the datapoints in the minibatch in each minibatch (leftmost datapoint at input layer 0).are changed. (a) Each pane shows a scatterplot of activations at a given layer fare changed. (a) Each pane shows a scatterplot of activations at a given layer for all datapoints throughout the network. However, due to batch normin the minibatch, projected onto the top two PCA directions. PCA directions arein the minibatch, projected onto the top two PCA directions. PCA directions are computed using change nonlinearly as the datapoints in the minibatchthe concatenation of the two minibatches. Due to the batch norm nonlinearity,the concatenation of the two minibatches. Due to the batch norm nonlinearity, minibatches that rplot of activations at a given layer for all datapointsare nearly identical in input space grow increasingly dissimilar with depth. Intuitiare nearly identical in input space grow increasingly dissimilar with depth. Intuitively, this chaotic PCA directions. PCA directions are computed usinginput-output map can be understood as the source of exploding gradients when batcinput-output map can be understood as the source of exploding gradients when batch norm is applied (a)Due to the batch norm nonlinearity, minibatches thatto very deep networks, since very small changes in an input correspond to very larto very deep networks, since very small changes in an input correspond to very large movements in network outputs. (b) The correlation between the two minibatches, as a functionnetwork outputs. (b) The correlation between the two minibatches, as a function of layer, for the nges in an input correspond to very large movements insame network. Despite having a correlation near one at the input layer, the two misame network. Despite having a correlation near one at the input layer, the two minibatches rapidly en the two minibatches, as a function of layer, for thedecorrelate with depth. See for a theoretical treadecorrelate with depth. See Appendix H for a theoretical treatment. + +A.1 VGG19 WITH BATCHNORM ON CIFAR100that for most nonlinearities used in practice like ReLU, or even for the identity function, ie no (b)ON CIFAR100pointwise nonlinearity, the global fixed point of this dynamics is cross-batch BSB1, with diagonal atch norm leads to a chaotic input-output map with increasing depth. A lineam causes gradient explosion, after the first few epochs,Even though at initialization time batchnorm causeBSB1 blocks, and off-diagonal entries all equal to the same constant $c _ { \mathrm { C B } } ^ { * }$ ad: + +$$ +\widetilde { \Sigma } ^ { * } = \left( { \begin{array} { c c c c } { \Sigma ^ { * } } & { c _ { \mathrm { C B } } ^ { * } } & { c _ { \mathrm { C B } } ^ { * } } & { \cdot \cdot } \\ { c _ { \mathrm { C B } } ^ { * } } & { \Sigma ^ { * } } & { c _ { \mathrm { C B } } ^ { * } } & { \cdot \cdot } \\ { c _ { \mathrm { C B } } ^ { * } } & { c _ { \mathrm { C B } } ^ { * } } & { \Sigma ^ { * } } & { \cdot \cdot } \\ { \vdots } & { \vdots } & { \vdots } & { \ddots } \end{array} } \right) . +$$ + +To interpret this phenomenon, note that, after mean centering of the preactivations, this covariance matrix becomes a multiple of identity. Thus, deep embedding of batches loses the mutual information between them in the input space. Qualitatively, this implies that two batches that are similar at the input to the network will become increasingly dissimilar — i.e. chaotic — as the signal propagates deep into the network. This loss in fact happens exponentially fast, as illustrated in Fig. 2, and as shown theoretically by Thm H.13, at a rate of $\bar { \lambda } _ { \widetilde { \mathbb { M } } } ^ { \uparrow } < 1$ . Formally, the linearized dynamics of Eq. (20) has an eigenspace $\smash { \widetilde { \mathbb { M } } }$ given by all block matrices with zero diagonal blocks. The associated eigenvalue is $\lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow }$ . We shall return to $\lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow }$ shortly and give its Gegenbauer expansion $( \mathrm { T h m } 3 . 1 1 )$ ). + +Backward The gradients of two batches of input are correlated throughout the course of backpropagation. We hence also study the generalization of Eq. (17) to the $k$ -batch setting (see Appendix I): + +$$ +\begin{array}{c} \begin{array} { r l } & { \tilde { \Pi } ^ { l } = \mathrm { V } _ { B _ { \phi } ^ { \Phi 2 } } ( \widetilde \Sigma ^ { l } ) ^ { \dagger } \{ \tilde { \Pi } ^ { l + 1 } \} } \\ { \mathrm { V } _ { B _ { \phi } ^ { \Phi 2 } } / \widetilde \Sigma ^ { * } ) ^ { \dagger } \{ ( \begin{array} { l l } { \Pi _ { 1 } } & { \Xi } \\ { \Xi ^ { T } } & { \Pi _ { 2 } } \end{array} ) \} = \underset { ( x , y ) \sim \mathcal { N } ( 0 , \widetilde \Sigma ^ { * } ) } { \mathbb { E } } [ \binom { \mathcal { B } _ { \phi } ^ { \prime } ( x ) ^ { T } \Pi _ { 1 } \mathcal { B } _ { \phi } ^ { \prime } ( x ) } { \mathcal { B } _ { \phi } ^ { \prime } ( y ) ^ { T } \Xi ^ { T } \mathcal { B } _ { \phi } ^ { \prime } ( x ) } } & { \mathcal { B } _ { \phi } ^ { \prime } ( x ) ^ { T } \Xi \mathcal { B } _ { \phi } ^ { \prime } ( y ) } \end{array} ) ] \end{array} +$$ + +where for simplicity of exposition and without loss of generality, we take $k = 2$ . Like in Eq. (17), we have assumed that $\widetilde { \Sigma } ^ { l }$ has converged to its limit $\widetilde { \Sigma } ^ { * }$ . It is not hard to see that each block of $\tilde { \Pi } ^ { l }$ evolves independently, with the diagonal blocks in particular evolving according to Eq. (17). Analyzing the off-diagonal blocks might seem unwiby scalar multiplication by a constant $\lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow }$ at first, but it turns out that its dynamics is given simply(Thm I.4). Even more surprisingly, + +Theorem 3.11 (Batchnorm causes information loss). The convergence rate of cross-batch preactivation covariance $\lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow }$ is equal to the decay rate of the cross-batch gradient covariance $\lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow }$ . If $\phi ( { \sqrt { B - 1 } } x )$ has Gegenbauer expansion $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) , } \end{array}$ , then both are given by + +$$ +\lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } = \lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow } = \frac { \frac { 1 } { 2 \pi } B ( B - 1 ) a _ { 1 } ^ { 2 } \operatorname { B e t a } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { 2 } } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } . +$$ + +This quantity is always $< 1$ , and for any fixed $B$ , is maximized by $\phi = \mathrm { i d } .$ . Furthermore, for $\phi = \mathrm { i d } ,$ , $\begin{array} { r } { \lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } = \lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow } = \frac { B - 1 } { 2 } \mathrm { P } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { - 2 } } \end{array}$ and increase to $^ { l }$ from below as $B \infty$ . + +Thus, a deep batchnorm network loses correlation information between two input batches, exponentially fast in depth, no matter what nonlinearity (that induces fixed point of the form given by Eq. (21)). This again contrasts with the case for vanilla networks which can control this rate of information loss by tweaking the initialization variances for weights and biases Poole et al. (2016); Schoenholz et al. (2016). The absence of coordinatewise nonlinearity, i.e. $\phi = \mathrm { i d }$ , maximally suppresses both this loss of information as well as the gradient explosion, and in the ideal, infinite batch scenario, can cure both problems. + +# 4 EXPERIMENTS + +Having developed a theory for neural networks with batch normalization at initialization, we now explore the relationship between the properties of these random networks and their learning dynamics. We will see that the trainability of networks with batch normalization is controlled by gradient explosion. We quantify the depth scale over which gradients explode by $\xi = 1 / \log \lambda _ { G } ^ { \downarrow }$ where, as above, $\lambda _ { G } ^ { \downarrow }$ is the largest eigenvalue of the jacobian. Across many different experiments we will see strong agreement between $\xi$ and the maximum trainable depth. + +We first investigate the relationship between trainability and initialization for rectified linear networks as a function of batch size. The results of these experiments are shown in Fig. 3 where in each case we plot the test accuracy after training as a function of the depth and the batch size and overlay $1 6 \xi$ in white dashed lines. In Fig. 3 (a) we consider networks trained using SGD on MNIST where we observe that networks deeper than about 50 layers are untrainable regardless of batch size. In (b) we compare standard batch normalization with a modified version in which the batch size is held fixed but batch statistics are computed over subsets of size $B$ . This removes subtle gradient fluctuation effects noted in Smith & Le (2018). In (c) we do the same experiment with RMSProp and in (d) we train the networks on CIFAR10. In all cases we observe a nearly identical trainable region. + +![](images/06fb6d785031a6f5279637e6f03b5a8e24426114f3d6c70dee999e09b27a7afe.jpg) +Figure 3: Batch normalization strongly limits the maximum trainable depth. Colors show test accuracy for rectified linear networks with batch normalization and $\gamma = 1$ , $\beta = 0$ , $\epsilon = 1 0 ^ { - 3 }$ , $N = 3 8 4$ , and $\eta \ : = \ : 1 0 ^ { - 5 } B$ . (a) trained on MNIST for 10 epochs (b) trained with fixed batch size 1000 and batch statistics computed over sub batches of size $B$ . (c) trained using RMSProp. (d) Trained on CIFAR10 for 50 epochs. In each case, White dashed line indicates a theoretical prediction of trainable depth, as discussed in the text. + +![](images/a5f30b83245a38e0f886a3bb347ed85b4c6ef7d3156584dc8f17e6f2ec118c68.jpg) +Figure 4: Gradients in networks with batch normalization quickly achieve dynamical equilibrium. Plots of the relative magnitudes of (a) the weights (b) the gradients of the loss with respect to the pre-activations and (c) the gradients of the loss with respect to the weights for rectified linear networks of varying depths during the first 10 steps of training. Colors show step number from 0 (black) to 10 (green). + +It is counter intuitive that training can occur at intermediate depths, from 10 to 50 layers, where there is significant gradient explosion. To gain insight into the behavior of the network during learning we record the magnitudes of the weights, the gradients with respect to the pre-activations, and the gradients with respect to the weights for the first 10 steps of training for networks of different depths. The result of this experiment is shown in Fig. 4. Here we see that before learning, as expected, the norm of the weights is constant and independent of layer while the gradients feature exponential explosion. However, we observe that two related phenomena occur after a single step of learning: the weights grow exponentially in the depth and the magnitude of the gradients are stable up to some threshold after which they vanish exponentially in the depth. This is as the result of the scaling property of batchnorm, where $B _ { \phi } ^ { \prime } ( \alpha \bar { h } ) = \alpha ^ { - 1 } \bar { B } _ { \phi } ^ { \prime } ( h )$ : The first-step gradients dominate the weights due to gradient explosion, hence the exponential growth in weight norms, and thereafter, the gradients are scaled down commensurately. Thus, it seems that although the gradients of batch normalized networks at initialization are ill-conditioned, the gradients appear to quickly reach a stable dynamical equilibrium. While this appears to be beneficial for shallower networks, in deeper ones, the relative gradient vanishing can in fact be so severe as to cause lower layers to mostly stay constant during training. Aside from numerical issues, this seems to be the primary mechanism through which gradient explosion causes training problems for networks deeper than 50 layers. + +As discussed in the theoretical exposition above, batch normalization necessarily features exploding gradients for any nonlinearity that converges to a BSB1 fixed point. We performed a number of experiments exploring different ways of ameliorating this gradient explosion. These experiments are shown in Fig. 5 with theoretical predictions for the maximum trainable depth overlaid; in all cases we see exceptional agreement. In Fig. 5 (a,b) we explore two different ways of tuning the degree to which activation functions in a network are nonlinear. In Fig. 5 (a) we tune $\gamma \in [ 0 , 2 ]$ for networks with tanh-activations and note that in the $\gamma 0$ limit the function is linear. In Fig. 5 (b) we tune $\beta \in [ 0 , 2 ]$ for networks with rectified linear activations and we note, similarly, that in the $\beta \to \infty$ limit the function is linear. As expected, we see the maximum trainable depth increase significantly with decreasing $\gamma$ and increasing $\beta$ . In Fig. 5 (c,d) we vary $\epsilon$ for tanh and rectified linear networks respectively. In both cases, we observe a critical point at large $\epsilon$ where gradients do not explode and very deep networks are trainable. + +![](images/719edc40217e94666f7845a907bb79a6913bae0e8aebc61292cfa994eedb0428.jpg) +Figure 5: Three techniques for counteracting gradient explosion. Test accuracy on MNIST as a function of different hyperparameters along with theoretical predictions (white dashed line) for the maximum trainable depth. (a) tanh network changing the overall scale of the pre-activations, here $\gamma 0$ corresponds to the linear regime. (b) Rectified linear network changing the mean of the preactivations, here $\beta \to \infty$ corresponds to the linear regime. (c,d) tanh and rectified linear networks respectively as a function of $\epsilon$ , here we observe a well defined phase transition near $\epsilon \sim 1$ . Note that in the case of rectified linear activations we use $\beta = 2$ so that the function is locally linear about 0. We also find initializing $\beta$ and/or setting $\epsilon > 0$ having positive effect on VGG19 with batchnorm. See Figs. 8 and 9. + +# 5 CONCLUSION + +In this work we have presented a theory for neural networks with batch normalization at initialization. In the process of doing so, we have uncovered a number of counterintuitive aspects of batch normalization and – in particular – the fact that at initialization it unavoidably causes gradients to explode with depth. 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Deep mean field theory: Layerwise variance and width variation as methods to control gradient explosion. 2018. + +Barret Zoph, Vijay Vasudevan, Jonathon Shlens, and Quoc V Le. Learning transferable architectures for scalable image recognition. + +![](images/cfaf38a6f9689f6177902ab53192b119e12ff08a39f90e76ac5c89eb19aa50b8.jpg) +Figure 6: Batch norm leads to a chaotic input-output map with increasing depth. A linear network with batch norm is shown acting on two minibatches of size 64 after random orthogonal initialization. The datapoints in the minibatch are chosen to form a 2d circle in input space, except for one datapoint that is perturbed separately in each minibatch (leftmost datapoint at input layer 0). Because the network is linear, for a given minibatch it performs an affine transformation on its inputs – a circle in input space remains an ellipse throughout the network. However, due to batch norm the coefficients of that affine transformation change nonlinearly as the datapoints in the minibatch are changed. (a) Each pane shows a scatterplot of activations at a given layer for all datapoints in the minibatch, projected onto the top two PCA directions. PCA directions are computed using the concatenation of the two minibatches. Due to the batch norm nonlinearity, minibatches that are nearly identical in input space grow increasingly dissimilar with depth. Intuitively, this chaotic input-output map can be understood as the source of exploding gradients when batch norm is applied to very deep networks, since very small changes in an input correspond to very large movements in network outputs. $( b )$ The correlation between the two minibatches, as a function of layer, for the same network. Despite having a correlation near one at the input layer, the two minibatches rapidly decorrelate with depth. See Appendix H for a theoretical treatment. + +# A VGG19 WITH BATCHNORM ON CIFAR100 + +Even though at initialization time batchnorm causes gradient explosion, after the first few epochs, the relative gradient norms $\| \nabla _ { \theta } L \| / \| \theta \|$ for weight parameters $\theta = W$ or BN scale parameter $\theta = \gamma$ , equilibrate to about the same magnitude. See Fig. 7. + +![](images/1271c37190d4a136d372e3f3a01c0fdc8b9c25f650bc101aedd4505499dee4d8.jpg) +Figure 7: relative gradient norms of different parameters in layer order (input to output from left to right), with $\gamma$ and $W$ interleaving. From dark to light blue, each curve is separated by (a) 3, (b) 5, or (c) 10 epochs. We see that after 10 epochs, the relative gradient norms of both $\gamma$ and $W$ for all layers become approximately equal despite gradient explosion initially. + +We find acceleration effects, especially in initial training, due to setting $\epsilon > 0$ and/or initializing $\beta > 0$ . See Figs. 8 and 9. + +![](images/053708b11fbf0db9b129fca4bc8269325ed80518835b7c1df3d1b09d39879d2d.jpg) +Figure 8: We sweep over different values of learning rate, $\beta$ initialization, and $\epsilon$ , in training VGG19 with batchnorm on CIFAR100 with data augmentation. We use 8 random seeds for each combination, and assign to each combination the median training/validation accuracy over all runs. We then aggregate these scores here. In the first row we look at training accuracy with different learning rate vs $\beta$ initialization at different epochs of training, presenting the max over $\epsilon$ . In the second row we do the same for validation accuracy. In the third row, we look at the matrix of training accuracy for learning rate vs $\epsilon$ , taking max over $\beta$ . In the fourth row, we do the same for validation accuracy. + +# B GRADIENT INDEPENDENCE ASSUMPTION + +Following prior literature Schoenholz et al. (2016); Yang & Schoenholz (2017); Xiao et al. (2018), in this paper, in regards to computations involving backprop, we assume +Assumption 2. During backpropagation, whenever we multiply by $W ^ { T }$ for some weight matrix $W$ , we multiply by an iid copy instead. + +![](images/8831ac5316669b9a2076ca0903ff531318abb9c8e1e8e0d2529f5e15666bad87.jpg) +Figure 9: In the same setting as Fig. 8, except we don’t take the max over the unseen hyperparameter but rather set it to 0 (the default value). + +As in these previous works, we find excellent agreement between computations made under this assumption and the simulations (see Fig. 1). Yang (2019) in fact recently rigorously justified this assumption as used in the computation of moments, for a wide variety of architectures, like multilayer perceptron (Schoenholz et al., 2016), residual networks (Yang & Schoenholz, 2017), and convolutional networks (Xiao et al., 2018) studied previously, but without batchnorm. The reason that their argument does not extend to batchnorm is because of the singularity in its Jacobian at 0. However, as observed in our experiments, we expect that a proof can be found to extend Yang & Schoenholz (2017)’s work to batchnorm. + +# C NOTATIONS + +Definition C.1. Let $S _ { B }$ be the space of PSD matrices of size $B \times B$ . Given a measurable function $\Phi : \mathbb { R } ^ { B } \mathbb { R } ^ { A }$ , define the integral transform $\mathrm { V } _ { \Phi } : { \mathcal { S } } _ { B } \to { \mathcal { S } } _ { A }$ by $\mathrm { V } _ { \Phi } ( \Sigma ) = \mathbb { E } [ \Phi ( h ) ^ { \otimes 2 } : h \sim$ $\mathcal { N } ( 0 , \Sigma ) ]$ .3 When $\phi : \mathbb { R } \mathbb { R }$ and $B$ is clear from context, we also write $\mathrm { V } _ { \phi }$ for $\mathrm { V }$ applied to the function acting coordinatewise by $\phi$ . + +Definition C.2. For any $\phi ~ : ~ \mathbb { R } ~ \to ~ \mathbb { R }$ , let $B _ { \phi } \quad : \quad \mathbb { R } ^ { B } \quad \to \quad \mathbb { R } ^ { B }$ be batchnorm (applied to a batch of neuronal activations) followed by coordinatewise applications of $\phi$ , $B _ { \phi } ( x ) _ { j } ~ =$ $\begin{array} { r } { \phi \left( \left( x _ { j } - \operatorname { A v g } x \right) / \sqrt { \frac { 1 } { B } \sum _ { i = 1 } ^ { B } \left( x _ { i } - \operatorname { A v g } x \right) ^ { 2 } } \right) } \end{array}$ (here $\begin{array} { r } { \operatorname { A v g } x = \frac { 1 } { B } \sum _ { i = 1 } ^ { B } x _ { i } ) } \end{array}$ . When $\phi = \mathrm { i d }$ we will also write $\boldsymbol { B } = \boldsymbol { B } _ { \mathrm { i d } }$ . + +Definition C.3. Define the matrix $\begin{array} { r } { G ^ { B } = I - \frac { 1 } { B } \mathbf { 1 } \mathbf { 1 } ^ { T } } \end{array}$ . Let $\mathcal { S } _ { B } ^ { G }$ be the space of PSD matrices of size $B \times B$ with zero mean across rows and columns, $S _ { B } ^ { G } : = \{ \Sigma \in { \cal S } _ { B } : G ^ { B } \Sigma G ^ { B } = \Sigma \} = \{ \Sigma \in { \cal S } _ { B } :$ $\Sigma \mathbf { 1 } = 0 \}$ . + +When $B$ is clear from context, we will suppress the subscript/superscript $B$ . In short, for $h \in \mathbb { R } ^ { B }$ , $G h$ zeros the sample mean of $h$ . $G$ is a projection matrix to the subspace of vectors $\boldsymbol { h } \in \mathbb { R } ^ { B }$ of zero coordinate sum. With the above definitions, we then have $B _ { \phi } ( h ) = \bar { \phi } ( \sqrt { B } G h / \| G h \| )$ . + +We use $\Gamma$ to denote the Gamma function, $\mathrm { P } \left( a , b \right) : = \Gamma ( a + b ) / \Gamma ( a )$ to denote the Pochhammer symbol, and $\begin{array} { r } { \mathrm { B e t a } ( a , b ) = \frac { \Gamma ( a ) \Gamma ( b ) } { \Gamma ( a + b ) } } \end{array}$ to denote the Beta function. We use $\langle \cdot , \cdot \rangle$ to denote the dot product for vectors and trace inner product for matrices. Sometimes when it makes for simpler notation, we will also use $\cdot$ for dot product. + +We adopt the matrix convention that, for a multivariate function $f : \mathbb { R } ^ { n } \mathbb { R } ^ { m }$ , the Jacobian is + +$$ +\frac { \mathrm { d } f } { \mathrm { d } x } = \left( \begin{array} { c c c c } { \frac { \partial f _ { 1 } } { \partial x _ { 1 } } } & { \frac { \partial f _ { 1 } } { \partial x _ { 2 } } } & { \cdot \cdot \cdot } & { \frac { \partial f _ { 1 } } { \partial x _ { n } } } \\ { \frac { \partial f _ { 2 } } { \partial x _ { 1 } } } & { \frac { \partial f _ { 2 } } { \partial x _ { 2 } } } & { \cdot \cdot \cdot } & { \frac { \partial f _ { 2 } } { \partial x _ { n } } } \\ { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { \frac { \partial f _ { m } } { \partial x _ { 1 } } } & { \frac { \partial f _ { m } } { \partial x _ { 2 } } } & { \cdot \cdot \cdot } & { \frac { \partial f _ { m } } { \partial x _ { n } } } \end{array} \right) , +$$ + +so that $\begin{array} { r } { f ( x + \Delta x ) = \frac { \mathrm { d } f } { \mathrm { d } x } \Delta x + o ( \Delta x ) } \end{array}$ , where $x$ and $\Delta x$ are treated as column vectors. In what follows we further abbreviate $\begin{array} { r } { \mathcal { B } _ { \phi } ^ { \prime } ( x ) = \frac { \mathrm { d } \mathcal { B } _ { \phi } ( z ) } { \mathrm { d } z } \big | _ { z = x } , \mathcal { B } _ { \phi } ^ { \prime } ( x ) \Delta x = \frac { \mathrm { d } \mathcal { B } _ { \phi } ( z ) } { \mathrm { d } z } \big | _ { z = x } \Delta x } \end{array}$ . + +# D A GUIDE TO THE REST OF THE APPENDIX + +As discussed in the main text, we are interested in several closely related dynamics induced by batchnorm in a fully-connected network with random weights. One is the forward propagation equation Eq. (43) + +$$ +\Sigma ^ { l } = \mathrm { V } _ { \mathcal { B _ { \phi } } } ( \Sigma ^ { l - 1 } ) +$$ + +studied in Appendix F. Another is the backward propagation equation Eq. (51) + +$$ +\Pi ^ { l } = \mathrm { V } _ { B _ { \phi } ^ { \prime } } ( \Sigma ^ { * } ) ^ { \dagger } \{ \Pi ^ { l + 1 } \} +$$ + +studied in Appendix G. We will also study their generalizations to the simultaneous propagation of multiple batches in Appendices $\cdot$ and I. + +In general, we discover several ways to go about such analyses, each with their own benefits and drawbacks: + +• The Laplace Method (Appendix E.1): If the nonlinearity $\phi$ is positive homogeneous, we can use Schwinger’s parametrization to turn $\mathrm { V } _ { B _ { \phi } }$ and $\mathrm { V } _ { B _ { \phi } ^ { \prime } }$ into nicer forms involving only 1 or 2 integrals, as mentioned in the main text. All relevant quantities, such as eigenvalues of the backward dynamics, can be further simplified to closed forms. Lemma E.2 gives the master equation for the Laplace method. Because the Laplace method requires $\phi$ to be positive homogeneous to apply, we review relevant results of such functions in Appendices E.1.1 and E.1.2. + +The Fourier Method (Appendix E.2): For polynomially bounded and continuous nonlinearity $\phi$ (for the most general set of conditions, see Thm E.25), we can obtain similar simplifications using the Fourier expansion of the delta function, with the penalty of an additional complex integral. + +• Spherical Integration (Appendix E.3): Batchnorm can naturally be interpreted as a linear projection followed by projection to a sphere of radius $\sqrt { B }$ . Assuming that the forward dynamics converges to a BSB1 fixed point, one can express $\mathrm { V } _ { B _ { \phi } }$ and $\mathrm { V } _ { B _ { \phi } ^ { \prime } }$ as spherical integrals, for all measurable $\phi$ . One can reduce this $( B - 1 )$ -dimensional integral, in spherical angular coordinates, into 1 or 2 dimensions by noting that the integrand depending on $\phi$ only depends on 1 or 2 angles. Thus this method is very suitable for numerical evaluation of quantities of interest for general $\phi$ . + +• The Gegenbauer Method (Appendix E.4): Gegenbauer polynomials are orthogonal polynomials under the weight $( 1 - x ^ { 2 } ) ^ { \alpha }$ for some $\alpha$ . They correspond to a special type of spherical harmonics called zonal harmonics (Defn E.44) that has the very useful reproducing property (Fact E.46). By expressing the nonlinearity $\phi$ in this basis, we can see easily that the relevant eigenvalues of the forward and backward dynamics are ratios of two quadratic forms of $\phi$ ’s Gegenbauer coefficients. In particular, for the largest eigenvalue of the backward dynamics, the two quadratic forms are both diagonal, and in a way that makes apparent the necessity of gradient explosion (under the BSB1 fixed point assumption). In regards to numerical computations, the Gegenbauer method has the benefit of only requiring 1-dimensional integrals to obtain the coefficients, but if $\phi$ has slowlydecaying coefficients, then a large number of such integrals may be required to get an accurate answer. + +In each of the following sections, we will attempt to conduct an analysis with each method when it is feasible. + +When studying the convergence rate of the forward dynamics and the gradient explosion of the backward dynamics, we encounter linear operators on matrices that satisfy an abundance of symmetries, called ultrasymmetric operators. We study these operators in Appendix E.5 and contribute several structural results Thms E.61, E.62, E.72 and E.72 on their eigendecomposition that are crucial to deriving the asymptotics of the dynamics. + +All of the above will be explained in full in the technique section Appendix E. We then proceed to study the forward and backward dynamics in detail (Appendices F to I), now armed with tools we need. Up to this point, we only consider the dynamics assuming Eq. (43) converges to a BSB1 fixed point, and $B \geq 4$ . In Appendix J we discuss what happens outside this regime, and in Appendix K we show our current understanding of BSB2 fixed point dynamics. + +Note that the backward dynamics studied in these sections is that of the gradient with respect to the hidden preactivations. Nevertheless, we can compute the moments of the weight gradients from this, which is of course what is eventually used in gradient descent. We do so in the final section Appendix L. + +Main Technical Results Corollary F.3 establishes the global convergence of Eq. (43) to BSB1 fixed point for $\phi = \mathrm { i d }$ (as long as the initial covariance is nondegenerate), but we are not able to prove such a result for more general $\phi$ . We thus resort to studying the local convergence properties of BSB1 fixed points. + +First, Thms F.5, F.8 and F.13 respectively compute the BSB1 fixed point from the perspectives of spherical integration, the Laplace method, and the Gegenbauer method. Thms F.22 and F.24 give the Gegenbauer expansion of the BSB1 local convergence rate, and Thm F.33 gives a more succinct form of the same for positive homogeneous $\phi$ . Appendix F.3.1 discusses how to do this computation via spherical integration. + +Next, we study the gradient dynamics, and Thms G.5 and G.12 yield the gradient explosion rates with the Gegenbauer method and the Laplace method (for positive homogeneous $\phi ,$ , respectively. Appendix G.1.1 discusses how to compute this for general $\phi$ using spherical integration. + +We then turn to the dynamics of cross batch covariances. Thm H.3 gives the form of the cross batch fixed point via spherical integration and the Gegenbauer method, and Thm H.6 yields a more specific form for positive homogeneous $\phi$ via the Laplace method. The local convergence rate to this fixed point is given by Thms H.13 and H.14, respectively using the Gegenbauer method and the Laplace method. Finally, the correlation between gradients of the two batches is shown to decrease exponentially in Corollary I.6 using the Gegenbauer method, and the decay rate is given succinctly for positive homogeneous $\phi$ in Thm I.3. + +# E TECHNIQUES + +We introduce the key techniques and tools in our analysis in this section. + +As discussed above, the Laplace method is useful for deriving closed form expressions for positive homogeneous $\phi$ . The key insight here is to apply Schwinger parametrization to deal with normalization. + +Lemma E.1 (Schwinger parametrization). For $z > 0$ and $c > 0$ , + +$$ +c ^ { - z } = \Gamma ( z ) ^ { - 1 } \int _ { 0 } ^ { \infty } x ^ { z - 1 } e ^ { - c x } \mathrm { d } x +$$ + +The following is the key lemma in the Laplace method. + +Lemma E.2 (The Laplace Method Master Equation). For $A , A ^ { \prime } \in \mathbb { N } ,$ , let $f : \mathbb { R } ^ { A } \to \mathbb { R } ^ { A ^ { \prime } }$ and let $k \geq 0$ . Suppose $\| f ( \bar { y } ) \| \leq h ( \| y \| )$ for some nondecreasing function $h : \mathbb { R } ^ { \geq 0 } \mathbb { R } ^ { \geq 0 }$ such that $\mathbb { E } [ h ( r \| y \| ) : y \in \dot { \mathcal { N } } ( 0 , \dot { I } _ { A } ) ]$ exists for every $r \geq 0$ . Define $\wp ( \bar { \Sigma } ) : = \mathbb { E } [ \| y \| ^ { - 2 k } f ( y ) : y \sim \mathcal { N } ( 0 , \Sigma ) ]$ . Then on $\left\{ \Sigma \in { \mathcal { S } } _ { A } : { \mathrm { r a n k } } \Sigma > 2 k \right\}$ , $\wp ( \Sigma )$ is well-defined and continuous, and furthermore satisfies + +$$ +\wp ( \Sigma ) = \Gamma ( k ) ^ { - 1 } \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { k - 1 } \operatorname* { d e t } ( I + 2 s \Sigma ) ^ { - 1 / 2 } \mathbb { E } [ f ( y ) : y \sim { \mathcal { N } } ( 0 , \Sigma ( I + 2 s \Sigma ) ^ { - 1 } ) ] +$$ + +Proof. $\wp ( \Sigma )$ is well-defined for full rank $\Sigma$ because the $\| y \| ^ { - 2 k }$ singularity at $y = 0$ is Lebesgueintegrable in a neighborhood of 0 in dimension $A > 2 k$ . + +We prove Eq. (25) in the case when $\Sigma$ is full rank and then apply a continuity argument. + +Proof of Eq. (25) for full rank $\Sigma$ . First, we will show that we can exchange the order of integration + +$$ +\int _ { 0 } ^ { \infty } \mathrm { d } s \ \int _ { \mathbb { R } ^ { A } } \mathrm { d } y \ s ^ { k - 1 } f ( y ) e ^ { - { \frac { 1 } { 2 } } y ^ { T } ( \Sigma ^ { - 1 } + 2 s I ) y } = \int _ { \mathbb { R } ^ { A } } \mathrm { d } y \ \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { k - 1 } f ( y ) e ^ { - { \frac { 1 } { 2 } } y ^ { T } ( \Sigma ^ { - 1 } + 2 s I ) y } +$$ + +by Fubini-Tonelli’s theorem. Observe that + +$$ +\begin{array} { r l } & { \quad ( 2 \pi ) ^ { - A / 2 } \operatorname* { d e t } \Sigma ^ { - 1 / 2 } \displaystyle \int _ { 0 } ^ { \infty } \operatorname { d } s ~ s ^ { k - 1 } \int _ { \mathbb R ^ { A } } \mathrm { d } y ~ \| f ( y ) \| e ^ { - \frac { 1 } { 2 } y ^ { T } ( \Sigma ^ { - 1 } + 2 s I ) y } } \\ & { = \displaystyle \int _ { 0 } ^ { \infty } \mathrm { d } s ~ s ^ { k - 1 } \operatorname* { d e t } ( \Sigma ( \Sigma ^ { - 1 } + 2 s I ) ) ^ { - 1 / 2 } \mathbb { E } [ \| f ( y ) \| : y \sim \mathcal { N } ( 0 , ( \Sigma ^ { - 1 } + 2 s I ) ^ { - 1 } ) ] } \\ & { = \displaystyle \int _ { 0 } ^ { \infty } \mathrm { d } s ~ s ^ { k - 1 } \operatorname* { d e t } ( I + 2 s \Sigma ) ^ { - 1 / 2 } \mathbb { E } [ \| f ( y ) \| : y \sim \mathcal { N } ( 0 , \Sigma ( I + 2 s \Sigma ) ^ { - 1 } ) ] } \end{array} +$$ + +For $\lambda = \| \Sigma \| _ { 2 }$ + +$$ +\| f ( { \sqrt { \Sigma ( I + 2 s \Sigma ) ^ { - 1 } } } y ) \| \leq h ( \| { \sqrt { \Sigma ( I + 2 s \Sigma ) ^ { - 1 } } } y \| \leq h \left( \left\| { \sqrt { \frac { \lambda } { 1 + 2 s \lambda } } } y \right\| \right) \leq h ( { \sqrt { \lambda } } \| y \| ) +$$ + +Because $\mathbb { E } [ h ( { \sqrt { \lambda } } \| y \| ) : y \in { \mathcal { N } } ( 0 , I ) ]$ exists by assumption, + +$$ +\begin{array} { r l } & { \quad \mathbb { E } [ \| f ( y ) \| : y \sim \mathcal { N } ( 0 , \Sigma ( I + 2 s \Sigma ) ^ { - 1 } ) ] } \\ & { = \mathbb { E } [ \| f ( \sqrt { \Sigma ( I + 2 s \Sigma ) ^ { - 1 } } y ) \| : y \sim \mathcal { N } ( 0 , I ) ] } \\ & { \mathbb { E } [ \| f ( 0 ) \| : y \sim \mathcal { N } ( 0 , I ) ] = \| f ( 0 ) \| } \end{array} +$$ + +as $s \to \infty$ , by dominated convergence with dominating function $h ( { \sqrt { \lambda } } \| y \| ) e _ { . } ^ { - { \frac { 1 } { 2 } } \| y \| ^ { 2 } } ( 2 \pi ) ^ { - A / 2 }$ . By the same reasoning, the function $s \mapsto \mathbb { E } [ \| f ( y ) \| : y \sim \mathcal { N } ( 0 , \Sigma ( \bar { I } + 2 s \dot { \Sigma } ) ^ { - 1 } ) ]$ is continuous. In particular this implies that $\begin{array} { r } { \operatorname* { s u p } _ { 0 < s < \infty } \mathbb { E } [ \| f ( y ) \| : y \sim \mathcal { N } ( 0 , \Sigma ( I + 2 s \Sigma ) ^ { - 1 } ) ] < \propto } \end{array}$ . Combined with the fact that $\operatorname* { d e t } ( I + 2 s \Sigma ) ^ { - 1 / 2 } = \mathbf { \bar { \Theta } } ( s ^ { - A / 2 } )$ as $s \to \infty$ , + +$$ +\begin{array} { r l } & { \quad \displaystyle \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { k - 1 } \operatorname* { d e t } ( I + 2 s \Sigma ) ^ { - 1 / 2 } \mathbb { E } [ \| f ( y ) \| : y \sim \mathcal { N } ( 0 , \Sigma ( I + 2 s \Sigma ) ^ { - 1 } ) ] } \\ & { \le \displaystyle \int _ { 0 } ^ { \infty } \mathrm { d } s \ \Theta ( s ^ { k - 1 - A / 2 } ) } \end{array} +$$ + +which is bounded by our assumption that $A / 2 > k$ . This shows that we can apply Fubini-Tonelli’s theorem to allow exchanging order of integration. + +Thus, + +$$ +\begin{array} { r l } & { \quad \mathbb { E } [ \| y \| ^ { - 2 k } f ( y ) \cdot y \cdot \mathcal { N } ( 0 , \Sigma ) ] } \\ & { = \mathbb { E } [ f ( y ) \int _ { 0 } ^ { \infty } \mathrm { d } s \ \Gamma ( k ) ^ { - 1 } s ^ { k - 1 } e ^ { - \| y \| ^ { 2 s } } : y \sim \mathcal { N } ( 0 , \Sigma ) ] } \\ & { = ( 2 \pi ) ^ { - A / 2 } \operatorname* { d e t } \Sigma ^ { - 1 / 2 } \int \mathrm { d } y \ e ^ { - \frac { 1 } { 2 } g ^ { T } \Sigma ^ { - 1 } s } f ( y ) \int _ { 0 } ^ { \infty } \mathrm { d } s \ \Gamma ( k ) ^ { - 1 } s ^ { k - 1 } e ^ { - \| y \| ^ { 2 s } } } \\ & { = ( 2 \pi ) ^ { - A / 2 } \Gamma ( k ) ^ { - 1 } \operatorname* { d e t } \Sigma ^ { - 1 / 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { k - 1 } \int \mathrm { d } y \ f ( y ) e ^ { - \frac { 1 } { 2 } g ^ { T } ( \Sigma ^ { - 1 } + 2 z / 3 ) } } \\ & { \qquad \mathrm { ( b y ~ F r u b i n i - I o r e l i ) } } \\ & { = \Gamma ( k ) ^ { - 1 } \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { k - 1 } \operatorname* { d e t } [ \Sigma ( \Sigma ^ { - 1 } + 2 s T ) ] ^ { - 1 / 2 } \mathbb { E } [ f ( y ) : y \sim \mathcal { N } ( 0 , ( \Sigma ^ { - 1 } + 2 s T ) ^ { - 1 } ) ] } \\ & { = \Gamma ( k ) ^ { - 1 } \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { k - 1 } \operatorname* { d e t } ( I + 2 s \Sigma ) ^ { - 1 / 2 } \mathbb { E } [ f ( y ) : y \sim \mathcal { N } ( 0 , \Sigma ( I + 2 s \Sigma ) ^ { - 1 } ) ] } \end{array} +$$ + +Domain and continuity of $\wp ( \Sigma )$ . The LHS of Eq. (25), $\wp ( \Sigma )$ , is defined and continuous on rank $\Sigma / 2 > k$ . Indeed, if $\begin{array} { r } { \begin{array} { r c l } { \dot { \Sigma } } & { = } & { M I _ { C } M ^ { T } } \end{array} } \end{array}$ , where $M$ is a full rank $A \times C$ matrix with rank $\Sigma = C \leq A$ , then + +$$ +\begin{array} { r l } & { \mathbb { E } [ \| y \| ^ { - 2 k } \| f ( y ) \| : y \sim { \mathcal { N } } ( 0 , \Sigma ) ] } \\ & { = \mathbb { E } [ \| M z \| ^ { - 2 k } \| f ( M z ) \| : z \sim { \mathcal { N } } ( 0 , I _ { C } ) ] . } \end{array} +$$ + +This is integrable in a neighborhood of 0 iff $C > 2 k$ , while it’s always integrable outside a ball around 0 because $\| f \|$ by itself already is. So $\wp ( \Sigma )$ is defined whenever rank $\Sigma > 2 k$ . Its continuity can be established by dominated convergence. + +Proof of Eq. (25) for rank $\Sigma > 2 k$ . Observe that $\operatorname* { d e t } ( I + 2 s \Sigma ) ^ { - 1 / 2 }$ is continuous in $\Sigma$ and, by an application of dominated convergence as in the above, $\mathbb { E } [ f ( y ) : y \sim \mathcal { N } ( 0 , \Sigma ( I + 2 s \Sigma ) ^ { - 1 } ) ]$ is continuous in $\Sigma$ . So the RHS of Eq. (25) is continuous in $\Sigma$ whenever the integral exists. By the reasoning above, $\mathbb { E } [ f ( y ) : y \sim \mathcal { N } ( 0 , \Sigma ( I + 2 s \Sigma ) ^ { - 1 } ) ]$ is bounded in $s$ and $\mathrm { d e t } \bar { ( } I + 2 s \Sigma ) ^ { - 1 } \bar { / } 2 =$ $\Theta ( s ^ { - \mathrm { r a n k } \Sigma / 2 } )$ , so that the integral exists iff rank $\Sigma / 2 > k$ . + +To summarize, we have proved that both sides of Eq. (25) are defined and continous for rank $\Sigma >$ $2 k$ . Because the full rank matrices are dense in this set, by continuity Eq. (25) holds for all rank $\Sigma >$ $2 k$ . □ + +If $\phi$ is degree- $\alpha$ positive homogeneous, i.e. $\phi ( r u ) = r ^ { \alpha } \phi ( u )$ for any $u \in \mathbb { R } , r \in \mathbb { R } ^ { + }$ , we can apply Lemma E.2 with $k = \alpha$ , + +$$ +\begin{array} { r l } & { { \mathrm { V } _ { B _ { \phi } } ( \Sigma ) } = \mathbb { E } [ \phi ( \sqrt { B } G h / \| G h \| ) ^ { \otimes 2 } : h \sim \mathcal { N } ( 0 , \Sigma ) ] = B ^ { \alpha } \mathbb { E } [ \phi ( G h ) ^ { \otimes 2 } / \| G h \| ^ { 2 \alpha } : h \sim \mathcal { N } ( 0 , \Sigma ) ] } \\ & { \qquad = B ^ { \alpha } \mathbb { E } [ \phi ( z ) ^ { \otimes 2 } / \| z \| ^ { 2 \alpha } : z \sim \mathcal { N } ( 0 , G \Sigma G ) ] } \\ & { \qquad = B ^ { \alpha } \Gamma ( \alpha ) ^ { - 1 } \displaystyle \int _ { 0 } ^ { \infty } \mathrm { d } s s ^ { \alpha - 1 } \operatorname* { d e t } ( I + 2 s \Sigma G ) ^ { - 1 / 2 } \mathrm { V } _ { \phi } ( G ( I + 2 s \Sigma G ) ^ { - 1 } \Sigma G ) . } \end{array} +$$ + +If $\phi = \mathrm { r e l u }$ , then + +$$ +\mathrm { V } _ { \phi } ( \Sigma ) _ { i j } = { \left\{ \begin{array} { l l } { { { \frac { 1 } { 2 } } \Sigma _ { i i } } } & { { \mathrm { i f ~ } } i = j } \\ { { \frac { 1 } { 2 } } \mathrm { J } _ { 1 } \left( \Sigma _ { i j } / { \sqrt { \Sigma _ { i i } \Sigma _ { j j } } } \right) { \sqrt { \Sigma _ { i i } \Sigma _ { j j } } } } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. } +$$ + +and, more succinctly, $\mathrm { V } _ { \phi } ( \Sigma ) ~ = ~ D ^ { 1 / 2 } \mathrm { V } _ { \phi } ( D ^ { - 1 / 2 } \Sigma D ^ { - 1 / 2 } ) D ^ { 1 / 2 }$ where $D \ = \ \operatorname { D i a g } ( \Sigma )$ . Here $\begin{array} { r } { \mathrm { J _ { 1 } } ( c ) : = \frac { 1 } { \pi } ( \sqrt { 1 - c ^ { 2 } } + ( \pi - \operatorname { a r c c o s } ( c ) ) c ) } \end{array}$ (Cho & Saul, 2009). + +Matrix simplification. We can simplify the expression $G ( I + 2 s \Sigma G ) ^ { - 1 } \Sigma G$ , leveraging the fact that $G$ is a projection matrix. + +Definition E.3. Let $\mathbb { P }$ be an $B \times ( B - 1 )$ matrix whose columns form an orthonormal basis of $\begin{array} { r } { \operatorname { i m } G : = \{ G v : v \in \mathbb { R } ^ { B } \} = \{ w \in \mathbb { R } ^ { B } : \sum _ { i } w _ { i } = 0 \} } \end{array}$ . Then the $B \times B$ matrix $\tilde { \mathbb { e } } = \left( \mathbb { e } | B ^ { - 1 / 2 } \mathbf { 1 } \right)$ is an orthogonal matrix. For much of this paper $\mathbb { P }$ can be any such basis, but at certain sections we will consider specific realizations of $\mathbb { P }$ for explicit computation. + +From easy computations it can be seen that $G = \tilde { \mathbb { e } } \left( \begin{array} { c c } { I _ { B - 1 } } & { 0 } \\ { 0 } & { 0 } \end{array} \right) \tilde { \mathbb { e } } ^ { T }$ . Suppose $\Sigma = \tilde { \mathbb { P } } \left( \stackrel { \Sigma } { v ^ { T } } \stackrel { v } { a } \right) \tilde { \mathbb { P } } ^ { T }$ where $\Sigma _ { \square }$ is $( B - 1 ) \times ( B - 1 )$ , $v$ is a column vector and $a$ is a scalar. Then $\Sigma _ { \bigstar } = \ L _ { \mathbb { e } } ^ { T } \Sigma _ { \mathbb { e } }$ and $\Sigma G = \tilde { \mathbb { e } } \left( \stackrel { \Sigma } { v ^ { T } } \stackrel { 0 } { 0 } \right) \tilde { \mathbb { e } } ^ { T }$ is block lower triangular, and + +$$ +\begin{array} { r l } & { \left( I + 2 s \Sigma G \right) ^ { - 1 } = \tilde { \Phi } \left( \begin{array} { c c } { ( I + 2 s \Sigma _ { \perp } ) ^ { - 1 } } & { 0 } \\ { \ast } & { 1 } \end{array} \right) \tilde { \Phi } ^ { T } } \\ & { ~ } \\ & { ( I + 2 s \Sigma G ) ^ { - 1 } \Sigma G = \tilde { \Phi } \left( \begin{array} { c c } { ( I + 2 s \Sigma _ { \perp } ) ^ { - 1 } \Sigma _ { \perp } } & { 0 } \\ { \ast } & { 0 } \end{array} \right) \tilde { \Phi } ^ { T } } \\ & { G ( I + 2 s \Sigma G ) ^ { - 1 } \Sigma G = \tilde { \Phi } \left( \begin{array} { c c } { ( I + 2 s \Sigma _ { \perp } ) ^ { - 1 } \Sigma _ { \perp } } & { 0 } \\ { 0 } & { 0 } \end{array} \right) \tilde { \Phi } ^ { T } } \\ & { ~ } \\ & { ~ = \varphi ( I + 2 s \Sigma _ { \perp } ) ^ { - 1 } \Sigma _ { \perp } \tilde { \Phi } ^ { T } } \\ & { ~ = G \Sigma G ( I + 2 s G \Sigma G ) ^ { - 1 } } \\ & { ~ = : \Sigma ^ { G } ( I + 2 s \Sigma ^ { G } ) ^ { - 1 } } \end{array} +$$ + +where + +Definition E.4. For any matrix $\Sigma$ , write $\Sigma ^ { G } : = G \Sigma G$ . + +Similarly, $\operatorname* { d e t } ( I _ { B } + 2 s \Sigma G ) = \operatorname* { d e t } ( I _ { B - 1 } + 2 s \Sigma _ { \Sigma } ) = \operatorname* { d e t } ( I _ { B } + 2 s \Sigma ^ { G } )$ . So, altogether, we have Theorem E.5. Suppose $\phi : \mathbb { R } \mathbb { R }$ is degree- $\alpha$ positive homogeneous. Then for any $B \times ( B - 1 )$ matrix $\mathbb { P }$ whose columns form an orthonormal basis of $\operatorname { i m } G : = \{ G v : v \in { \bar { \mathbb { R } } } ^ { B } \} \ = \{ w \in \mathbb { R } ^ { B } ^ { * }$ : $\begin{array} { r } { \sum _ { i } w _ { i } = 0 \} } \end{array}$ , with $\Sigma _ { \perp } = \mathrm { \bar { e } } ^ { T } \Sigma _ { \Phi }$ , + +$$ +\begin{array} { c } { { \displaystyle \mathrm { V } _ { \mathcal { B } _ { \phi } } ( \Sigma ) = B ^ { \alpha } \Gamma ( \alpha ) ^ { - 1 } \int _ { 0 } ^ { \infty } \mathrm { d } s ~ s ^ { \alpha - 1 } \operatorname * { d e t } ( I + 2 s \Sigma _ { \boxdot } ) ^ { - 1 / 2 } \mathrm { V } _ { \phi } ( \mathtt { e } ( I + 2 s \Sigma _ { \boxdot } ) ^ { - 1 } \Sigma _ { \boxdot } ) ^ { - 1 } \mathrm { e } ^ { T } ) } } \\ { { = B ^ { \alpha } \Gamma ( \alpha ) ^ { - 1 } \displaystyle \int _ { 0 } ^ { \infty } \mathrm { d } s ~ s ^ { \alpha - 1 } \operatorname * { d e t } ( I + 2 s \Sigma ^ { G } ) ^ { - 1 / 2 } \mathrm { V } _ { \phi } ( ( I + 2 s \Sigma ^ { G } ) ^ { - 1 } \Sigma ^ { G } ) ^ { - 1 } ) } } \end{array} +$$ + +Now, in order to use Laplace’s method, we require $\phi$ to be positive homogeneous. What do these functions look like? The most familiar example to a machine learning audience is most likely ReLU. It turns out that 1-dimensional positive homogeneous functions can always be described as a linear combination of powers of ReLU and its reflection across the y-axis. Below, we review known facts about these functions and their integral transforms, starting with the powers of ReLU in Appendix E.1.1 and then onto the general case in Appendix E.1.2. + +# E.1.1 $\alpha$ -RELU + +Recall that $\alpha$ -ReLUs (Yang & Schoenholz, 2017) are, roughly speaking, the $\alpha$ th power of ReLU. + +Definition E.6. The $\alpha$ -ReLU function $\rho _ { \alpha } : \mathbb { R } \to \mathbb { R }$ sends $x \mapsto x ^ { \alpha }$ when $x > 0$ and $x \mapsto 0$ otherwise. + +This is a continuous function for $\alpha > 0$ but discontinuous at 0 for all other $\alpha$ . + +We briefly review what is currently known about the $\mathrm { V }$ and W transforms of $\rho _ { \alpha }$ (Cho & Saul, 2009; +Yang & Schoenholz, 2017). + +Definition E.7. For any $\alpha > - \frac { 1 } { 2 }$ , define $\begin{array} { r } { \mathsf { c } _ { \alpha } = \frac { 1 } { \sqrt { \pi } } 2 ^ { \alpha - 1 } \Gamma ( \alpha + \frac { 1 } { 2 } ) } \end{array}$ + +When considering only 1-dimensional Gaussians, $\mathrm { V } _ { \rho _ { \alpha } }$ is very simple. + +Proposition E.8. If $\alpha > - \frac { 1 } { 2 }$ , then for any $q \in S _ { 1 } = \mathbb { R } ^ { \geq 0 }$ , $\mathrm { V } _ { \rho _ { \alpha } } ( q ) = \mathsf { c } _ { \alpha } q ^ { \alpha }$ + +To express results of $\mathrm { V } _ { \rho _ { \alpha } }$ on $\boldsymbol { S } _ { B }$ for higher $B$ , we first need the following Definition E.9. Define + +$$ +J _ { \alpha } ( \theta ) : = \frac { 1 } { 2 \pi \mathsf { c } _ { \alpha } } ( \sin \theta ) ^ { 2 \alpha + 1 } \Gamma ( \alpha + 1 ) \int _ { 0 } ^ { \pi / 2 } \frac { \mathrm { d } \eta \cos ^ { \alpha } \eta } { ( 1 - \cos \theta \cos \eta ) ^ { 1 + \alpha } } +$$ + +and $\mathrm { J } _ { \alpha } ( c ) = J _ { \alpha } ( \operatorname { a r c c o s } c )$ for $\alpha > - 1 / 2$ . + +Then + +Proposition E.10. For any $\Sigma \in { \cal S } _ { B }$ , let $D$ be the diagonal matrix with the same diagonal as $\Sigma$ . Then + +$$ +\mathrm { V } _ { \rho _ { \alpha } } ( \Sigma ) = \mathsf { c } _ { \alpha } D ^ { \alpha / 2 } \mathrm { J } _ { \alpha } ( D ^ { - 1 / 2 } \Sigma D ^ { - 1 / 2 } ) D ^ { \alpha / 2 } +$$ + +where $\mathrm { J } _ { \alpha }$ is applied entrywise. + +For example, $J _ { \alpha }$ and $\mathrm { J } _ { \alpha }$ for the first few integral $\alpha$ are + +$$ +\begin{array} { l } { { J _ { 0 } ( \theta ) = \displaystyle \frac { \pi - \theta } { \pi } } } \\ { { J _ { 1 } ( \theta ) = \displaystyle \frac { \sin \theta + ( \pi - \theta ) \cos \theta } { \pi } } } \\ { { J _ { 2 } ( \theta ) = \displaystyle \frac { 3 \sin \theta \cos \theta + ( \pi - \theta ) ( 1 + 2 \cos ^ { 2 } \theta ) } { 3 \pi } } } \\ { { J _ { 0 } ( c ) = \displaystyle \frac { \pi - \mathrm { a r c c o s c } } { \pi } } } \\ { { J _ { 1 } ( c ) = \displaystyle \frac { \sqrt { 1 - c ^ { 2 } + ( \pi - \mathrm { a r c c o s c } ) c } } { \pi } } } \\ { { { \bf J } _ { 2 } ( c ) = \displaystyle \frac { 3 c \sqrt { 1 - c ^ { 2 } + ( \pi - \mathrm { a r c c o s c } ) ( 1 + 2 c ^ { 2 } ) } } { 3 \pi } } } \end{array} +$$ + +One can observe very easily that (Daniely et al., 2016; Yang & Schoenholz, 2017) + +Proposition E.11. For each $\alpha > - 1 / 2 , \mathrm { J } _ { \alpha } ( c )$ is an increasing and convex function on $c \in [ 0 , 1 ]$ , and is continous on $c \in [ 0 , 1 ]$ and smooth on $c \in ( 0 , 1 )$ . $\mathrm { J } _ { \alpha } ( 1 ) = 1$ , $\begin{array} { r } { \mathrm { J } _ { \alpha } ( 0 ) = \frac { 1 } { 2 \sqrt { \pi } } \frac { \Gamma ( \frac { \alpha } { 2 } + \frac { 1 } { 2 } ) ^ { 2 } } { \Gamma ( \alpha + \frac { 1 } { 2 } ) } } \end{array}$ , and ${ \mathrm J } _ { \alpha } ( - 1 ) = 0$ . + +Yang & Schoenholz (2017) also showed the following fixed point structure + +Theorem E.12. For $\alpha \in [ 1 / 2 , 1 )$ , $\mathrm { J } _ { \alpha } ( c ) = c$ has two solutions: an unstable solution at $I$ (”unstable” meaning $\mathrm { J } _ { \alpha } ^ { \prime } ( 1 ) > 1$ ) and a stable solution in $c ^ { * } \in ( 0 , 1 )$ (”stable” meaning $\mathrm { J } _ { \alpha } ^ { \prime } ( c ^ { * } ) < 1 \}$ ). + +The $\alpha$ -ReLUs satisfy very interesting relations amongst themselves. For example, + +Lemma E.13. Suppose $\alpha > 1$ . Then + +$$ +\begin{array} { l } { { J _ { \alpha } ( \theta ) = \cos \theta J _ { \alpha - 1 } ( \theta ) + ( \alpha - 1 ) ^ { 2 } ( 2 \alpha - 1 ) ^ { - 1 } ( 2 \alpha - 3 ) ^ { - 1 } \sin ^ { 2 } \theta J _ { \alpha - 2 } ( \theta ) } } \\ { { \mathrm J _ { \alpha } ( c ) = c \mathrm J _ { \alpha - 1 } ( c ) + ( \alpha - 1 ) ^ { 2 } ( 2 \alpha - 1 ) ^ { - 1 } ( 2 \alpha - 3 ) ^ { - 1 } ( 1 - c ^ { 2 } ) \mathrm J _ { \alpha - 2 } ( c ) } } \end{array} +$$ + +In additional, surprisingly, one can use differentiation to go from $\alpha$ to $\alpha + 1$ and from $\alpha$ to $\alpha - 1$ ! Proposition E.14 (Yang & Schoenholz (2017)). Suppose $\alpha > 1 / 2$ . Then + +$$ +\begin{array} { r l } & { J _ { \alpha } ^ { \prime } ( \theta ) = - \alpha ^ { 2 } ( 2 \alpha - 1 ) ^ { - 1 } J _ { \alpha - 1 } ( \theta ) \sin \theta } \\ & { \mathrm { J } _ { \alpha } ^ { \prime } ( c ) = \alpha ^ { 2 } ( 2 \alpha - 1 ) ^ { - 1 } \mathrm { J } _ { \alpha - 1 } ( c ) } \end{array} +$$ + +so that + +$$ +\operatorname { J } _ { \alpha - n } ( c ) = { \Bigg [ } \prod _ { \beta = \alpha - n + 1 } ^ { \alpha } \beta ^ { - 2 } ( 2 \beta - 1 ) { \Bigg ] } ( \partial / \partial c ) ^ { n } \mathrm { J } _ { \alpha } ( c ) +$$ + +We have the following from Cho & Saul (2009) + +Proposition E.15 (Cho & Saul (2009)). For all $\alpha \geq 0$ and integer $n \geq 1$ + +$$ +\begin{array} { l } { \displaystyle \mathrm { J } _ { n + \alpha } ( c ) = \frac { \mathrm { c } _ { \alpha } } { \mathrm { c } _ { n + \alpha } } ( 1 - c ^ { 2 } ) ^ { n + \alpha + \frac { 1 } { 2 } } ( \partial / \partial c ) ^ { n } \mathrm { ( J } _ { \alpha } ( c ) / ( 1 - c ^ { 2 } ) ^ { \alpha + \frac { 1 } { 2 } } ) } \\ { \displaystyle \qquad = \left[ \prod _ { \beta = \alpha } ^ { \alpha + n - 1 } ( 2 \beta + 1 ) \right] ^ { - 1 } ( 1 - c ^ { 2 } ) ^ { n + \alpha + \frac { 1 } { 2 } } ( \partial / \partial c ) ^ { n } \mathrm { ( J } _ { \alpha } ( c ) / ( 1 - c ^ { 2 } ) ^ { \alpha + \frac { 1 } { 2 } } ) } \end{array} +$$ + +This implies in particular that we can obtain $\mathrm { J } _ { \alpha } ^ { \prime }$ from $\mathrm { J } _ { \alpha }$ and $\mathrm { J } _ { \alpha + 1 }$ . + +Proposition E.16. For all $\alpha \geq 0$ , + +$$ +\mathrm { J } _ { \alpha } ^ { \prime } ( c ) = ( 2 \alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( \mathrm { J } _ { 1 + \alpha } ( c ) - c \mathrm { J } _ { \alpha } ( c ) ) ) +$$ + +Proof. + +$$ +{ \begin{array} { r l } & { \operatorname { J } _ { 1 + \alpha } ( c ) = { \frac { c _ { \alpha } } { c _ { 1 + \alpha } } } ( 1 - c ^ { 2 } ) ^ { 1 + \alpha + { \frac { 1 } { 2 } } } ( \partial / \partial c ) ( \operatorname { J } _ { \alpha } ( c ) / ( 1 - c ^ { 2 } ) ^ { \alpha + { \frac { 1 } { 2 } } } ) } \\ & { \qquad = ( 2 \alpha + 1 ) ^ { - 1 } ( 1 - c ^ { 2 } ) ^ { \alpha + 3 / 2 } ( \operatorname { J } _ { \alpha } ^ { \prime } ( c ) / ( 1 - c ^ { 2 } ) ^ { \alpha + 1 / 2 } + 2 c ( \alpha + 1 / 2 ) \operatorname { J } _ { \alpha } ( c ) / ( 1 - c ^ { 2 } ) ^ { \alpha + 3 / 2 } ) } \\ & { \qquad = ( 2 \alpha + 1 ) ^ { - 1 } \operatorname { J } _ { \alpha } ^ { \prime } ( c ) ( 1 - c ^ { 2 } ) + c \operatorname { J } _ { \alpha } ( c ) } \\ & { \qquad \operatorname { J } _ { \alpha } ^ { \prime } ( c ) = ( 2 \alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( \operatorname { J } _ { 1 + \alpha } ( c ) - c \operatorname { J } _ { \alpha } ( c ) ) } \end{array} } +$$ + +Note that we can also obtain this via Lemma E.13 and Proposition E.14. + +# E.1.2 POSITIVE-HOMOGENEOUS FUNCTIONS IN 1 DIMENSION + +Suppose for some $\alpha \in \mathbb { R } , \phi : \mathbb { R } \to \mathbb { R }$ is degree $\alpha$ positive-homogeneous, i.e. $\phi ( r x ) = r ^ { \alpha } \phi ( x )$ for any $x \in R , r > 0$ . The following simple lemma says that we can always express $\phi$ as linear combination of powers of $\alpha$ -ReLUs. + +Proposition E.17. Any degree α positive-homogeneous function $\phi : \mathbb { R } \mathbb { R }$ with $\phi ( 0 ) = 0$ can be written as $x \mapsto a \rho _ { \alpha } ( x ) - b \rho _ { \alpha } ( - x )$ . + +Proof. Take $a = \phi ( 1 )$ and $b = \phi ( - 1 )$ . Then positive-homogeneity determines the value of $\phi$ on $\mathbb { R } \backslash \{ 0 \}$ and it coincides with $x \mapsto a \rho _ { \alpha } ( x ) - b \rho _ { \alpha } ( - x )$ . □ + +As a result we can express the $\mathrm { V }$ and W transforms of any positive-homogeneous function in terms of those of $\alpha$ -ReLUs. + +Proposition E.18. Suppose $\phi : \mathbb { R } \mathbb { R }$ is degree α positive-homogeneous. By Proposition E.17, $\phi$ restricted to $\mathbb { R } \backslash \{ 0 \}$ can be written as $x \mapsto a \rho _ { \alpha } ( x ) - b \rho _ { \alpha } ( - x ) .$ for some a and $b$ . Then for any PSD $2 \times 2$ matrix $M$ , + +$$ +\begin{array} { l } { { \mathrm { V } _ { \phi } ( M ) _ { 1 1 } = ( a ^ { 2 } + b ^ { 2 } ) \mathrm { V } _ { \rho _ { \alpha } } ( M ) _ { 1 1 } = \mathrm { c } _ { \alpha } ( a ^ { 2 } + b ^ { 2 } ) M _ { 1 1 } ^ { \alpha } } } \\ { { \mathrm { V } _ { \phi } ( M ) _ { 2 2 } = ( a ^ { 2 } + b ^ { 2 } ) \mathrm { V } _ { \rho _ { \alpha } } ( M ) _ { 2 2 } = \mathrm { c } _ { \alpha } ( a ^ { 2 } + b ^ { 2 } ) M _ { 2 2 } ^ { \alpha } } } \\ { { \mathrm { } / _ { \phi } ( M ) _ { 1 2 } = \mathrm { V } _ { \phi } ( M ) _ { 2 1 } = ( a ^ { 2 } + b ^ { 2 } ) \mathrm { V } _ { \rho _ { \alpha } } ( M ) _ { 1 2 } - 2 a b \mathrm { V } _ { \rho _ { \alpha } } ( M ^ { \prime } ) _ { 1 2 } } } \\ { { = \mathrm { c } _ { \alpha } ( M _ { 1 1 } M _ { 2 2 } ) ^ { \alpha / 2 } ( ( a ^ { 2 } + b ^ { 2 } ) ] _ { \alpha } ( M _ { 1 2 } / \sqrt { M _ { 1 1 } M _ { 2 2 } } ) - 2 a b \mathrm { J } _ { \alpha } ( - M _ { 1 2 } / \sqrt { M _ { 1 1 } M _ { 2 2 } } ) } } \\ { { \mathrm { } } } \\ { \mathrm { } \nu h e r e \ M ^ { \prime } : = \left( { 1 0 \ \begin{array} { l l } { { \ } & { { 0 } } \\ { 0 \ } & { { - 1 } \end{array} } } M \left( { 0 } & { - 1 } \right) . } } \end{\right)array} \end{array} +$$ + +Proof. We directly compute, using the expansion of $\phi$ into $\rho _ { \alpha }$ s: + +$$ +\begin{array} { r l } & { \mathrm { V } _ { \phi } ( M ) _ { 1 1 } = \mathbb { E } [ \phi ( x ) ^ { 2 } : x \sim \mathcal { N } ( 0 , M _ { 1 1 } ) ] } \\ & { \qquad = \mathbb { E } [ a ^ { 2 } \rho _ { \alpha } ( x ) ^ { 2 } + b ^ { 2 } \rho _ { \alpha } ( - x ) ^ { 2 } + 2 a b \rho _ { \alpha } ( x ) \rho _ { \alpha } ( - x ) ] } \\ & { \qquad = a ^ { 2 } \mathbb { E } [ \rho _ { \alpha } ( x ) ^ { 2 } ] + b ^ { 2 } \mathbb { E } [ \rho _ { \alpha } ( - x ) ^ { 2 } ] } \\ & { \qquad = ( a ^ { 2 } + b ^ { 2 } ) \mathbb { E } [ \rho _ { \alpha } ( x ) ^ { 2 } : x \sim \mathcal { N } ( 0 , M _ { 1 1 } ) ] } \\ & { \qquad = \mathsf { c } _ { \alpha } ( a ^ { 2 } + b ^ { 2 } ) M _ { 1 1 } ^ { \alpha } } \end{array} +$$ + +where in Eq. (28) we used negation symmetry of centered Gaussians. The case of $\mathrm { V } _ { \phi } ( M ) _ { 2 2 }$ is similar. + +$$ +\begin{array} { r l } & { \mathrm { V } _ { \phi } ( M ) _ { 1 2 } = \mathbb { E } [ \phi ( x ) \phi ( y ) : ( x , y ) \sim \mathcal { N } ( 0 , M ) ] } \\ & { \qquad = \mathbb { E } [ a ^ { 2 } \rho _ { \alpha } ( x ) \rho _ { \alpha } ( y ) + b ^ { 2 } \rho _ { \alpha } ( - x ) \rho _ { \alpha } ( - y ) - a b \rho _ { \alpha } ( x ) \rho _ { \alpha } ( - y ) - a b \rho _ { \alpha } ( - x ) \rho _ { \alpha } ( y ) ] } \\ & { \qquad = ( a ^ { 2 } + b ^ { 2 } ) \mathrm { V } _ { \rho _ { \alpha } } ( M ) _ { 1 2 } - 2 a b \mathrm { V } _ { \rho _ { \alpha } } ( M ^ { \prime } ) _ { 1 2 } } \\ & { \qquad = \mathrm { c } _ { \alpha } ( M _ { 1 1 } M _ { 2 2 } ) ^ { \alpha / 2 } ( ( a ^ { 2 } + b ^ { 2 } ) \mathrm { J } _ { \alpha } ( M _ { 1 2 } / \sqrt { M _ { 1 1 } M _ { 2 2 } } ) - 2 a b \mathrm { J } _ { \alpha } ( - M _ { 1 2 } / \sqrt { M _ { 1 1 } M _ { 2 2 } } ) ) } \end{array} +$$ + +where in the last equation we have applied Proposition E.10. + +This then easily generalizes to PSD matrices of arbitrary dimension: + +Corollary E.19. Suppose $\phi : \mathbb { R } \mathbb { R }$ is degree $\alpha$ positive-homogeneous. By Proposition E.17, $\phi$ restricted to $\mathbb { R } \backslash \{ 0 \}$ can be written as $x \mapsto a \rho _ { \alpha } ( x ) - b \rho _ { \alpha } ( - x )$ for some a and $b$ . Let $\Sigma \in S _ { B }$ . Then + +$$ +\mathrm { V } _ { \phi } ( \Sigma ) = \mathsf { c } _ { \alpha } D ^ { \alpha / 2 } \mathrm { J } _ { \phi } ( D ^ { - 1 / 2 } \Sigma D ^ { - 1 / 2 } ) D ^ { \alpha / 2 } +$$ + +where $\mathrm { J } _ { \phi }$ is defined below and is applied entrywise. Explicitly, this means that for all $i ,$ + +$$ +\begin{array} { r l } & { \mathrm { V } _ { \phi } ( \Sigma ) _ { i i } = \mathsf { c } _ { \alpha } \Sigma _ { i i } ^ { \alpha } \mathrm { J } _ { \phi } ( 1 ) } \\ & { \mathrm { V } _ { \phi } ( \Sigma ) _ { i j } = \mathsf { c } _ { \alpha } \mathrm { J } _ { \phi } ( \Sigma _ { i j } / \sqrt { \Sigma _ { i i } \Sigma _ { j j } } ) \Sigma _ { i i } ^ { \alpha / 2 } \Sigma _ { j j } ^ { \alpha / 2 } } \end{array} +$$ + +Definition E.20. Suppose $\phi : \mathbb { R } \mathbb { R }$ is degree $\alpha$ positive-homogeneous. By Proposition E.17, $\phi$ restricted to $\mathbb { R } \ \backslash \ \{ 0 \}$ can be written as $x \mapsto a \rho _ { \alpha } ( x ) - b \rho _ { \alpha } ( - x )$ for some $a$ and $b$ . Define $\mathrm { J } _ { \phi } ( c ) : = ( a ^ { 2 } + b ^ { 2 } ) \mathrm { \bar { J } } _ { \alpha } ( \bar { c } ) - 2 a b \mathrm { J } _ { \alpha } ( - c )$ . + +Let us immediately make the following easy but important observations. + +$$ +\begin{array} { r } { \mathrm { J } _ { \phi } ( 0 ) = ( a ^ { 2 } + b ^ { 2 } - 2 a b ) \mathrm { J } _ { \alpha } ( 0 ) = ( a - b ) ^ { 2 } \frac { 1 } { 2 \sqrt { \pi } } \frac { \Gamma ( \frac { \alpha } { 2 } + \frac { 1 } { 2 } ) ^ { 2 } } { \Gamma ( \alpha + \frac { 1 } { 2 } ) } , } \end{array} +$$ + +Proof. Use the fact that $\mathrm { J } _ { \alpha } ( - 1 ) = 0$ and $\begin{array} { r } { \mathrm { J } _ { \alpha } ( 0 ) = \frac { 1 } { 2 \sqrt { \pi } } \frac { \Gamma ( \frac { \alpha } { 2 } + \frac { 1 } { 2 } ) ^ { 2 } } { \Gamma ( \alpha + \frac { 1 } { 2 } ) } } \end{array}$ by Proposition E.11. + +As a sanity check, we can easily compute that $\mathrm { J _ { i d } } ( c ) = 2 \mathrm { J _ { 1 } } ( c ) - 2 \mathrm { J _ { 1 } } ( - c ) = 2 c$ because $\operatorname { i d } ( x ) =$ $\mathrm { r e l u } ( x ) { - } \mathrm { r e l u } ( - x )$ . By Corollary E.19 and $\mathsf { c } _ { 1 } = \textstyle { \frac { 1 } { 2 } }$ , this recovers the obvious fact that $\mathrm { V } _ { \mathrm { ( i d ) } } ( \Sigma ) = \Sigma$ . + +We record the partial derivatives of $\mathrm { V } _ { \phi }$ + +Proposition E.22. Let $\phi$ be positive homogeneous of degree $\alpha$ . Then for all $i$ with $\Sigma _ { i i } \neq 0$ , + +$$ +\frac { \partial \mathrm { V } _ { \phi } ( \Sigma ) _ { i i } } { \partial \Sigma _ { i i } } = \mathsf { c } _ { \alpha } \alpha \Sigma _ { i i } ^ { \alpha - 1 } \mathrm { J } _ { \phi } ( 1 ) +$$ + +For all $i \neq j$ with $\Sigma _ { i i } , \Sigma _ { j j } \neq 0$ , + +$$ +\begin{array} { r l } & { \frac { \partial \nabla _ { \phi } \left( \Sigma \right) _ { i i } } { \partial \Sigma _ { i j } } = 0 } \\ & { \frac { \partial \nabla _ { \phi } \left( \Sigma \right) _ { i j } } { \partial \Sigma _ { i j } } = \mathsf { c } _ { \alpha } \Sigma _ { i i } ^ { ( \alpha - 1 ) / 2 } \Sigma _ { j j } ^ { ( \alpha - 1 ) / 2 } \mathbf { J } _ { \phi } ^ { \prime } ( { c } _ { i j } ) } \\ & { \frac { \partial \nabla _ { \phi } \left( \Sigma \right) _ { i j } } { \partial \Sigma _ { i i } } = \frac { 1 } { 2 } \mathsf { c } _ { \alpha } \Sigma _ { i i } ^ { \frac { 1 } { 2 } ( \alpha - 2 ) } \Sigma _ { j j } ^ { \frac { 1 } { 2 } \alpha } ( \alpha \mathbf { J } _ { \phi } ( { c } _ { i j } ) - { c } _ { i j } \mathbf { J } _ { \phi } ^ { \prime } ( { c } _ { i j } ) ) } \end{array} +$$ + +where $c _ { i j } = \Sigma _ { i j } / \sqrt { \Sigma _ { i i } \Sigma _ { j j } }$ and $\mathrm { J } _ { \phi } ^ { \prime }$ denotes its derivative. + +Proposition E.23. If $\phi ( c ) = a \rho _ { \alpha } ( c ) - b \rho _ { \alpha } ( - c )$ with $\alpha > 1 / 2$ on $\mathbb { R } \backslash \{ 0 \}$ , then + +$$ +\begin{array} { l } { { { \bf J } _ { \phi } ^ { \prime } ( c ) = ( 2 \alpha - 1 ) ^ { - 1 } { \bf J } _ { \phi ^ { \prime } } ( c ) } } \\ { { \mathrm { ~ \alpha ~ } = ( 2 \alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( ( a ^ { 2 } + b ^ { 2 } ) { \bf J } _ { \alpha + 1 } ( c ) + 2 a b { \bf J } _ { \alpha + 1 } ( - c ) ) - c { \bf J } _ { \phi } ( c ) } } \\ { { { \bf J } _ { \phi } ^ { \prime } ( 0 ) = ( a + b ) ^ { 2 } \displaystyle \frac { 1 } { \sqrt { \pi } } \displaystyle \frac { \Gamma ( \frac { \alpha } { 2 } + 1 ) ^ { 2 } } { \Gamma ( \alpha + \frac { 1 } { 2 } ) } } } \end{array} +$$ + +Proof. We have $\phi ^ { \prime } ( c ) = a \alpha \rho _ { \alpha - 1 } ( c ) + b \alpha \rho _ { \alpha - 1 } ( - c )$ . On the other hand, $\mathrm { J } _ { \phi } ^ { \prime } = ( a ^ { 2 } + b ^ { 2 } ) \mathrm { J } _ { \alpha } ^ { \prime } ( c ) +$ $2 a b \mathbf { J } _ { \alpha } ^ { \prime } ( - c )$ which by Proposition E.14 is $\alpha ^ { 2 } ( 2 \alpha - 1 ) ^ { - 1 } ( ( a ^ { 2 } + b ^ { 2 } ) \mathrm { J } _ { \alpha - 1 } ( \dot { c } ) + 2 a b \mathrm { J } _ { \alpha - 1 } ( - c ) ) =$ $( 2 \alpha - \mathrm { i } ) ^ { - 1 } \mathrm { J } _ { \phi ^ { \prime } } ( c )$ . This proves the first equation. + +With Proposition E.16, + +$$ +\begin{array} { l } { \mathrm { J } _ { \phi } ^ { \prime } ( c ) = ( a ^ { 2 } + b ^ { 2 } ) \mathrm { J } _ { \alpha } ^ { \prime } ( c ) + 2 a b \mathrm { J } _ { \alpha } ^ { \prime } ( - c ) } \\ { = ( a ^ { 2 } + b ^ { 2 } ) ( 2 \alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( \mathrm { J } _ { \alpha + 1 } ( c ) - c \mathrm { J } _ { \alpha } ( c ) ) } \\ { \qquad + 2 a b ( 2 \alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( \mathrm { J } _ { \alpha + 1 } ( - c ) + c \mathrm { J } _ { \alpha } ( - c ) ) } \\ { \qquad = ( 2 \alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( ( a ^ { 2 } + b ^ { 2 } ) \mathrm { J } _ { \alpha + 1 } ( c ) + 2 a b \mathrm { J } _ { \alpha + 1 } ( - c ) ) - c \mathrm { J } _ { \phi } ( c ) } \end{array} +$$ + +This gives the second equation. + +Expanding $\mathrm { J } _ { \alpha + 1 } ( 0 )$ With Proposition E.11, we get + +$$ +\begin{array} { l } { \displaystyle \mathrm { J } _ { \phi } ^ { \prime } ( 0 ) = ( 2 \alpha + 1 ) ( ( a ^ { 2 } + b ^ { 2 } ) \mathrm { J } _ { \alpha + 1 } ( 0 ) + 2 a b \mathrm { J } _ { \alpha + 1 } ( 0 ) ) } \\ { = ( 2 \alpha + 1 ) ( a + b ) ^ { 2 } \displaystyle \frac { 1 } { 2 \sqrt { \pi } } \frac { \Gamma ( \frac { \alpha } { 2 } + 1 ) ^ { 2 } } { \Gamma ( \alpha + \frac { 3 } { 2 } ) } } \end{array} +$$ + +Unpacking the definition of $\Gamma ( \alpha + \frac { 3 } { 2 } )$ then yields the third equation. + +In general, we can factor diagonal matrices out of $\mathrm { V } _ { \phi }$ . + +Proposition E.24. For any $\Sigma \in { \cal S } _ { B }$ , $D$ any diagonal matrix, and $\phi$ positive-homogeneous with degree $\alpha$ , + +$$ +\mathrm { V } _ { \phi } ( D \Sigma D ) = D ^ { \alpha } \mathrm { V } _ { \phi } ( \Sigma ) D ^ { \alpha } +$$ + +Proof. For any $i$ + +$$ +\begin{array} { r l } & { { \mathbf V } _ { \phi } ( D \Sigma D ) _ { i i } = \mathbb E [ \phi ( x ) ^ { 2 } : x \sim { \mathcal N } ( 0 , D _ { i i } ^ { 2 } \Sigma _ { i i } ) ] } \\ & { \qquad = \mathbb E [ \phi ( D _ { i i } x ) ^ { 2 } : x \sim { \mathcal N } ( 0 , \Sigma _ { i i } ) ] } \\ & { \qquad = \mathbb E [ D _ { i i } ^ { 2 \alpha } \phi ( x ) ^ { 2 } : x \sim { \mathcal N } ( 0 , \Sigma _ { i i } ) ] } \\ & { \qquad = D _ { i i } ^ { 2 \alpha } { \mathbf V } _ { \phi } ( \Sigma ) _ { i i } } \end{array} +$$ + +For any $i \neq j$ + +$$ +\begin{array} { r l } & { { \mathrm { V } _ { \phi } } ( D \Sigma D ) _ { i j } = \mathbb { E } [ \phi ( x ) \phi ( y ) : ( x , y ) \sim \mathcal { N } ( 0 , \left( \begin{array} { c c } { D _ { i i } ^ { 2 } \Sigma _ { i i } } & { D _ { i i } D _ { j j } \Sigma _ { i j } } \\ { D _ { i i } D _ { j j } \Sigma _ { j i } } & { D _ { j j } ^ { 2 } \Sigma _ { j j } } \end{array} \right) ) } \\ & { \quad \quad \quad \quad = \mathbb { E } [ \phi ( D _ { i i } x ) \phi ( D _ { j j } y ) : ( x , y ) \sim \mathcal { N } ( 0 , \left( \begin{array} { c c } { \Sigma _ { i i } } & { \Sigma _ { i j } } \\ { \Sigma _ { j i } } & { \Sigma _ { j j } } \end{array} \right) ) ] } \\ & { \quad \quad \quad = D _ { i i } ^ { \alpha } D _ { j j } ^ { \alpha } \mathbb { E } [ \phi ( x ) \phi ( y ) : ( x , y ) \sim \mathcal { N } ( 0 , \left( \begin{array} { c c } { \Sigma _ { i i } } & { \Sigma _ { i j } } \\ { \Sigma _ { j i } } & { \Sigma _ { j j } } \end{array} \right) ) ] } \\ & { \quad \quad \quad = D _ { i i } ^ { \alpha } D _ { j j } ^ { \alpha } \nabla _ { \phi } ( \Sigma ) _ { i j } } \end{array} +$$ + +# E.2 FOURIER METHOD + +The Laplace method crucially used the fact that we can pull out the norm factor $\| G h \|$ out of $\phi$ , so that we can apply Schwinger parametrization. For general $\phi$ this is not possible, but we can apply + +some + +$$ +\begin{array}{c} \begin{array} { r l } & { \mathrm { ( d e r a t ) ~ t o ~ E q r i e n ) } } \\ & { \mathrm { ( b y ~ t o ~ E q r i e n ) } } \\ & { \mathrm { ( b y ~ t o ~ E q r i e n ) } } \\ & { = \mathrm { ( d e r a t ) } } \\ & { \mathrm { ( b y ~ t o ~ E q r i e n ) } } \\ & { = \mathrm { ( d e r a t ) } } \\ & { = \mathrm { ( d e r a t ) } } \\ & { \mathrm { ( b y ~ t o ~ E q r i e n ) } } \\ & { = \mathrm { ( d e r a t ) } } \\ & { \mathrm { ( b y ~ t o ~ E q r i e n ) } } \\ & { = \mathrm { ( d e r a t ) } } \\ & { \mathrm { ( c r a t e n ) } } \\ & { = \mathrm { ( d e r a t ) } } \\ & { \mathrm { ( c r a t e n ) } } \end{array} \quad \begin{array} { r l } { \mathrm { ( d ) } } & { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \end{array} \\ & { \quad } \\ & { \quad + \mathrm { ( d ) } } \\ & { \quad { ( d ) } } \\ & { = \mathrm { ( d e r a t ) } } \\ & { \quad + \mathrm { ( d ) } } \\ & { \quad { ( d ) } } \end{array} \\ & { \quad = \begin{array} { r l } { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \end{array} } \\ & { \quad = \begin{array} { r l } { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \end{array} } \\ & \quad = \begin{array} { r l } { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \\ { \mathrm { ( d ) } } & { \mathrm { ( d ) } } \end{array} +$$ + +Here all steps are legal except for possibly Eq. (29), Eq. (30), and Eq. (31). In Eq. (29), we “Fourier expanded” the delta function — this could possibly be justified as a principal value integral, but this cannot work in combination with Eq. (30) where we have switched the order of integration, integrating over $h$ first. Fubini’s theorem would not apply here because $e ^ { i \lambda ( r ^ { 2 } - \| G h \| ^ { 2 } / B ) }$ has norm 1 and is not integrable. Finally, in Eq. (31), we need to extend the definition of Gaussian to complex covariance matrices, via complex Gaussian integration. This last point is no problem, and we will just define + +$$ +\underset { h \sim \mathcal { N } ( 0 , \Sigma ) } { \mathbb { E } } f ( h ) : = \operatorname* { d e t } ( 2 \pi \Sigma ) ^ { - 1 / 2 } \int _ { \mathbb { R } ^ { B } } \mathrm { d } h f ( h ) e ^ { - \frac { 1 } { 2 } h ^ { T } \Sigma ^ { - 1 } h } +$$ + +for general complex $\Sigma$ , whenever $\Sigma$ is nonsingular and this integral is exists, and + +$$ +\underset { h \sim \mathcal { N } ( 0 , G \Sigma G ) } { \mathbb { E } } f ( h ) : = \operatorname* { d e t } ( 2 \pi ( \mathfrak { e } ^ { T } \Sigma \mathfrak { e } ) ) ^ { - 1 / 2 } \int _ { \mathbb { R } ^ { B - 1 } } \mathrm { d } z f ( \mathfrak { e } z ) e ^ { - \frac { 1 } { 2 } z ^ { T } ( \mathfrak { e } ^ { T } \Sigma \mathfrak { e } ) ^ { - 1 } z } +$$ + +if $\Phi ^ { T } \Sigma \Phi$ is nonsingular (as in the case above). The definition of $\mathrm { V } _ { \phi }$ stays the same given the above extension of the definition of Gaussian expectation. + +Nevertheless, the above derivation is not correct mathematically due to the other points. However, its end result can be justified rigorously by carefully expressing the delta function as the limit of mollifiers. + +Theorem E.25. Let $\Sigma$ be a positive-definite $D \times D$ matrix. Let $\phi : \mathbb { R } \mathbb { R }$ . Suppose for any ${ \mathsf b } > { \mathsf a } > 0$ , + +1. $\begin{array} { r } { \int _ { \mathsf { a } } ^ { \mathsf { b } } \mathrm { d } ( r ^ { 2 } ) | \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) | } \end{array}$ exists and is finite for any $z _ { a } , z _ { b } \in \mathbb { R }$ (i.e. $\phi ( z _ { a } / r ) \phi ( z _ { b } / r )$ is locally integrable in $r$ ). + +2. there exists an $\epsilon > 0$ such that any $\gamma \in [ - \epsilon , \epsilon ]$ and for each $a , b \in [ D ]$ , + +$$ +\int _ { \| z \| ^ { 2 } \in [ \mathbf { a } , \mathbf { b } ] } \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } | \phi ( z _ { a } / \sqrt { \| z \| ^ { 2 } + \gamma } ) \phi ( z _ { b } / \sqrt { \| z \| ^ { 2 } + \gamma } ) | +$$ + +exists and is uniformly bounded by some number possibly depending on a and b. + +3. for each $\begin{array} { r } { a , b \in [ D ] , \int _ { \mathbb { R } ^ { D } } \mathrm { d } z \int _ { \mathsf { a } } ^ { \mathsf { b } } \mathrm { d } ( r ^ { 2 } ) e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } | \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) | } \end{array}$ exists and is finite. + +4. $\begin{array} { r } { \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \left| \operatorname* { d e t } ( I + 2 i \lambda \Sigma ) ^ { - 1 / 2 } \mathrm { V } _ { \phi } ( ( \Sigma ^ { - 1 } + 2 i \lambda I ) ^ { - 1 } r ^ { - 2 } ) \right| } \end{array}$ exists and is finite for each $r > 0$ + +$I f e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \phi ( z / \| z \| ) ^ { \otimes 2 }$ is integrable over $z \in \mathbb { R } ^ { D }$ , then + +$$ +\begin{array} { l } { { \displaystyle ( 2 \pi ) ^ { - D / 2 } \operatorname* { d e t } \Sigma ^ { - 1 / 2 } \int \mathrm { d } z e ^ { - \frac 1 2 z ^ { T } \Sigma ^ { - 1 } z } \phi ( z / \| z \| ) ^ { \otimes 2 } } } \\ { { \displaystyle = \frac 1 { 2 \pi } \operatorname* { l i m } _ { a \searrow 0 , \flat \ne \infty } \int _ { a } ^ { \flat } \mathrm { d } s \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda e ^ { i \lambda s } \operatorname* { d e t } ( I + 2 i \lambda \Sigma ) ^ { - 1 / 2 } \nabla _ { \phi } \big ( ( \Sigma ^ { - 1 } + 2 i \lambda I ) ^ { - 1 } s ^ { - 1 } \big ) . } } \end{array} +$$ + +Similarly, + +$$ +\begin{array} { l } { { ( 2 \pi ) ^ { - D / 2 } \operatorname* { d e t } \Sigma ^ { - 1 / 2 } \displaystyle \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \phi ( \sqrt { D } z / \| z \| ) ^ { \otimes 2 } } } \\ { { = \displaystyle \frac { 1 } { 2 \pi } \operatorname* { l i m } _ { a \searrow 0 , \flat , \ne \infty } \int _ { \mathrm { a } } ^ { \flat } \mathrm { d } s \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda e ^ { i \lambda s } \operatorname* { d e t } ( I + 2 i \lambda \Sigma ) ^ { - 1 / 2 } \nabla _ { \phi } ( ( \Sigma ^ { - 1 } + 2 i \lambda I / D ) ^ { - 1 } s ^ { - 1 } ) . } } \end{array} +$$ + +If $G$ is the mean-centering projection matrix and $\Sigma ^ { G } = G \Sigma G$ , then by the same reasoning as in Thm E.5, + +$$ +\begin{array} { l } { { \displaystyle \gamma _ { B _ { \phi } } ( \Sigma ) = ( 2 \pi ) ^ { - D / 2 } \operatorname * { d e t } \Sigma ^ { - 1 / 2 } \int \operatorname * { d } z e ^ { - \frac 1 2 z ^ { T } \Sigma ^ { - 1 } z } \phi ( \sqrt { D } G z / \| G z \| ) ^ { \otimes 2 } } } \\ { { \displaystyle \qquad = \frac 1 { 2 \pi } \operatorname* { l i m } _ { a \searrow 0 , 6 , 7 \infty } \int _ { \mathbb { a } } ^ { \mathbb { b } } \mathrm { d } s \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda e ^ { i \lambda s } \operatorname * { d e t } ( I + 2 i \lambda \Sigma ^ { G } / D ) ^ { - 1 / 2 } { \mathrm { V } } _ { \phi } ( \Sigma ^ { G } ( I + 2 i \lambda \Sigma ^ { G } / D ) ^ { - 1 } s ^ { - 1 / 2 } ) , } } \end{array} +$$ + +Note that assumption (1) is satisfied if $\phi$ is continuous; assumption (4) is satisfied if $D \geq 3$ and for all $\Pi$ , $\| \mathrm { V } _ { \phi } ( \bar { \Pi } ) \| \leq \| \Pi \| ^ { \alpha }$ for some $\alpha \geq 0$ ; this latter condition, as well as assumptions (2) and (3), will be satisfied if $\phi$ is polynomially bounded. Thus the common coordinatewise nonlinearities ReLU, identity, tanh, etc all satisfy these assumptions. + +Warning: in general, we cannot swap the order of integration as $\int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \int _ { - \infty } ^ { \infty }$ ds. For example, if $\phi = \mathrm { i d }$ , then + +$$ +\begin{array} { r l } & { \quad ( 2 \pi ) ^ { - D / 2 } \operatorname* { d e t } \Sigma ^ { - 1 / 2 } \displaystyle \int \mathrm { d } z \ e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \phi ( z / \| z \| ) ^ { \otimes 2 } } \\ & { = \displaystyle \frac { 1 } { 2 \pi } \operatorname* { l i m } _ { \mathrm { a } \searrow 0 , \mathrm { b } \ne \infty } \int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { d } s \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda e ^ { i \lambda s } \operatorname* { d e t } ( I + 2 i \lambda \Sigma ) ^ { - 1 / 2 } ( \Sigma ^ { - 1 } + 2 i \lambda I ) ^ { - 1 } s ^ { - 1 } } \\ & { \ne \displaystyle \frac { 1 } { 2 \pi } \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \int _ { 0 } ^ { \infty } \mathrm { d } s e ^ { i \lambda s } \operatorname* { d e t } ( I + 2 i \lambda \Sigma ) ^ { - 1 / 2 } ( \Sigma ^ { - 1 } + 2 i \lambda I ) ^ { - 1 } s ^ { - 1 } } \end{array} +$$ + +because the $s$ -integral in the latter diverges (in a neighborhood of 0). + +Proof. We will only prove the first equation; the others follow similarly. + +By dominated convergence, + +$$ +\begin{array} { r l } & { \displaystyle \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \phi ( z / \| z \| ) ^ { \otimes 2 } } \\ & { = \displaystyle \operatorname* { l i m } _ { \mathfrak { a } \searrow 0 , \mathfrak { b } , \nearrow \infty } \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \phi ( z / \| z \| ) ^ { \otimes 2 } \mathbb { I } ( \| z \| ^ { 2 } \in [ \mathsf { a } , \mathfrak { b } ] ) . } \end{array} +$$ + +Let $\eta : \mathbb { R } \mathbb { R }$ be a nonnegative bump function (i.e. compactly supported and smooth) with support $[ - 1 , 1 ]$ and integral 1 such that $\eta ^ { \prime } ( 0 \bar { ) } = \eta ^ { \prime \prime } ( 0 ) = 0$ . Then its Fourier transform $\hat { \eta } ( t )$ decays like $\bar { \boldsymbol { O } } ( t ^ { - 2 } )$ . Furthermore, $\eta _ { \epsilon } ( x ) : = \epsilon ^ { - 1 } \eta ( x / \epsilon )$ is a mollifier, i.e. for all $f \in L ^ { 1 } ( \mathbb { R } )$ , $f * \eta _ { \epsilon } \to f$ in $L ^ { 1 }$ and pointwise almost everywhere. + +Now, we will show that + +$$ +\begin{array} { l } { { \displaystyle \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \mathbb { I } ( | | z | | ^ { 2 } \in [ \mathbf { a } , \mathbf { b } ] ) } } \\ { { \displaystyle \qquad = \operatorname* { l i m } _ { \epsilon \searrow 0 } \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \int _ { \mathbf { a } } ^ { \mathbf { b } } \mathrm { d } ( r ^ { 2 } ) \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \eta _ { \epsilon } ( | | z | | ^ { 2 } - r ^ { 2 } ) } } \end{array} +$$ + +by dominated convergence. Pointwise convergence is immediate, because + +$$ +\begin{array} { r l } { \displaystyle \int _ { a } ^ { \mathsf { b } } \mathrm { d } ( r ^ { 2 } ) \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \eta _ { \epsilon } ( r ^ { 2 } - \| z \| ^ { 2 } ) = \int _ { - \infty } ^ { \infty } \mathrm { d } ( r ^ { 2 } ) \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \mathbb { I } ( r ^ { 2 } \in [ \mathsf { a } , \mathsf { b } ] ) \eta _ { \epsilon } ( \| z \| ^ { 2 } - r ^ { 2 } ) } & { } \\ { = [ ( r ^ { 2 } \mapsto \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \mathbb { I } ( r ^ { 2 } \in [ \mathsf { a } , \mathsf { b } ] ) ) * \eta _ { \epsilon } ] ( \| z \| ^ { 2 } ) } & { } \\ { \to \phi ( z _ { a } / \| z \| ) \phi ( z _ { b } / \| z \| ) \mathbb { I } ( \| z \| ^ { 2 } \in [ \mathsf { a } , \mathsf { b } ] ) } & { } \end{array} +$$ + +as $\epsilon 0$ (where we used assumption (1) that $\phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \mathbb { I } ( r ^ { 2 } \in [ \mathsf { a } , \mathsf { b } ] ) \mathrm { i s } L ^ { 1 } )$ . + +Finally, we construct a dominating integrable function. Observe + +$$ +\begin{array} { r l } & { \quad \int \mathrm { d } s e ^ { - \frac { 1 } { 2 } s ^ { 2 } \tau ^ { 2 } s ^ { 3 } } \int _ { s } ^ { s } \mathrm { d } s ( \gamma ) \mathrm { d } s ( \omega _ { s } \gamma ) \mathrm { d } s ( | \gamma | \geq s _ { 1 } \gamma ) } \\ & { = \int \mathrm { d } s ( e ^ { - \frac { 1 } { 2 } s ^ { 2 } \tau ^ { 3 } } ) ^ { s } \int _ { s } ^ { s } \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( | \gamma | \geq - \gamma ^ { 2 } ) } \\ & { = \int \mathrm { d } s \mathrm { e } ^ { - \frac { 1 } { 2 } s ^ { 2 } \tau ^ { 3 } } \int _ { s } ^ { s } \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma - \gamma ) } \\ & { = \int \mathrm { d } s \mathrm { e } ^ { - \frac { 1 } { 2 } s ^ { 2 } \tau ^ { 3 } } \int _ { s } ^ { s } \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma - \gamma ) } \\ & { = \int \mathrm { d } s \mathrm { e } ^ { - \frac { 1 } { 2 } s ^ { 2 } \tau ^ { 3 } } \int _ { s } ^ { s } \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma - \gamma ) } \\ & { = \int \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) \mathrm { d } s ( \gamma ) } \\ & \qquad \mathrm { d } s \mathrm { e } ^ { - \frac { 1 } { 2 } s ^ { 2 } } \mathrm { e } ^ { \frac { 1 } { 2 } s ^ { 2 } } \end{array} +$$ + +For small enough $\epsilon$ then, this is integrable by assumption (2), and yields a dominating integrable function for our application of dominated convergence. + +In summary, we have just proven that + +$$ +\begin{array} { l } { { \displaystyle \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \phi ( z / \| z \| ) ^ { \otimes 2 } } } \\ { { \displaystyle = \operatorname* { l i m } _ { \mathrm { a \setminus { } 0 , b \setminus { \infty } } } \operatorname* { l i m } _ { \epsilon \searrow 0 } \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { d } ( r ^ { 2 } ) \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \eta _ { \epsilon } ( \| z \| ^ { 2 } - r ^ { 2 } ) } } \end{array} +$$ + +Now, + +$$ +\begin{array} { l } { { \displaystyle \int \mathrm { d } z e ^ { - \frac 1 2 z ^ { T } \Sigma ^ { - 1 } z } \int _ { 0 } ^ { \mathbf { b } } \mathrm { d } ( r ^ { 2 } ) \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \eta _ { \epsilon } ( \| z \| ^ { 2 } - r ^ { 2 } ) } } \\ { { \displaystyle = \int \mathrm { d } z e ^ { - \frac 1 2 z ^ { T } \Sigma ^ { - 1 } z } \int _ { \mathbf { a } } ^ { \mathbf { b } } \mathrm { d } ( r ^ { 2 } ) \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \frac 1 { 2 \pi } \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \hat { \eta } _ { \epsilon } ( \lambda ) e ^ { - i \lambda ( \| z \| ^ { 2 } - r ^ { 2 } ) } } } \\ { { \displaystyle = \frac 1 { 2 \pi } \int \mathrm { d } z e ^ { - \frac 1 2 z ^ { T } \Sigma ^ { - 1 } z } \int _ { \mathbf { a } } ^ { \mathbf { b } } \mathrm { d } ( r ^ { 2 } ) \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \hat { \eta } ( \epsilon \lambda ) e ^ { - i \lambda ( \| z \| ^ { 2 } - r ^ { 2 } ) } } } \end{array} +$$ + +Note that the absolute value of the integral is bounded above by + +$$ +\begin{array} { l } { { \displaystyle \frac 1 { 2 \pi } \int \mathrm { d } z e ^ { - \frac 1 2 z ^ { T } \Sigma ^ { - 1 } z } \int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { d } ( r ^ { 2 } ) | \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) | \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda | \hat { \eta } ( \epsilon \lambda ) e ^ { - i \lambda ( \| z \| ^ { 2 } - r ^ { 2 } ) } | } } \\ { { \displaystyle = \frac 1 { 2 \pi } \int \mathrm { d } z e ^ { - \frac 1 2 z ^ { T } \Sigma ^ { - 1 } z } \int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { d } ( r ^ { 2 } ) | \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) | \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda | \hat { \eta } ( \epsilon \lambda ) | } } \\ { { \displaystyle \le C \int \mathrm { d } z e ^ { - \frac 1 2 z ^ { T } \Sigma ^ { - 1 } z } \int _ { \mathrm { a } } ^ { \mathrm { b } } \mathrm { d } ( r ^ { 2 } ) | \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) | } } \end{array} +$$ + +for some $C$ , by our construction of $\eta$ that $\hat { \eta } ( t ) = O ( t ^ { - 2 } )$ for large $| t |$ . By assumption (3), this integral exists. Therefore we can apply the Fubini-Tonelli theorem and swap order of integration. + +$$ +\begin{array} { c } { { \displaystyle \frac 1 { 2 \pi } \int \mathrm { d } z e ^ { - \frac 1 2 z ^ { 2 } \sum ^ { - 1 } z } \int _ { a } ^ { \mathbf b } \mathrm { d } ( r ^ { 2 } ) \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \hat { \eta } ( c ) \ : e ^ { - i \lambda ( | z | ^ { 2 } - r ^ { 2 } ) } } } \\ { { \displaystyle = \frac 1 { 2 \pi } \int _ { a } ^ { \mathbf b } \mathrm { d } ( r ^ { 2 } ) \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \hat { \eta } ( c \lambda ) e ^ { - i \lambda ( | z | ^ { 2 } - r ^ { 2 } ) } \int \mathrm { d } z \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) e ^ { - \frac 1 2 z ^ { 2 } \tau \sum ^ { - 1 } z } } } \\ { { \displaystyle = \frac 1 { 2 \pi } \int _ { a } ^ { \mathbf b } \mathrm { d } ( r ^ { 2 } ) \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \hat { \eta } ( c ) \ : e ^ { i \lambda r ^ { 2 } } \int \mathrm { d } z \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) e ^ { - \frac 1 2 z ^ { 2 } \tau ( \mathbb S ^ { - 1 } + 2 i \lambda I ) z } } } \\ { { \displaystyle = \frac 1 { 2 \pi } \int _ { a } ^ { \mathbf b } \mathrm { d } ( r ^ { 2 } ) r ^ { D } \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \hat { \eta } ( c \lambda ) e ^ { i \lambda r ^ { 2 } } \int \mathrm { d } z \phi ( z _ { a } ) \phi ( z _ { a } ) \ : e ^ { - \frac 1 2 r ^ { 2 } z ^ { 2 } \tau ( \mathbb S ^ { - 1 } + 2 i \lambda I ) z } } } \\ { \displaystyle = \frac 1 { 2 \pi } \int _ { a } ^ { \mathbf b } \mathrm { d } ( r ^ { 2 } ) r ^ { 2 } \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \hat { \eta } ( c ) \ : e ^ { i \lambda r ^ { 2 } } \int \mathrm { d } z \phi ( z _ { a } ) \ : \phi ( z _ { b } ) e ^ { - \frac 1 2 r ^ { 2 } z ^ { 2 } \tau \left( \mathbb S ^ { - 1 } + 2 i \lambda I \right) z } } \\ \displaystyle = ( 2 \pi ) ^ { D / 2 - 1 } \mathrm { d } \mathbf e \mathrm { t } \ : \Sigma ^ \end{array} +$$ + +By assumption (4), + +$$ +\begin{array} { r l } & { \quad \displaystyle \int _ { \ b { \mathrm { \ b { a } } } } ^ { \ b } \mathrm { d } ( r ^ { 2 } ) \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \ \left| \hat { \eta } ( \epsilon \lambda ) e ^ { i \lambda r ^ { 2 } } \operatorname* { d e t } ( I + 2 i \lambda \Sigma ) ^ { - 1 / 2 } \nabla _ { \phi } ( ( \Sigma ^ { - 1 } + 2 i \lambda I ) ^ { - 1 } r ^ { - 2 } ) \right| } \\ & { \leq C \int _ { \ b { \mathrm { \ b { a } } } } ^ { \ b } \mathrm { d } ( r ^ { 2 } ) \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda \ \left| \operatorname* { d e t } ( I + 2 i \lambda \Sigma ) ^ { - 1 / 2 } \nabla _ { \phi } ( ( \Sigma ^ { - 1 } + 2 i \lambda I ) ^ { - 1 } r ^ { - 2 } ) \right| } \end{array} +$$ + +is integrable (where we used the fact that $\hat { \eta }$ is bounded — which is true for all $\eta \in C _ { 0 } ^ { \infty } ( { \mathbb { R } } ) )$ , so by dominated convergence (applied to $\epsilon \searrow 0$ ), + +$$ +\begin{array} { r l } & { \quad \displaystyle \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \phi ( z / \| z \| ) ^ { \otimes 2 } } \\ & { = \underset { \mathrm { \tiny ~ a \sim \sim \Theta , b , \sim \infty } } { \operatorname* { l i m } } \underset { \epsilon \searrow } { \operatorname* { l i m } } \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \int _ { \mathrm { a } } ^ { \mathbf { b } } \mathrm { d } ( r ^ { 2 } ) \phi ( z _ { a } / r ) \phi ( z _ { b } / r ) \eta _ { \epsilon } ( \| z \| ^ { 2 } - r ^ { 2 } ) } \\ & { = ( 2 \pi ) ^ { D / 2 - 1 } \operatorname* { d e t } \Sigma ^ { 1 / 2 } \underset { \mathrm { \tiny ~ a \sim \Theta , b , \sim \infty } } { \operatorname* { l i m } } \int _ { \mathrm { a } } ^ { \mathbf { b } } \mathrm { d } ( r ^ { 2 } ) \int _ { - \infty } ^ { \infty } \mathrm { d } \lambda e ^ { i \lambda r ^ { 2 } } \operatorname* { d e t } ( I + 2 i \lambda \Sigma ) ^ { - 1 / 2 } \mathrm { V } _ { \phi } ( ( \Sigma ^ { - 1 } + 2 i \lambda I ) ^ { - 1 } r ^ { - 2 } ) , } \end{array} +$$ + +where we used the fact that $\begin{array} { r } { \hat { \eta } ( 0 ) = \int _ { \mathbb { R } } \eta ( x ) \mathrm { d } x = 1 } \end{array}$ by construction. This gives us the desired result after putting back in some constants and changing $r ^ { 2 } \mapsto s$ . □ + +# E.3 SPHERICAL INTEGRATION + +As mentioned above, the scale invariance of batchnorm combined with the BSB1 fixed point assumption will naturally lead us to consider spherical integration. Here we review several basic facts. + +Definition E.26. We define the spherical angular coordinates in $\mathbb { R } ^ { B }$ , $( r , \theta _ { 1 } , \ldots , \theta _ { B - 2 } ) \in [ 0 , \infty ) \times$ $\mathsf { A } = : [ 0 , \pi ] ^ { B - 3 } \times [ 0 , 2 \pi ]$ , by + +$$ +\begin{array} { c } { x _ { 1 } = r \cos \theta _ { 1 } } \\ { x _ { 2 } = r \sin \theta _ { 1 } \cos \theta _ { 2 } } \\ { x _ { 3 } = r \sin \theta _ { 1 } \sin \theta _ { 2 } \cos \theta _ { 3 } } \\ { \vdots } \\ { x _ { B - 2 } = r \sin \theta _ { 1 } \cdot \cdot \cdot \sin \theta _ { B - 3 } \cos \theta _ { B - 2 } } \\ { x _ { B - 1 } = r \sin \theta _ { 1 } \cdot \cdot \cdot \sin \theta _ { B - 3 } \sin \theta _ { B - 2 } . } \end{array} +$$ + +Then we set + +$$ +\begin{array} { r } { v : = \left( \begin{array} { c } { \cos \theta _ { 1 } } \\ { \sin \theta _ { 1 } \cos \theta _ { 2 } } \\ { \sin \theta _ { 1 } \sin \theta _ { 2 } \cos \theta _ { 3 } } \\ { \vdots } \\ { \sin \theta _ { 1 } \cdot \cdot \cdot \sin \theta _ { B - 3 } \cos \theta _ { B - 2 } } \\ { \sin \theta _ { 1 } \cdot \cdot \cdot \sin \theta _ { B - 3 } \sin \theta _ { B - 2 } } \end{array} \right) } \end{array} +$$ + +as the unit norm version of $x$ . The integration element in this coordinate satisfies + +$$ +\mathrm { d } x _ { 1 } \dots \mathrm { d } x _ { B - 1 } = r ^ { B - 2 } \sin ^ { B - 3 } \theta _ { 1 } \sin ^ { B - 4 } \theta _ { 2 } \cdot \cdot \cdot \sin \theta _ { B - 3 } \mathrm { d } r \mathrm { d } \theta _ { 1 } \cdot \cdot \cdot \mathrm { d } \theta _ { B - 2 } . +$$ + +Fact E.27. If $F : S ^ { B - 2 } \to \mathbb { R }$ , then its expectation on the sphere can be written as + +$$ +\underset { v \sim S ^ { B - 2 } } { \mathbb { E } } F ( v ) = \frac { 1 } { 2 } \pi ^ { \frac { 1 - B } { 2 } } \Gamma \left( \frac { B - 1 } { 2 } \right) \int _ { \mathbb { A } } F ( v ) \sin ^ { B - 3 } \theta _ { 1 } \cdot \cdot \cdot \sin \theta _ { B - 3 } \mathrm { d } \theta _ { 1 } \cdot \cdot \cdot \mathrm { d } \theta _ { B - 2 } +$$ + +where $\mathsf { A } = [ 0 , \pi ] ^ { B - 3 } \times [ 0 , 2 \pi ]$ and $v$ are as in Defn E.26. + +The following integrals will be helpful for simplifying many expressions involving spherical integration using angular coordinates. + +Lemma E.28. For $j , k \geq 0$ and $0 \leq s \leq t \leq \pi / 2$ , + +$$ +\int _ { s } ^ { t } \sin ^ { j } \theta \cos ^ { k } \theta \mathrm { { d } } \theta = { \frac { 1 } { 2 } } \operatorname { B e t a } \left( \cos ^ { 2 } t , \cos ^ { 2 } s ; { \frac { k + 1 } { 2 } } , { \frac { j + 1 } { 2 } } \right) . +$$ + +By antisymmetry of cos with respect to $\theta \mapsto \pi - \theta , i f \pi / 2 \leq s \leq t \leq \pi ,$ , + +$$ +\int _ { s } ^ { t } \sin ^ { j } \theta \cos ^ { k } \theta \mathrm { d } \theta = { \frac { 1 } { 2 } } ( - 1 ) ^ { k } \operatorname { B e t a } \left( \cos ^ { 2 } s , \cos ^ { 2 } t ; { \frac { k + 1 } { 2 } } , { \frac { j + 1 } { 2 } } \right) . +$$ + +Proof. Set $x : = \cos ^ { 2 } \theta \implies \mathrm { d } x = - 2 \cos \theta \sin \theta \mathrm { d } \theta$ . So the integral in question is + +$$ +\begin{array} { r l } & { \quad \displaystyle \frac { - 1 } { 2 } \int _ { \cos ^ { 2 } s } ^ { \cos ^ { 2 } t } ( 1 - x ) ^ { \frac { j - 1 } { 2 } } x ^ { \frac { k - 1 } { 2 } } \mathrm { d } x } \\ & { = \displaystyle \frac { 1 } { 2 } ( \mathrm { B e t a } ( \cos ^ { 2 } s ; \frac { k + 1 } { 2 } , \frac { j + 1 } { 2 } ) - \mathrm { B e t a } ( \cos ^ { 2 } t ; \frac { k + 1 } { 2 } , \frac { j + 1 } { 2 } ) ) } \end{array} +$$ + +As consequences, + +Lemma E.29. For $j , k \geq 0$ , + +$$ +\int _ { 0 } ^ { \pi } \sin ^ { j } \theta \cos ^ { k } \theta \mathrm { d } \theta = { \frac { 1 + ( - 1 ) ^ { k } } { 2 } } \operatorname { B e t a } ( { \frac { j + 1 } { 2 } } , { \frac { k + 1 } { 2 } } ) = { \frac { 1 + ( - 1 ) ^ { k } } { 2 } } { \frac { \Gamma ( { \frac { 1 + j } { 2 } } ) \Gamma ( { \frac { 1 + k } { 2 } } ) } { \Gamma ( { \frac { 2 + j + k } { 2 } } ) } } . +$$ + +# Lemma E.30. + +$$ +\int _ { 0 } ^ { \pi } \sin \theta _ { 1 } \mathrm { d } \theta _ { 1 } \int _ { 0 } ^ { \pi } \sin ^ { 2 } \theta _ { 2 } \mathrm { d } \theta _ { 2 } \cdot \cdot \cdot \int _ { 0 } ^ { \pi } \sin ^ { k } \theta _ { k } \mathrm { d } \theta _ { k } = \pi ^ { k / 2 } / \Gamma ( \frac { k + 2 } { 2 } ) . +$$ + +Proof. By Lemma E.29, $\begin{array} { r } { \int _ { 0 } ^ { \pi } \sin ^ { r } \theta _ { r } \mathrm { d } \theta _ { r } = \mathrm { B e t a } \big ( \frac { r + 1 } { 2 } , \frac { 1 } { 2 } \big ) = \frac { \Gamma ( \frac { r + 1 } { 2 } ) \sqrt { \pi } } { \Gamma ( \frac { r + 2 } { 2 } ) } } \end{array}$ Γ( r+12 ) πr+2 . Thus this product of integrals is equal to + +$$ +\prod _ { r = 1 } ^ { k } \sqrt { \pi } \frac { \Gamma \big ( \frac { r + 1 } { 2 } \big ) } { \Gamma \big ( \frac { r + 2 } { 2 } \big ) } = \pi ^ { k / 2 } \Gamma ( 1 ) / \Gamma \big ( \frac { k + 2 } { 2 } \big ) = \pi ^ { k / 2 } / \Gamma \big ( \frac { k + 2 } { 2 } \big ) . +$$ + +$B > 1 , \mu > 0$ , $\begin{array} { r } { \int _ { 0 } ^ { \infty } r ^ { B } e ^ { - r ^ { 2 } / 2 \mu } \mathrm { d } r = 2 ^ { \frac { B - 1 } { 2 } } \mu ^ { \frac { B + 1 } { 2 } } \Gamma ( \frac { B + 1 } { 2 } ) } \end{array}$ + +Proof. Apply change of coordinates $z = r ^ { 2 } / 2 \mu$ + +Definition E.32. Let $K ( v ; \Sigma , \beta ) \ : = \ ( \operatorname * { d e t } \circledast ^ { T } \Sigma \circledast ^ { - 1 / 2 } ( v ^ { T } ( \operatorname { \mathbb { e } } ^ { T } \Sigma \circledast ^ { - 1 } v ) ^ { - \frac { B - 1 + \beta } { 2 } }$ and $\kappa ( v ; \Sigma ) : =$ $\kappa ( v ; \Sigma , 0 )$ . + +The following proposition allows us to express integrals involving batchnorm as expectations over the uniform spherical measure. + +Proposition E.33. If $f : \mathbb { R } ^ { B - 1 } \mathbb { R } ^ { B ^ { \prime } }$ , $B > 1 - \beta$ , and $\Sigma \in \mathbb { R } ^ { ( B - 1 ) \times ( B - 1 ) }$ is nonsingular, then + +$$ +\underset { h \sim \bar { N } ( 0 , \Sigma ) } { \mathbb { E } } \Vert h \Vert ^ { \beta } f ( h / \Vert h \Vert ) = 2 ^ { \beta / 2 } \mathrm { P } \left( \frac { B - 1 } { 2 } , \frac { \beta } { 2 } \right) \underset { v \sim \bar { S } ^ { B - 2 } } { \mathbb { E } } f ( v ) ( \operatorname* { d e t } \Sigma ) ^ { - 1 / 2 } ( v ^ { T } \Sigma ^ { - 1 } v ) ^ { - \frac { B - 1 + \beta } { 2 } } . +$$ + +If $f : \mathbb { R } ^ { B } \mathbb { R } ^ { B ^ { \prime } }$ and $\Sigma \in \mathbb { R } ^ { B \times B }$ is such that $\Phi ^ { T } \Sigma \Phi$ is nonsingular (where e is as in Defn E.3) and $G \Sigma G = \Sigma$ , + +$$ +\underset { h \sim \mathcal { N } ( 0 , \Sigma ) } { \mathbb { E } } \Vert G h \Vert ^ { \beta } f ( G h / \Vert G h \Vert ) = 2 ^ { \beta / 2 } \mathrm { P } \left( \frac { B - 1 } { 2 } , \frac { \beta } { 2 } \right) \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } f ( \mathfrak { e } v ) K ( v ; \Sigma , \beta ) . +$$ + +In particular, + +$$ +\mathbb { E } [ f ( B ( h ) ) : h \sim \mathcal { N } ( 0 , \Sigma ) ] = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } f ( \mathbb { e } v ) K ( v ; \Sigma ) +$$ + +Proof. We will prove the second statement. The others follow trivially. + +$$ +\begin{array} { r l } & { \quad \mathbb { E } [ \| G h \| ^ { \beta } f ( \mathcal { B } ( h ) ) : h \sim \mathcal { N } ( 0 , \Sigma ) ] } \\ & { = \mathbb { E } [ \| x \| ^ { \beta } f ( \mathbf { e x } / \| x \| ) : x \sim \mathcal { N } ( 0 , \mathbf { e } ^ { T } \Sigma \mathbf { e } ) ] } \\ & { = ( 2 \pi ) ^ { \frac { 1 - B } { 2 } } ( \operatorname* { d e t } \mathbb { e } ^ { T } \Sigma \mathbf { e } ) ^ { - 1 / 2 } \int _ { \mathbb { R } ^ { B - 1 } } \| x \| ^ { \beta } f ( \mathbf { e x } / \| x \| ) e ^ { - \frac { 1 } { 2 } x ^ { T } ( \mathbf { e } ^ { T } \Sigma \mathbf { e } ) ^ { - 1 } x } \mathrm { d } x } \\ & { = ( 2 \pi ) ^ { \frac { 1 - B } { 2 } } ( \operatorname* { d e t } \mathbb { e } ^ { T } \Sigma \mathbf { e } ) ^ { - 1 / 2 } \int _ { \mathbb { A } } f ( \mathbf { e } v ) \sin ^ { B - 3 } \theta _ { 1 } \cdot \cdot \cdot \sin \theta _ { B - 3 } \mathrm { d } r \mathrm { d } \theta _ { 1 } \cdot \cdot \cdot \mathrm { d } \theta _ { B - 2 } \times } \\ & { \quad \quad \quad \int _ { 0 } ^ { \infty } r ^ { B - 2 + \beta } e ^ { - \frac { 1 } { 2 } r ^ { 2 } \cdot v ^ { T } ( \mathbf { e } ^ { T } \Sigma \mathbf { e } ) ^ { - 1 } v } } \end{array} +$$ + +(spherical coordinates Defn E.26) + +$$ +\begin{array} { r l } & { = ( 2 \pi ) ^ { \frac { 1 - 2 } { 2 } } ( \dot { \mathrm { f o r } } \theta ^ { \top } \nabla \mathrm { s } \mathrm { i } ) ^ { - 1 / 2 } \times 2 ^ { \frac { n - 3 } { 2 } } \Gamma ( \frac { B - 1 } { 2 } ) \times \frac { 2 ^ { \frac { n - 3 } { 2 } } \Gamma ( \frac { B - 1 - \rho \xi } { 2 } ) } { 2 ^ { \frac { n - 3 } { 2 } } \Gamma ( \frac { B - 1 } { 2 } ) } } \\ & { \qquad \times \int _ { \mathrm { a } } \frac { f ( \mathrm { e v } ) \sin ^ { \frac { \theta - 3 } { 2 } } \theta _ { 1 } - \sin \theta _ { B - 3 } \sin \theta _ { 1 } - \sin \theta _ { B - 4 } \sin \theta _ { B - 2 } } { ( \sigma ^ { \prime } ( \mathrm { e } ^ { \gamma } \mathrm { s } \mathrm { e } ^ { \gamma } ) ) ^ { \frac { n - 3 } { 2 } } \frac { \mathrm { d } \theta _ { 1 } - \cos \theta _ { B - 2 } } { 1 / \sigma ^ { \prime } } } } \\ & { = \frac { 1 } { 2 } \pi ^ { \frac { 1 - \rho } { 2 } } \Gamma ( \frac { B - 1 } { 2 } ) \times 2 ^ { \rho / 2 } \Gamma ( \frac { B - 1 } { 2 } + \frac { \beta } { 2 } ) \Gamma ( \frac { B - 1 } { 2 } ) ^ { - 1 } } \\ & { \qquad \times \int _ { \mathrm { a } } \frac { f ( \mathrm { e v } ) \sin ^ { \frac { \theta - 3 } { 2 } } \theta _ { 1 } - \sin \theta _ { B - 3 } \sin \theta _ { 1 } - \sin \theta _ { 1 } - \sin \theta _ { B - 2 } } { ( \sigma ^ { \prime } ( \mathrm { e } ^ { \gamma } \mathrm { e } ^ { \gamma } ) ) ^ { \frac { n - 3 } { 2 } - \frac { n + 3 } { 2 } } } f ( \mathrm { e } \mathrm { e } ^ { \gamma } \mathrm { e } ^ { \gamma } ) } \\ & = 2 ^ { 3 / 2 } \Gamma ( \frac { B - 1 + \beta } { 2 } ) \Gamma ( \frac { B - 1 } { 2 } ) ^ { - 1 } \frac { \Gamma } \mathrm { e } ^ { \gamma } \mathrm { e } ^ { \gamma } \mathrm { e } ^ { \gamma } \mathrm { e } ^ { \gamma } \mathrm { f o r } ( \mathrm { e } ^ { \gamma } \mathrm { e } ^ { \gamma } \end{array} +$$ + +As a straightforward consequence, + +Proposition E.34. If $\Phi ^ { T } \Sigma \Phi$ is nonsingular, + +$$ +\mathrm { V } _ { \mathcal { B } _ { \phi } } ( \Sigma ) = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathbb { e } v ) ^ { \otimes 2 } \mathcal { K } ( v ; \Sigma ) +$$ + +For numerical calculations, it is useful to realize $\mathbb { P }$ as the matrix whose columns are $\begin{array} { l l } { \Phi _ { B - m } } & { : = } \end{array}$ $m$ $( m ( m + 1 ) ) ^ { - 1 / 2 } ( 0 , \ldots , 0 , - m , \overbrace { 1 , 1 , \ldots , 1 } ^ { \cdots } ) ^ { T }$ , for each $m = 1 , \ldots , B - 1$ . + +$$ +\begin{array} { r } { \boldsymbol { \Phi } ^ { T } = \left( \begin{array} { l l l l l l } { \frac { 1 } { \sqrt { B ( B - 1 ) } } } & & & & & \\ & { \frac { 1 } { \sqrt { ( B - 1 ) ( B - 2 ) } } } & & & & \\ & & { \ddots } & & & \\ & & & & { 6 ^ { - 1 / 2 } } & \\ & & & & & { 2 ^ { - 1 / 2 } } \end{array} \right) } \end{array} +$$ + +$$ +\times \left( \begin{array} { c c c c c c c } { { - ( B - 1 ) } } & { { 1 } } & { { 1 } } & { { \cdots } } & { { 1 } } & { { 1 } } & { { 1 } } \\ { { 0 } } & { { - ( B - 2 ) } } & { { 1 } } & { { \cdots } } & { { 1 } } & { { 1 } } & { { 1 } } \\ { { \vdots } } & { { } } & { { \vdots } } & { { \ddots } } & { { \vdots } } & { { \vdots } } & { { \vdots } } \\ { { 0 } } & { { 0 } } & { { 0 } } & { { \cdots } } & { { - 2 } } & { { 1 } } & { { 1 } } \\ { { 0 } } & { { 0 } } & { { 0 } } & { { \cdots } } & { { 0 } } & { { - 1 } } & { { 1 } } \end{array} \right) +$$ + +We have + +$$ +\begin{array} { l } { \displaystyle ( \mathtt { e } v ) _ { 1 } = - \sqrt { \frac { B - 1 } { B } } \cos \theta _ { 1 } } \\ { \displaystyle ( \mathtt { e } v ) _ { 2 } = \cos \theta _ { 1 } - \sqrt { \frac { B - 2 } { B - 1 } } \sin \theta _ { 1 } \cos \theta _ { 2 } } \\ { \displaystyle ( \mathtt { e } v ) _ { 3 } = \cos \theta _ { 1 } + \sin \theta _ { 1 } \cos \theta _ { 2 } - \sqrt { \frac { B - 3 } { B - 2 } } \sin \theta _ { 1 } \sin \theta _ { 2 } \cos \theta _ { 3 } } \\ { \displaystyle ( \mathtt { e } v ) _ { 4 } = \cos \theta _ { 1 } + \sin \theta _ { 1 } \cos \theta _ { 2 } + \sin \theta _ { 1 } \sin \theta _ { 2 } \cos \theta _ { 3 } - \sqrt { \frac { B - 4 } { B - 3 } } \sin \theta _ { 1 } \sin \theta _ { 2 } \sin \theta _ { 3 } \cos \theta _ { 4 } } \end{array} +$$ + +so in particular they only depend on $\theta _ { 1 } , \ldots , \theta _ { 4 }$ . + +Angular coordinates In many situations, the sparsity of this realization of $\mathbb { P }$ combined with sufficient symmetry allows us to simplify the high-dimensional spherical integral into only a few dimensions, using one of the following lemmas + +Lemma E.35. Let $f : \mathbb { R } ^ { 4 } \to \mathbb { R } ^ { A }$ for some $A \in \mathbb { N }$ . Suppose $B \geq 6$ . Then for any $k < B - 1$ + +$$ +\begin{array} { l } { { \ \mathbb { E } [ r ^ { - k } f ( v _ { 1 } , \dots , v _ { 4 } ) : x \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] } } \\ { { = { \frac { \Gamma ( ( B - 1 - k ) / 2 ) } { \Gamma ( ( B - 5 ) / 2 ) } } 2 ^ { - k / 2 } \pi ^ { - 2 } \displaystyle \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } \cdots \displaystyle \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 4 } \ f ( v _ { 1 } ^ { \theta } , \dots , v _ { 4 } ^ { \theta } ) \sin ^ { B - 3 } \theta _ { 1 } \cdots \sin ^ { B - 6 } \theta _ { 4 } } } \end{array} +$$ + +where $v _ { i } = x _ { i } / \| x \|$ and $r = \| x \|$ , and + +$$ +\begin{array} { r l } & { v _ { 1 } ^ { \theta } = \cos \theta _ { 1 } } \\ & { v _ { 2 } ^ { \theta } = \sin \theta _ { 1 } \cos \theta _ { 2 } } \\ & { v _ { 3 } ^ { \theta } = \sin \theta _ { 1 } \sin \theta _ { 2 } \cos \theta _ { 3 } } \\ & { v _ { 4 } ^ { \theta } = \sin \theta _ { 1 } \sin \theta _ { 2 } \sin \theta _ { 3 } \cos \theta _ { 4 } . } \end{array} +$$ + +Lemma E.36. Let $f : \mathbb { R } ^ { 2 } \to \mathbb { R } ^ { A }$ for some $A \in \mathbb { N }$ . Suppose $B \geq 4$ . Then for any $k < B - 1$ + +$$ +\begin{array} { c l } { { } } & { { \mathbb { E } [ r ^ { - k } f ( v _ { 1 } , v _ { 2 } ) : x \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] } } \\ { { } } & { { = \displaystyle \frac { \Gamma ( ( B - 1 - k ) / 2 ) } { \Gamma ( ( B - 3 ) / 2 ) } 2 ^ { - k / 2 } \pi ^ { - 1 } \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 2 } f ( v _ { 1 } ^ { \theta } , v _ { 2 } ^ { \theta } ) \sin ^ { B - 3 } \theta _ { 1 } \sin ^ { B - 4 } \theta _ { 2 } } } \end{array} +$$ + +where $v _ { i } = x _ { i } / \| x \|$ and $r = \| x \|$ , and + +$$ +\begin{array} { l } { { v _ { 1 } ^ { \theta } = \cos \theta _ { 1 } } } \\ { { v _ { 2 } ^ { \theta } = \sin \theta _ { 1 } \cos \theta _ { 2 } } } \end{array} +$$ + +Lemma E.37. Let $f : \mathbb { R } \to \mathbb { R } ^ { A }$ for some $A \in \mathbb { N }$ . Suppose $B \geq 3 .$ . Then for any $k < B - 1$ + +$$ +\begin{array} { r l } & { \quad \mathbb { E } [ r ^ { - k } f ( v ) : x \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] } \\ & { = \frac { \Gamma ( ( B - 1 - k ) / 2 ) } { \Gamma ( ( B - 2 ) / 2 ) } 2 ^ { - k / 2 } \pi ^ { - 1 / 2 } \int _ { 0 } ^ { \pi } \mathrm { d } \theta \ f ( \cos \theta ) \sin ^ { B - 3 } \theta } \end{array} +$$ + +where $v _ { i } = x _ { i } / \| x \|$ and $r = \| x \|$ . + +We will prove Lemma E.35; those of Lemma E.36 and Lemma E.37 are similar. + +Proof of Lemma E.35. + +$$ +\begin{array} { l } { { \mathbb { E } [ r ^ { - k } f ( v _ { 1 } , \dots , v _ { 4 } ) : x \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] } } \\ { { = ( 2 \pi ) ^ { - ( B - 1 ) / 2 } \int \mathrm { d } x \ r ^ { - k } f ( v _ { 1 } , \dots , v _ { 4 } ) e ^ { - r ^ { 2 } / 2 } } } \\ { { = ( 2 \pi ) ^ { - ( B - 1 ) / 2 } \int \left( r ^ { B - 2 } \sin ^ { B - 3 } \theta _ { 1 } \cdots \sin \theta _ { B - 3 } \mathrm { d } r \mathrm { d } \theta _ { 1 } \cdots \mathrm { d } \theta _ { B - 2 } \right) r ^ { - k } f ( v _ { 1 } ^ { \theta } , \dots , v _ { 4 } ^ { \theta } ) e ^ { - r ^ { 2 } / 2 } } } \\ { { = ( 2 \pi ) ^ { - ( B - 1 ) / 2 } \int f ( v _ { 1 } ^ { \theta } , \dots , v _ { 4 } ^ { \theta } ) \sin ^ { B - 3 } \theta _ { 1 } \cdots \sin \theta _ { B - 3 } \mathrm { d } \theta _ { 1 } \cdots \mathrm { d } \theta _ { B - 2 } \int _ { 0 } ^ { \infty } r ^ { B - 2 - k } e ^ { - r ^ { 2 } / 2 } \mathrm { d } r } } \\ { { = ( 2 \pi ) ^ { - ( B - 1 ) / 2 } \int f ( v _ { 1 } ^ { \theta } , \dots , v _ { 4 } ^ { \theta } ) \sin ^ { B - 3 } \theta _ { 1 } \cdots \sin \theta _ { B - 3 } \mathrm { d } \theta _ { 1 } \cdots \mathrm { d } \theta _ { B - 2 } ( ^ { B - 3 - k } ) ^ { / 2 } \Gamma ( ( B - 1 - k ) / 2 ) } } \end{array} +$$ + +$$ += \Gamma ( ( B - 1 - k ) / 2 ) 2 ^ { - ( 2 + k ) / 2 } \pi ^ { - ( B - 1 ) / 2 } \int f ( v _ { 1 } ^ { \theta } , \dots , v _ { 4 } ^ { \theta } ) \sin ^ { B - 3 } \theta _ { 1 } \cdot \cdot \cdot \sin \theta _ { B - 3 } \mathrm { d } \theta _ { 1 } \cdot \cdot \cdot \mathrm { d } \theta _ { B - 2 } +$$ + +Because $v _ { 1 } ^ { \theta } , \ldots , v _ { 4 } ^ { \theta }$ only depends on $\theta _ { 1 } , \ldots , \theta _ { 4 }$ , we can integrate out $\theta _ { 5 } , \ldots , \theta _ { B - 2 }$ . By applying Lemma E.30 to $\theta _ { 5 } , \ldots \theta _ { B - 3 }$ and noting that $\begin{array} { r } { \int _ { 0 } ^ { 2 \pi } \mathrm { d } \theta _ { B - 2 } = 2 \pi } \end{array}$ , we get + +$$ +\begin{array} { r l } & { \quad \mathbb { E } [ r ^ { - k } f ( v _ { 1 } ^ { \theta } , \dots , v _ { d } ^ { \theta } ) : x \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] } \\ & { = \Gamma ( ( B - 1 - k ) / 2 ) 2 ^ { - ( 2 + k ) / 2 } \pi ^ { - ( B - 1 ) / 2 } \int f ( v _ { 1 } ^ { \theta } , \dots , v _ { 4 } ^ { \theta } ) \sin ^ { B - 3 } \theta _ { 1 } \cdots \sin ^ { B - 6 } \theta _ { 4 } \pi ^ { ( B - 7 ) / 2 } \Gamma ( ( B - 1 , \dots , I _ { B } ) ) } \\ & { = \frac { \Gamma ( ( B - 1 - k ) / 2 ) } { \Gamma ( ( B - 5 ) / 2 ) } 2 ^ { - k / 2 } \pi ^ { - 2 } \int f ( v _ { 1 } ^ { \theta } , \dots , v _ { 4 } ^ { \theta } ) \sin ^ { B - 3 } \theta _ { 1 } \cdots \sin ^ { B - 6 } \theta _ { 4 } } \end{array} +$$ + +Cartesian coordinates. We can often also simplify the high dimensional spherical integrals without trigonometry, as the next two lemmas show. Both are proved easily using change of coordinates. + +Lemma E.38. For any $u \in S ^ { n }$ , + +$$ +\underset { v \sim S ^ { n } } { \mathbb { E } } f ( u \cdot v ) = \frac { \omega _ { n - 1 } } { \omega _ { n } } \int _ { - 1 } ^ { 1 } \mathrm { d } h \left( 1 - h ^ { 2 } \right) ^ { \frac { n - 2 } { 2 } } f ( h ) +$$ + +where $\omega _ { n - 1 }$ is hypersurface of the $( n - 1 )$ -dimensional unit hypersphere. + +Lemma E.39. For any $u _ { 1 } , u _ { 2 } \in S ^ { n }$ + +$$ +\underbrace { \mathbb { E } } _ { v \sim S ^ { n } } f ( u _ { 1 } \cdot v , u _ { 2 } \cdot v ) = { \frac { \omega _ { n - 2 } } { \omega _ { n } } } \int _ { \underbrace { v \in \mathrm { s p a n } ( u _ { 1 } , u _ { 2 } ) } _ { v \in \mathrm { s p a n } ( u _ { 1 } , u _ { 2 } ) } } \mathrm { d } v ( 1 - \| v \| ^ { 2 } ) ^ { \frac { n - 3 } { 2 } } f ( u _ { 1 } \cdot v , u _ { 2 } \cdot v ) +$$ + +where $\omega _ { n - 1 }$ is hypersurface of the $( n - 1 )$ -dimensional unit hypersphere. + +# E.4 GEGENBAUER EXPANSION + +Definition E.40. The Gegenbauer polynomials $\{ C _ { l } ^ { ( \alpha ) } ( x ) \} _ { l = 0 } ^ { \infty }$ , with $\deg C _ { l } ^ { ( \alpha ) } ( x ) ~ = ~ l$ , are the set of orthogonal polynomials with respect to the weight function $( 1 - x ^ { 2 } ) ^ { \alpha - 1 / 2 }$ (i.e. $\int _ { - 1 } ^ { 1 } ( 1 ~ -$ $x ^ { 2 } ) ^ { \alpha - 1 / 2 } C _ { l } ^ { ( \alpha ) } ( x ) C _ { m } ^ { ( \alpha ) } ( x ) = 0$ if $l \neq m$ ). By convention, they are normalized so that + +$$ +\int _ { - 1 } ^ { 1 } \left[ C _ { n } ^ { ( \alpha ) } ( x ) \right] ^ { 2 } ( 1 - x ^ { 2 } ) ^ { \alpha - \frac { 1 } { 2 } } d x = \frac { \pi 2 ^ { 1 - 2 \alpha } \Gamma ( n + 2 \alpha ) } { n ! ( n + \alpha ) \Gamma ( \alpha ) ^ { 2 } } . +$$ + +Here are several examples of low degree Gegenbauer polynomials + +$$ +\begin{array} { l } { { C _ { 0 } ^ { ( \alpha ) } ( x ) = 1 } } \\ { { C _ { 1 } ^ { ( \alpha ) } ( x ) = 2 \alpha x } } \\ { { C _ { 2 } ^ { ( \alpha ) } ( x ) = - \alpha + 2 \alpha ( 1 + \alpha ) x ^ { 2 } } } \\ { { C _ { 3 } ^ { ( \alpha ) } ( x ) = - 2 \alpha ( 1 + \alpha ) x + \displaystyle \frac { 4 } { 3 } \alpha ( 1 + \alpha ) ( 2 + \alpha ) x ^ { 3 } . } } \end{array} +$$ + +They satisfy a few identities summarized below + +Fact E.41 (Suetin). $C _ { n } ^ { ( \alpha ) } ( \pm 1 ) = ( \pm 1 ) ^ { n } { \binom { n + 2 \alpha - 1 } { n } }$ + +Fact E.42 (Weisstein). + +$$ +\begin{array} { r l } & { \ C _ { n } ^ { ( \lambda ) \prime } ( x ) = 2 \lambda C _ { n - 1 } ^ { ( \lambda + 1 ) } ( x ) } \\ & { x C _ { n } ^ { ( \lambda ) \prime } ( x ) = n C _ { n } ^ { ( \lambda ) } ( x ) + { C _ { n - 1 } ^ { ( \lambda ) } } ^ { \prime } ( x ) } \\ & { C _ { n + 1 } ^ { ( \lambda ) } { ' } ( x ) = ( n + 2 \lambda ) C _ { n } ^ { ( \lambda ) } ( x ) + x C _ { n } ^ { ( \lambda ) \prime } ( x ) } \\ & { n C _ { n } ^ { ( \lambda ) } ( x ) = 2 ( n - 1 + \lambda ) x C _ { n - 1 } ^ { ( \lambda ) } ( x ) - ( n - 2 + 2 \lambda ) C _ { n - 2 } ^ { ( \lambda ) } ( x ) , \forall n \geq 2 } \end{array} +$$ + +By repeated applications of Eq. (38), + +# Proposition E.43. + +$$ +\begin{array} { l } { { \displaystyle x C _ { n } ^ { ( \lambda ) } ( x ) = \frac { 1 } { 2 ( n + \lambda ) } \left( ( n + 1 ) C _ { n + 1 } ^ { ( \lambda ) } ( x ) + ( n - 1 + 2 \lambda ) C _ { n - 1 } ^ { ( \lambda ) } ( x ) \right) , \forall n \geq 1 } } \\ { { \displaystyle x ^ { 2 } C _ { n } ^ { ( \lambda ) } ( x ) = \kappa _ { 2 } ( n , \lambda ) C _ { n + 2 } ^ { ( \lambda ) } ( x ) + \kappa _ { 0 } ( n , \lambda ) C _ { n } ^ { ( \lambda ) } ( x ) + \kappa _ { - 2 } \left( n , \lambda \right) C _ { n - 2 } ^ { ( \lambda ) } ( x ) , \forall n \geq 2 } } \\ { { \displaystyle x C _ { 0 } ^ { ( \lambda ) } ( x ) = x = \frac { 1 } { 2 \lambda } C _ { 1 } ^ { ( \lambda ) } ( x ) } } \\ { { \displaystyle x ^ { 2 } C _ { 0 } ^ { ( \lambda ) } ( x ) = \frac { 1 } { 4 \lambda ( 1 + \lambda ) } \left( 2 C _ { 2 } ^ { ( \lambda ) } ( x ) + 2 \lambda C _ { 0 } ^ { ( \lambda ) } ( x ) \right) } } \\ { { \displaystyle \quad \quad = \kappa _ { 2 } ( 0 , \lambda ) C _ { 2 } ^ { ( \lambda ) } ( x ) + \kappa _ { 0 } ( 0 , \lambda ) C _ { 0 } ^ { ( \lambda ) } ( x ) } } \\ { { \displaystyle x ^ { 2 } C _ { 1 } ^ { ( \lambda ) } ( x ) = \kappa _ { 2 } ( 1 , \lambda ) C _ { 3 } ^ { ( \lambda ) } ( x ) + \kappa _ { 0 } ( 1 , \lambda ) C _ { 1 } ^ { ( \lambda ) } ( x ) } } \\ { { \displaystyle \kappa _ { 0 } ( n , \lambda ) : = - \frac { ( n + 1 ) ( n + 2 ) } { 4 \lambda } , \kappa _ { 0 } ( n , \lambda ) : = - \frac { ( n + 1 ) ( n + 2 \lambda ) } { 4 } - \frac { ( n - 1 + 2 \lambda ) n } { 4 } , \kappa _ { - } \rho ( n , \lambda ) : = - \frac { 1 } { 4 \lambda } C _ { 0 } ^ { ( \lambda ) } ( x ) } } \end{array} +$$ + +where κ2(n, λ) := (n+1)(n+2)4(n+λ)(n+1+λ) , κ0(n, λ) := 4 (n−1+2λ)n4(n+λ)(n−1+λ) , κ−2(n, λ) := $\frac { ( n - 1 + 2 \lambda ) ( n - 2 + 2 \lambda ) } { 4 ( n + \lambda ) ( n - 1 + \lambda ) }$ + +This proposition is useful for the Gegenbauer expansion of the local convergence rate of Eq. (43). + +Zonal harmonics. The Gegenbauer polynomials are closely related to a special type of spherical harmonics called zonal harmonics, which are intuitively those harmonics that only depend on the “height” of the point along some fixed axis. + +Definition E.44. Let $Z _ { u } ^ { N - 1 , ( l ) }$ denote the degree $l$ zonal spherical harmonics on the $( N - 1 )$ - dimensional sphere with axis $u \in S ^ { N - 1 }$ , defined by $Z _ { u } ^ { N - 1 , ( l ) } ( v ) = c _ { N , l } ^ { - 1 } C _ { l } ^ { ( \frac { N - 2 } { 2 } ) } ( u \cdot v )$ for any u, v ∈ SN−1, where cN,l = N−2N+2l−2 . + +The first few zonal spherical harmonics are $Z _ { u } ^ { N - 1 , ( 0 ) } ( v ) = 1 , Z _ { u } ^ { N - 1 , ( 1 ) } ( v ) = N u \cdot v , \ldots .$ We note a trivial lemma that is useful for simplying many expressions in Gegenbauer expansions. + +Lemma E.45. $\begin{array} { r } { \frac { c _ { B + 1 , l - 1 } } { c _ { B - 1 , l } } = \frac { B - 1 } { B - 3 } } \end{array}$ . + +Let $\langle - , - \rangle _ { S ^ { N } }$ denote the inner product induced by the uniform measure on $S ^ { N }$ . One of the most important properties of zonal harmonics is their + +Fact E.46 (Reproducing property (Suetin)). + +$$ +\begin{array} { r l } & { \langle Z _ { u } ^ { N - 1 , ( l ) } , Z _ { v } ^ { N - 1 , ( l ) } \rangle _ { S ^ { N - 1 } } = Z _ { u } ^ { N - 1 , ( l ) } ( v ) = c _ { N , l } ^ { - 1 } C _ { l } ^ { ( \frac { N - 2 } { 2 } ) } ( u \cdot v ) } \\ & { \langle Z _ { u } ^ { N - 1 , ( l ) } , Z _ { v } ^ { N - 1 , ( m ) } \rangle _ { S ^ { N - 1 } } = 0 , i f l \neq m . } \end{array} +$$ + +In our applications of the reproducing property, $u$ and $v$ will always be $\hat { \mathbb { E } } _ { a , : } : = \mathbb { e } _ { a , : } / \Vert \mathbb { e } _ { a , : } \Vert \ =$ $\mathbb { e } _ { a , : } \sqrt { \frac { B } { B - 1 } }$ BB−1 , where ea,: is the ath row of e. + +Next, we present a new (to the best of our knowledge) identity showing that a certain quadratic form of $\phi$ , reminiscent of Dirichlet forms, depending only on the derivative $\phi ^ { \prime }$ through a spherical integral, is diagonalized in the Gegenbauer basis. This will be crucial to proving the necessity of gradient explosion under the BSB1 fixed point assumption. + +Theorem E.47 (Dirichlet-like form of $\phi$ diagonalizes over Gegenbauer basis). Let $u _ { 1 } , u _ { 2 } \in S ^ { B - 2 }$ . Suppose $\phi ( \sqrt { B - 1 } x ) , \phi ^ { \prime } ( \sqrt { B - 1 } x ) \in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \frac { B - 3 } { 2 } } )$ and $\phi ( \sqrt { B - 1 } x )$ has Gegenbauer expansion $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ . Then + +$$ +\begin{array} { r l } { { \underset { v \in S ^ { B - 2 } } { \mathbb { E } } ( u _ { 1 } \cdot u _ { 2 } - ( u _ { 1 } \cdot v ) ( u _ { 2 } \cdot v ) ) \phi ^ { \prime } ( \sqrt { B - 1 } u _ { 1 } \cdot v ) \phi ^ { \prime } ( \sqrt { B - 1 } u _ { 2 } \cdot v ) } } \\ & { = \displaystyle \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } ( l + B - 3 ) l c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) } \end{array} +$$ + +This is proved using the following lemmas. + +Lemma E.48. + +$$ +\begin{array} { l } { { \displaystyle x C _ { l } ^ { ( \alpha ) \prime } ( x ) = l C _ { l } ^ { ( \alpha ) } ( x ) + \sum _ { j \in [ l - 2 ] _ { 2 } } 2 ( j + \alpha ) C _ { j } ^ { ( \alpha ) } ( x ) } } \\ { { \displaystyle C _ { l } ^ { ( \alpha ) \prime } ( x ) = \sum _ { j \in [ l - 1 ] _ { 2 } } 2 ( j + \alpha ) C _ { j } ^ { ( \alpha ) } ( x ) } } \end{array} +$$ + +where $[ n ] _ { 2 }$ is the sequence $\{ ( n \% 2 ) , ( n \% 2 ) + 1 , . . . , n - 2 , n \}$ and $n \% 2$ is the remainder of $n / 2$ + +Proof. Apply Eq. (36) and Eq. (37) alternatingly. + +Lemma E.49. + +$$ +l C _ { l } ^ { ( \alpha ) } ( t ) + \sum _ { j \in [ l - 2 ] _ { 2 } } 2 ( j + \alpha ) C _ { j } ^ { ( \alpha ) } ( t ) - t \sum _ { j \in [ l - 1 ] _ { 2 } } 2 ( j + \alpha ) C _ { j } ^ { ( \alpha ) } ( t ) = 0 +$$ + +Proof. Apply Eq. (38) to $l C _ { l } ^ { ( \alpha ) } ( t ) , ( l - 2 ) C _ { l - 2 } ^ { ( \alpha ) } ( t ) , . . . .$ + +Proof of Thm E.47. We have + +$$ +\begin{array} { l } { \displaystyle \phi ^ { \prime } ( \sqrt { B - 1 } x ) = \frac { 1 } { \sqrt { B - 1 } } \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) \prime } ( x ) } \\ { \displaystyle \quad = \frac { 1 } { \sqrt { B - 1 } } \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } \sum _ { j \in [ l - 1 ] _ { 2 } } ( 2 j + B - 3 ) C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \\ { \displaystyle \quad = \frac { 1 } { \sqrt { B - 1 } } \sum _ { l = 0 } ^ { \infty } \sum _ { j \in [ l - 1 ] _ { 2 } } a _ { l } \frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } ( 2 j + B - 3 ) c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \\ { \displaystyle \quad = \frac { 1 } { \sqrt { B - 1 } } \sum _ { j = 0 } ^ { \infty } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) \sum _ { l \in [ j + 1 , \infty ] _ { 2 } } a _ { l } ^ { - } \frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } ( 2 j + B - 3 ) } \end{array} +$$ + +where $[ n , m ] _ { 2 }$ is the sequence $n , n + 2 , \ldots , m$ . So for any $u _ { 1 } , u _ { 2 } \in S ^ { B - 2 }$ , + +$$ +\begin{array} { r l } & { \quad \underset { v \in S ^ { B - 2 } } { \mathbb { E } } \phi ^ { \prime } ( \sqrt { B - 1 } u _ { 1 } \cdot v ) \phi ^ { \prime } ( \sqrt { B - 1 } u _ { 2 } \cdot v ) } \\ & { = \frac { 1 } { B - 1 } { \displaystyle \sum _ { j = 0 } ^ { \infty } } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) \left( \underset { l \in [ j + 1 , \infty ] _ { 2 } } { \sum } a _ { l } \frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } ( 2 j + B - 3 ) \right) ^ { 2 } } \\ & { = \frac { 1 } { B - 1 } { \displaystyle \sum _ { j = 0 } ^ { \infty } } c _ { B - 1 , j } ( 2 j + B - 3 ) ^ { 2 } C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) \left( \underset { l , m \in [ j + 1 , \infty ] _ { 2 } } { \sum } \frac { a _ { l } } { c _ { B - 1 , l } } \frac { a _ { m } } { c _ { B - 1 , m } } \right) . } \end{array} +$$ + +For $l < m$ and $m - l$ even, the coefficient of $a _ { l } a _ { m }$ is + +$$ +\begin{array} { r l r } & { } & { \frac { 2 c _ { B - 1 , l } ^ { - 1 } c _ { B - 1 , m } ^ { - 1 } } { B - 1 } \displaystyle \sum _ { j \in [ l - 1 ] _ { 2 } } c _ { B - 1 , j } ( 2 j + B - 3 ) ^ { 2 } C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) } \\ & { } & { = \frac { 2 c _ { B - 1 , l } ^ { - 1 } c _ { B - 1 , m } ^ { - 1 } } { B - 1 } \displaystyle \sum _ { j \in [ l - 1 ] _ { 2 } } ( B - 3 ) ( 2 j + B - 3 ) C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) . } \end{array} +$$ + +For each $l$ , the coefficient of $a _ { l } ^ { 2 }$ is + +$$ +\begin{array} { c } { { \displaystyle \frac { c _ { B - 1 , l } ^ { - 2 } } { B - 1 } \sum _ { j \in [ l - 1 ] _ { 2 } } c _ { B - 1 , j } ( 2 j + B - 3 ) ^ { 2 } C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) } } \\ { { = \displaystyle \frac { c _ { B - 1 , l } ^ { - 2 } } { B - 1 } \sum _ { j \in [ l - 1 ] _ { 2 } } ( B - 3 ) ( 2 j + B - 3 ) C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) . } } \end{array} +$$ + +Similarly, if $\phi ^ { \prime } ( x ) \in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \frac { B - 3 } { 2 } } )$ , we have + +$$ +\begin{array} { l } { { \displaystyle x \phi ^ { \prime } ( \sqrt { B - 1 } x ) = \frac { 1 } { \sqrt { B - 1 } } \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } x C _ { l } ^ { ( \frac { B - 3 } { 2 } ) \prime } ( x ) } \ ~ } \\ { { \displaystyle ~ = \frac { 1 } { \sqrt { B - 1 } } \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } \left( l C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) + \sum _ { j \in [ l - 2 ] _ { 2 } } ( 2 j + B - 3 ) C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) \right) } } \\ { { \displaystyle ~ = \frac { 1 } { \sqrt { B - 1 } } \sum _ { j = 0 } ^ { \infty } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) \left( a _ { j } j + \sum _ { l \in [ j + 2 , \infty ] _ { 2 } } a _ { l } \frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } ( 2 j + B - 3 ) \right) } } \end{array} +$$ + +Then + +$$ +\begin{array} { r l } & { \quad \underset { v \in S ^ { B - 2 } } { \mathbb { E } } ( u _ { 1 } \cdot v ) ( u _ { 2 } \cdot v ) \phi ^ { \prime } ( \sqrt { B - 1 } u _ { 1 } \cdot v ) \phi ^ { \prime } ( \sqrt { B - 1 } u _ { 2 } \cdot v ) } \\ & { = \frac { 1 } { B - 1 } { \displaystyle \sum _ { j = 0 } ^ { \infty } } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) \left( a _ { j } j + { \displaystyle \sum _ { l \in [ j + 2 , \infty ] _ { 2 } } } a _ { l } \frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } ( 2 j + B - 3 ) \right) ^ { 2 } } \end{array} +$$ + +For $l < m$ and $m - l$ even, the coefficient of $a _ { l } a _ { m }$ is + +$$ +\begin{array} { l } { \displaystyle \frac { 1 } { B - 1 } \left( 2 l ( 2 l + B - 3 ) \frac { c _ { B - 1 , l } } { c _ { B - 1 , m } } c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) \right. } \\ { \displaystyle \qquad \left. + 2 \sum _ { j \in [ l - 2 ] _ { 2 } } \frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } \frac { c _ { B - 1 , j } } { c _ { B - 1 , m } } ( 2 j + B - 3 ) ^ { 2 } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) \right) } \\ { \displaystyle = 2 c _ { B - 1 , l } ^ { - 1 } c _ { B - 1 , m } ^ { - 1 } \frac { B - 3 } { B - 1 } \left( l C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) + \sum _ { j \in [ l - 2 ] _ { 2 } } ( 2 j + B - 3 ) C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) \right) } \end{array} +$$ + +For each $l$ , the coefficient of $a _ { l } ^ { 2 }$ is + +$$ +\begin{array} { r l r } & { } & { \frac { 1 } { B - 1 } \left( l ^ { 2 } c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) + \displaystyle \sum _ { j \in [ l - 2 ] _ { 2 } } \left( \frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } \right) ^ { 2 } ( 2 j + B - 3 ) ^ { 2 } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) \right. } \\ & { } & { \left. = c _ { B - 1 , l } ^ { - 2 } \frac { B - 3 } { B - 1 } \left( \frac { l ^ { 2 } } { 2 l + B - 3 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) + \displaystyle \sum _ { j \in [ l - 2 ] _ { 2 } } ( 2 j + B - 3 ) C _ { j } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) \right) ( 4 2 ) \right. } \end{array} +$$ + +By Lemma E.49 and Eqs. (39) and (41), the coefficient of $a _ { l } a _ { m }$ with $m > l$ in $\mathbb { E } _ { v \in S ^ { B - 2 } } ( u _ { 1 } \cdot u _ { 2 } - ( u _ { 1 }$ $v ) ( u _ { 2 } \cdot v ) ) \phi ^ { \prime } ( \sqrt { B - 1 } u _ { 1 } \cdot v ) \phi ^ { \prime } ( \sqrt { B - 1 } u _ { 2 } \cdot v )$ is 0. Similarly, by Lemma E.49 and Eqs. (40) and (42), the coefficient of $a _ { l } ^ { 2 }$ is $c _ { B - 1 , l } ^ { - 2 } \textstyle { \frac { { B } - 3 } { { B } - 1 } } \textstyle { \frac { l + B - 3 } { 2 l + B - 3 } } l C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } ) = c _ { B - 1 , l } ^ { - 1 } \textstyle { \frac { l + B - 3 } { B - 1 } } l C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( u _ { 1 } \cdot u _ { 2 } )$ . + +# E.5 SYMMETRY AND ULTRASYMMETRY + +As remarked before, all commonly used nonlinearities ReLU, tanh, and so on induce the dynamics Eq. (43) to converge to BSB1 fixed points, which we formally define as follows. + +Definition E.50. We say a matrix $\Sigma \in S _ { B }$ is BSB1 (short for “1-Block Symmetry Breaking”) if $\Sigma$ has one common entry on the diagonal and one common entry on the off-diagonal, i.e. + +$$ +\left( \begin{array} { c c c c } { { a } } & { { b } } & { { b } } & { { \ldots } } \\ { { b } } & { { a } } & { { b } } & { { \ldots } } \\ { { b } } & { { b } } & { { a } } & { { \ldots } } \\ { { \vdots } } & { { \vdots } } & { { \vdots } } & { { \ddots } } \end{array} \right) +$$ + +We will denote such a matrix as $\mathrm { B S B 1 } ( a , b )$ . Note that $\mathrm { B S B 1 } ( a , b )$ can be written as $( a - b ) I + b \mathbf { 1 } \mathbf { 1 } ^ { T }$ . +Its spectrum is given below. + +Lemma E.51. A $B \times B$ matrix of the form $\mu I + \nu \mathbf { 1 1 } ^ { T }$ has two eigenvalues $\mu$ and $B \nu + \mu$ , each with eigenspaces $\textstyle \{ x : \sum _ { i } x _ { i } = 0 \}$ and $\{ x : x _ { 1 } = \cdot \cdot \cdot = x _ { B } \}$ . Equivalently, if it has a on the diagonal and b on the off-diagonal, then the eigenvalues are $a - b$ and $( B - 1 ) b + a$ . + +The following simple lemma will also be very useful + +Lemma E.52. Let $\Sigma : = \mathrm { B S B } 1 _ { B } ( a , b )$ . Then $G \Sigma G = ( a - b ) G .$ . + +Proof. $G$ and $\mathrm { B S B } 1 _ { B } ( a , b )$ can be simultaneously diagonalized by Lemma E.51. Note that $G$ zeros out the eigenspace $\mathbb { R } \mathbf { 1 }$ , and is identity on its orthogonal complement. The result then follows from easy computations. □ + +The symmetry of BSB1 covariance matrices, especially the fact that they are isotropic in a codimension 1 subspace, is crucial to much of our analysis. + +Ultrasymmetry. Indeed, as mentioned before, when investigating the asymptotics of the forward or backward dynamics, we encounter 4-tensor objects, such as $\mathrm { \frac { \bar { d } V _ { } } { d \Sigma } }$ or $\mathrm { V } _ { B _ { \phi } ^ { \prime } }$ , that would seem daunting to analyze at first glance. However, under the BSB1 fixed point assumption, these 4- tensors contain considerable symmetry that somehow makes this possible. We formalize the notion of symmetry arising from such scenarios: + +Definition E.53. Let $\mathcal { T } : \mathbb { R } ^ { B \times B } \mathbb { R } ^ { B \times B }$ be a linear operator. Let $\delta _ { i } \in \mathbb { R } ^ { B }$ be the vector with 0 everywhere except 1 in coordinate $i$ ; then $\delta _ { i } \delta _ { j } ^ { T }$ is the matrix with 0 everywhere except 1 in position $( i , j )$ . Write $[ k l | i j ] : = \mathcal { T } ( \delta _ { i } \delta _ { j } ^ { T } ) _ { k l }$ . Suppose $\tau$ has the property that for all $i , j , k , l \in [ B ] =$ $\{ 1 , \ldots , B \}$ + +• $[ k l | i j ] = [ \pi ( k ) \pi ( l ) | \pi ( i ) \pi ( j ) ]$ for all permutation $\pi$ on $[ B ]$ , and • $[ i j | k l ] = [ j i | l k ]$ . + +Then we say $\tau$ is ultrasymmetric. + +Remark E.54. In what follows, we will often “normalize” the representation “ $[ i j | k l ]$ ” to the unique “ $[ i ^ { \prime } j ^ { \prime } | k ^ { \prime } l ^ { \prime } ] ^ { \cdot }$ ” that is in the same equivalence class according to Defn E.53 and such that $i ^ { \prime } , j ^ { \prime } , k ^ { \prime } , l ^ { \bar { \prime } } \in$ [4] and $i ^ { \prime } \overset { \cdot } { \leq } j ^ { \prime } , k ^ { \prime } \overset { } { \leq } l ^ { \prime }$ unless $i ^ { \prime } = l ^ { \prime } , j ^ { \prime } = k ^ { \prime }$ , in which case the normalization is [12|21]. Explicitly, we have the following equivalence classes and their normalized representations + +We will study the eigendecomposition of an ultrasymmetric $\tau$ as well as the projection $G ^ { \otimes 2 } \circ \mathcal { T } \circ$ $G ^ { \otimes 2 } : \Sigma \mapsto { \bar { G } } ( { \mathcal { T } } \{ { \bar { G } } \Sigma G \} ) G$ , respectively with the following domains + +Definition E.55. Denote by $\mathcal { H } _ { B }$ the space of symmetric matrices of dimension $B$ . Also write $\mathcal { H } _ { B } ^ { G }$ for the space of symmetric matrices $\Sigma$ of dimension $B$ such that $G \Sigma G = \Sigma$ (which is equivalent to saying rows of $\Sigma$ sum up to 0). + +As in the case of $s$ , we omit subscript $B$ when it’s clear from context. + +The following subspaces of $\mathcal { H } _ { B }$ and $\mathcal { H } _ { B } ^ { G }$ will turn out to comprise the eigenspaces of $G ^ { \otimes 2 } \circ T \circ G ^ { \otimes 2 }$ . + +
class repr.
i=j=k=li=j=kori=j=li=jand k=l1111]
[1112
[11]22
i=j[11|23/
i=k=lorj=k=l1211j
i=k and j=l[12]|12]
i=kori=l[12|[13]
i=land j= k12|21j
k=l1233
all different1234
+ +Definition E.56. Let $\mathbb { L } _ { B } : = \{ G D G : D$ diagonal, $\mathrm { t r } D = 0 \} \subseteq \mathcal { H } _ { B } ^ { G }$ and $\mathbb { M } _ { B } : = \{ \Sigma \in \mathcal { H } _ { B } ^ { G } :$ $\mathrm { D i a g } \Sigma = 0 \}$ . Note that $\begin{array} { r } { \dim \mathbb { L } _ { B } = B - 1 , \dim \mathbb { M } _ { B } = \frac { B ( B - 3 ) } { 2 } } \end{array}$ and $\mathbb { R } G ^ { B } \oplus \mathbb { L } _ { B } \oplus \mathbb { M } _ { B } = \mathcal { H } _ { B } ^ { G }$ is an orthogonal decomposition w.r.t Frobenius inner product. + +For the eigenspaces of $\tau$ , we also need to define + +Definition E.57. For any nonzero $a , b \in \mathbb { R }$ , set + +$$ +\begin{array} { r } { \mathrm { L } _ { \boldsymbol { B } } ( \boldsymbol { a } , \boldsymbol { b } ) : = \left( \begin{array} { l l l l l } { \boldsymbol { a } } & { \boldsymbol { 0 } } & { - b } & { - b } & { \cdots } \\ { \boldsymbol { 0 } } & { - \boldsymbol { a } } & { \boldsymbol { b } } & { \boldsymbol { b } } & { \cdots } \\ { - \boldsymbol { b } } & { \boldsymbol { b } } & { \boldsymbol { 0 } } & { \boldsymbol { 0 } } & { \cdots } \\ { - \boldsymbol { b } } & { \boldsymbol { b } } & { \boldsymbol { 0 } } & { \boldsymbol { 0 } } & { \cdots } \\ { \vdots } & { \vdots } & { \vdots } & { \vdots } & { \ddots } \end{array} \right) \in \mathcal { H } _ { \boldsymbol { B } } . } \end{array} +$$ + +In general, we say a matrix $M$ is $\operatorname { L } _ { B } ( a , b )$ -shaped if $M = { \cal P } \mathrm { L } _ { B } ( a , b ) { \cal P } ^ { T }$ for some permutation matrix $P$ . + +Note that + +Proposition E.58. $\operatorname { L } ( B - 2 , 1 )$ -shaped matrices span $\mathbb { L } _ { B }$ + +$$ +\begin{array} { r l } { { \mathrm { L } ( B - 2 , 1 ) = \frac { 1 } { B } [ ( - B + 1 , 1 , 1 , \ldots , 1 ) ^ { \otimes 2 } - ( 1 , - B + 1 , 1 , \ldots , 1 ) ^ { \otimes 2 } ] = } } & { { } } \\ { { \otimes ^ { 2 } - ( 0 , 1 , 0 , \ldots , 0 ) ^ { \otimes 2 } ] G } } & { { } } \end{array} +$$ + +Proof. Straightforward computation. + +Lemma E.60. Let ${ \cal L } = \mathrm { L } _ { B } ( a , b )$ . Then $\begin{array} { r } { G ^ { \otimes 2 } \{ L \} = G L G = \frac { a + 2 b } { B } \mathrm { L } _ { B } ( B - 2 , 1 ) . } \end{array}$ + +Proof. $G L G$ can be written as the sum of outer products + +$$ +\begin{array} { r l } & { G L G = a G _ { 1 } \otimes G _ { 1 } - a G _ { 2 } \otimes G _ { 2 } + \underset { i = 3 } { \overset { B } { \sum } } - b G _ { 1 } \otimes G _ { i } + b G _ { 2 } \otimes G _ { i } - b G _ { 4 } \otimes G _ { 1 } + b G _ { 4 } \otimes G _ { 2 } } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{array} +$$ + +We are now ready to discuss the eigendecomposition of ultrasymmetric operators. + +Theorem E.61 (Eigendecomposition of an ultrasymmetric operator). Let $\mathcal { T } : \mathbb { R } ^ { B \times B } \mathbb { R } ^ { B \times B }$ be an ultrasymmetric linear operator. Then $\tau$ has the following eigendecomposition. + +1. Two $^ { l }$ -dimensional eigenspace $\mathbb { R } \cdot \mathrm { B S B } 1 ( \lambda _ { \mathrm { B S B } 1 , i } ^ { \mathcal { T } } - \alpha _ { 2 2 } , \alpha _ { 2 1 } )$ with eigenvalue $\lambda _ { \mathrm { B S B } 1 , i } ^ { \mathcal { T } }$ for $i = 1 , 2$ , where + +$$ +\begin{array} { l } { \alpha _ { 1 1 } = [ 1 1 | 1 1 ] + [ 1 1 | 2 2 ] ( B - 1 ) } \\ { \alpha _ { 1 2 } = 2 ( B - 1 ) [ 1 1 | 1 2 ] + ( B - 2 ) ( B - 1 ) [ 1 1 | 2 3 ] } \\ { \alpha _ { 2 1 } = 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] } \\ { \alpha _ { 2 2 } = [ 1 2 | 1 2 ] + 4 ( B - 2 ) [ 1 2 | 1 3 ] + [ 1 2 | 2 1 ] + ( B - 2 ) ( B - 3 ) [ 1 2 } \end{array} +$$ + +and $\lambda _ { \mathrm { B S B 1 , 1 } } ^ { \mathcal { T } }$ and $\lambda _ { \mathrm { B S B 1 , 2 } } ^ { \mathcal { T } }$ are the roots to the quadratic + +$$ +x ^ { 2 } - ( \alpha _ { 1 1 } + \alpha _ { 2 2 } ) x + \alpha _ { 1 1 } \alpha _ { 2 2 } - \alpha _ { 1 2 } \alpha _ { 2 1 } . +$$ + +2. Two $( B - 1 )$ -dimensional eigenspaces $\mathfrak { S } _ { B } \cdot \mathrm { L } ( \lambda _ { \mathbb { L } , i } ^ { \mathcal { T } } - \beta _ { 2 2 } , \beta _ { 2 1 } )$ with eigenvalue $\lambda _ { \mathbb { L } , i } ^ { \mathcal { T } }$ for $i = 1 , 2$ . Here $\mathfrak { S } _ { B } \cdot W$ denotes the linear span of the orbit of matrix $W$ under simultaneous permutation of its column and rows (by the same permutation), and + +$$ +\begin{array} { r l } & { \beta _ { 1 1 } = [ 1 1 | 1 1 ] - [ 1 1 | 2 2 ] } \\ & { \beta _ { 1 2 } = 2 ( B - 2 ) ( [ 1 1 | 2 3 ] - [ 1 1 | 1 2 ] ) } \\ & { \beta _ { 2 1 } = - [ 1 2 | 1 1 ] + [ 1 2 | 3 3 ] } \\ & { \beta _ { 2 2 } = [ 1 2 | 2 1 ] + [ 1 2 | 1 2 ] + 2 ( B - 4 ) [ 1 2 | 1 3 ] - 2 ( B - 3 ) [ 1 2 | 3 4 ] . } \end{array} +$$ + +and λT and $\lambda _ { \mathbb { L } , 2 } ^ { \mathcal { T } }$ are the roots to the quadratic + +$$ +x ^ { 2 } - ( \beta _ { 1 1 } + \beta _ { 2 2 } ) x + \beta _ { 1 1 } \beta _ { 2 2 } - \beta _ { 1 2 } \beta _ { 2 1 } . +$$ + +3. Eigenspace M (dimension $B ( B \mathrm { ~ - ~ } 3 ) / 2 )$ with eigenvalue $\lambda _ { \mathbb { M } } ^ { \mathcal { T } } : = [ 1 2 | 1 2 ] + [ 1 2 | 2 1 ] -$ $4 [ 1 2 | 1 3 ] + 2 [ 1 2 | 3 4 ]$ . + +The proof is by careful, but ultimately straightforward, computation. + +Proof. We will use the bracket notation of Defn E.53 to denote entries of $\tau$ , and implicitly simplify it according to Remark E.54. + +Item 1. Let $U \in \mathbb { R } ^ { B \times B }$ be the BSB1 matrix. By ultrasymmetry of $\tau$ and BSB1 symmetry of $A$ $\tau \{ U \}$ is also BSB1. So we proceed to calculate the diagonal and off-diagonal entries of ${ \mathcal { T } } \{ G \}$ . + +We have + +$$ +\begin{array} { r l } & { \Gamma \{ \mathrm { B S B 1 } ( a , b ) \} _ { 1 1 } = [ 1 1 | 1 1 ] a + 2 ( B - 1 ) [ 1 1 | 2 ] b + [ 1 1 | 2 2 ] ( B - 1 ) a + [ 1 1 | 2 3 ] ( B - 2 ) ( B - 1 ) b } \\ & { \qquad = ( [ 1 1 | 1 1 ] + [ 1 1 | 2 ] ( B - 1 ) ) a + ( 2 ( B - 1 ) [ 1 1 | 1 2 ] + ( B - 2 ) ( B - 1 ) [ 1 1 | 2 3 ] ) b } \\ & { \Gamma \{ \mathrm { B S B 1 } ( a , b ) \} _ { 1 2 } = [ 1 2 | 1 2 ] b + 2 [ 1 2 | 1 1 ] a + ( B - 2 ) [ 1 2 | 3 3 ] a + 2 ( B - 2 ) [ 1 2 | 1 3 ] b } \\ & { \qquad + [ 1 2 | 2 1 ] b + 2 ( B - 2 ) [ 1 2 | 2 3 ] b + ( B - 2 ) ( B - 3 ) [ 1 2 | 3 4 ] b } \\ & { \qquad = ( 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] ) a } \\ & { \qquad + ( [ [ 1 2 | 1 2 ] + 2 ( B - 2 ) [ 1 2 | 1 2 ] + [ 1 2 | 2 1 ] + 2 ( B - 2 ) [ 1 2 | 1 3 ] + ( B - 2 ) ( B - 3 ) } \\ & { \qquad = ( 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] ) a } \\ & { \qquad + ( [ [ 1 2 | 1 2 ] + 4 ( B - 2 ) [ 1 2 | 1 2 ] + [ 1 2 | 2 1 ] + ( B - 2 ) ( B - 3 ) [ 1 2 | 3 4 ] ) b } \end{array} +$$ + +Thus $\mathrm { B S B 1 } ( \omega _ { 1 } , \gamma _ { 1 } )$ and $\mathrm { B S B 1 } ( \omega _ { 2 } , \gamma _ { 2 } )$ are the eigenmatrices of $\tau$ , where $( \omega _ { 1 } , \gamma _ { 1 } )$ and $( \omega _ { 2 } , \gamma _ { 2 } )$ are the eigenvectors of the matrix + +$$ +\left( \begin{array} { c c } { { \alpha _ { 1 1 } } } & { { \alpha _ { 1 2 } } } \\ { { \alpha _ { 2 1 } } } & { { \alpha _ { 2 2 } } } \end{array} \right) +$$ + +with + +$$ +\begin{array} { l } { \alpha _ { 1 1 } = [ 1 1 | 1 1 ] + [ 1 1 | 2 2 ] ( B - 1 ) } \\ { \alpha _ { 1 2 } = 2 ( B - 1 ) [ 1 1 | 1 2 ] + ( B - 2 ) ( B - 1 ) [ 1 1 | 2 3 ] } \\ { \alpha _ { 2 1 } = 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] } \\ { \alpha _ { 2 2 } = [ 1 2 | 1 2 ] + 4 ( B - 2 ) [ 1 2 | 1 3 ] + [ 1 2 | 2 1 ] + ( B - 2 ) ( B - 3 ) [ 1 2 | } \end{array} +$$ + +The eigenvalues are the two roots $\lambda _ { \mathrm { B S B 1 , 1 } } ^ { \mathcal { T } } , \lambda _ { \mathrm { B S B 1 , 2 } } ^ { \mathcal { T } }$ to the quadratic + +$$ +x ^ { 2 } - ( \alpha _ { 1 1 } + \alpha _ { 2 2 } ) x + \alpha _ { 1 1 } \alpha _ { 2 2 } - \alpha _ { 1 2 } \alpha _ { 2 1 } +$$ + +and the corresponding eigenvectors are + +$$ +\begin{array} { r } { ( \omega _ { 1 } , \gamma _ { 1 } ) = ( \lambda _ { 1 } - \alpha _ { 2 2 } , \alpha _ { 2 1 } ) } \\ { ( \omega _ { 2 } , \gamma _ { 2 } ) = ( \lambda _ { 2 } - \alpha _ { 2 2 } , \alpha _ { 2 1 } ) . } \end{array} +$$ + +Item 2. We will study the image of $\operatorname { L } _ { B } ( a , b )$ (Defn E.57) under $\tau$ . We have + +$$ +\begin{array} { r l } & { \mathcal { T } [ \{ \Delta \phi _ { i } \} ] _ { n } = - \mathcal { T } [ \{ \Delta \phi _ { i } \} ] _ { n } } \\ & { \qquad - \{ [ 1 1 1 \{ 1 1 \{ 1 9 \} - 2 \{ 1 2 \} \} \Delta \phi - 2 2 \Lambda [ 1 1 2 \phi _ { i } ] \{ 1 2 \{ 1 2 \phi _ { i } - 8 \} + 1 1 2 4 \{ 2 \Lambda ^ { 2 } - 2 \} \Lambda \{ 1 1 2 \phi _ { i } \} \} } \\ & { \qquad - \{ [ 1 1 1 \{ 1 1 \{ 1 9 \} - 2 \{ 1 5 \} \} - 6 \Lambda ^ { 2 } - 2 \Lambda ^ { 2 } \{ 1 1 \} \{ 1 9 \} \Phi - 2 \Lambda ^ { 2 } \{ 1 1 \} \{ 1 1 2 \} \Phi - \delta ] } \\ & { \qquad - \mathcal { T } [ \{ \Delta \phi _ { i } \} ] _ { n } } \\ & { \mathcal { T } [ \{ \Delta \phi _ { i } , \phi _ { i } \} ] _ { n } = - \mathcal { T } [ \{ \Delta \phi _ { i } \} ] _ { n } } \\ & \qquad - \{ [ 2 1 1 \{ 0 \} - 1 2 \{ 1 2 \} \{ 1 9 \} + 1 2 1 \} \\ & { \qquad \mathcal { T } [ \{ \Delta \phi _ { i } \} ] _ { n } = \delta } \\ & { \qquad - \{ [ 1 1 1 \{ 1 1 \{ 1 \} \{ 1 1 ^ { 2 } \{ \Delta \phi _ { i } - 3 \phi _ { i } \} \} ] \qquad \mathrm { i . e . } \Lambda ^ { \prime } [ \{ \Delta \phi _ { i } \} ] _ { n } + \delta ^ { \prime } [ \{ \Delta \phi _ { i } \} ] _ { n } \{ 1 2 \{ 2 \Lambda ^ { 2 } - 1 6 \} \Lambda ^ { 2 } \{ 1 2 \phi _ { i } - 3 \phi _ { i } \} } \\ & \qquad - \{ [ 1 1 1 \{ 1 1 \{ 1 \} \{ 2 \Lambda ^ { 2 } \} \} ] \sin [ 2 \Lambda ^ { 2 } \phi _ { i } \} \} \\ & { \qquad - \mathcal { T } [ \{ \Delta \phi _ { i } \} ] _ { n } } \\ & \mathcal { T } [ \{ \Delta \phi _ { i } \} ] _ { n } = - \mathcal { T } [ \{ \Delta \phi _ { i } \} ] _ \end{array} +$$ + +Thus $\operatorname { L } ( a , b )$ transforms under $\tau$ by the matrix + +$$ +\operatorname { L } ( a , b ) \mapsto \operatorname { L } ( { \binom { \beta _ { 1 1 } } { \beta _ { 2 1 } } } \quad \beta _ { 1 2 } ) \left( { a \atop b } \right) ) +$$ + +with + +$$ +\begin{array} { r l } & { \beta _ { 1 1 } = [ 1 1 | 1 1 ] - [ 1 1 | 2 2 ] } \\ & { \beta _ { 1 2 } = 2 ( B - 2 ) ( [ 1 1 | 2 3 ] - [ 1 1 | 1 2 ] ) } \\ & { \beta _ { 2 1 } = - [ 1 2 | 1 1 ] + [ 1 2 | 3 3 ] } \\ & { \beta _ { 2 2 } = [ 1 2 | 2 1 ] + [ 1 2 | 1 2 ] + 2 ( B - 4 ) [ 1 2 | 1 3 ] - 2 ( B - 3 ) [ 1 2 | 3 4 ] . } \end{array} +$$ + +So if $\lambda _ { \mathbb { L } , 1 } ^ { \mathcal { T } }$ and $\lambda _ { \mathbb { L } , 2 } ^ { T }$ are the roots of the equation + +$$ +x ^ { 2 } - ( \beta _ { 1 1 } + \beta _ { 2 2 } ) x + \beta _ { 1 1 } \beta _ { 2 2 } - \beta _ { 1 2 } \beta _ { 2 1 } +$$ + +then + +$$ +\begin{array} { r } { \mathcal { T } \{ \mathrm { L } ( \lambda _ { \mathbb { L } , 1 } ^ { \mathcal { T } } - \beta _ { 2 2 } , \beta _ { 2 1 } ) \} = \lambda _ { \mathbb { L } , 1 } ^ { \mathcal { T } } \mathrm { L } ( \lambda _ { \mathbb { L } , 1 } ^ { \mathcal { T } } - \beta _ { 2 2 } , \beta _ { 2 1 } ) } \\ { \mathcal { T } \{ \mathrm { L } ( \lambda _ { \mathbb { L } , 2 } ^ { \mathcal { T } } - \beta _ { 2 2 } , \beta _ { 2 1 } ) \} = \lambda _ { \mathbb { L } , 2 } ^ { \mathcal { T } } \mathrm { L } ( \lambda _ { \mathbb { L } , 2 } ^ { \mathcal { T } } - \beta _ { 2 2 } , \beta _ { 2 1 } ) } \end{array} +$$ + +Similarly, any image of these eigenvectors under simultaneous permutation of rows and columns remains eigenvectors with the same eigenvalue. This derives Item 2. + +Item 3. Let $M \in \mathbb { M }$ . We first show that $\tau \{ M \}$ has zero diagonal. We have + +$$ +\begin{array} { l } { { \displaystyle { \mathcal T } \{ M \} _ { 1 1 } = [ 1 1 | 1 2 ] ( \sum _ { i = 2 } ^ { B } M _ { 1 i } + M _ { i 1 } ) + [ 1 1 | 2 3 ] ( \sum _ { i , j = 1 } ^ { B } M _ { i j } - \left( \sum _ { i = 2 } ^ { B } M _ { 1 i } + M _ { i 1 } \right) ) } } \\ { { \displaystyle \quad \quad = 0 + 0 = 0 } } \end{array} +$$ + +which follows from $M { \bf 1 } = 0$ by definition of $\mathbb { M }$ . Similarly $\mathcal { T } \{ M \} _ { i i } = 0$ for all $i$ . + +Now we show that $M$ is an eigenmatrix. + +$$ +\begin{array} { l } { { r \{ M \} _ { 1 2 } = [ 1 2 | 1 2 ] M _ { 1 2 } + [ 1 2 | 1 1 ] 0 + [ 1 2 | 3 3 ] 0 + [ 1 2 | 1 3 ] \left( \displaystyle \sum _ { i = 3 } ^ { B } M _ { 1 i } + \displaystyle \sum _ { i = 1 } ^ { B } M _ { i 2 } \right) } } \\ { { \qquad + [ 1 2 | 2 1 ] M _ { 2 1 } + [ 1 2 | 3 1 ] \left( \displaystyle \sum _ { i = 3 } ^ { B } M _ { i \mathrm { i } } + \displaystyle \sum _ { i = 1 } ^ { B } M _ { 2 i } \right) + [ 1 2 | 3 4 ] \displaystyle \sum _ { \qquad i \ge 3 , j \ge 3 } M _ { i j } } } \\ { { \qquad = [ 1 2 | 1 2 ] M _ { 1 2 } - 2 M _ { 1 2 } [ 1 2 | 1 3 ] + M _ { 2 1 } [ 1 2 | 2 1 ] + [ 1 2 | 3 1 ] ( - 2 M _ { 1 2 } ) + [ 1 2 | 3 4 ] ( \displaystyle \sum _ { i = 3 } ^ { B } M _ { \mathrm { i } 1 } - M _ { \mathrm { i } } } } \\ { { \qquad = M _ { 1 2 } ( [ 1 2 | 1 2 ] - 2 | 1 2 | 3 ] + [ 1 2 | 2 1 ] - 2 [ 1 2 | 3 1 ] + 2 [ 1 2 | 3 4 ] ) } } \\ { { \qquad = M _ { 1 2 } ( [ 1 2 | 1 2 ] + [ 1 2 | 2 1 ] - 4 | 1 2 | 3 | + 2 [ 1 2 | 3 4 ] ) } } \\ { { \qquad = M _ { 1 2 } M _ { 1 2 } } } \end{array} +$$ + +Similarly $\mathcal { T } \{ M \} _ { i j } = \lambda _ { \mathbb { M } } ^ { \mathcal { T } } M _ { i j }$ for all $i \neq j$ . + +Note that the eigenspaces described above are in general not orthogonal under trace inner product, so $\tau$ is not self-adjoint (relative to the trace inner product) typically. However, as we see next, after projection by $G ^ { \otimes 2 }$ , it is self-adjoint. + +Theorem E.62 (Eigendecomposition of a projected ultrasymmetric operator). Let $\mathcal { T } : \mathbb { R } ^ { B \times B } $ $\mathbb { R } ^ { B \times B }$ be an ultrasymmetric linear operator. We write $\dot { G } ^ { \otimes 2 } \circ \mathcal { T } \ \uparrow \ \dot { \mathcal { H } } _ { B } ^ { G }$ for the operator $\mathcal { H } _ { B } ^ { G } $ $\mathcal { H } _ { B } ^ { G } , \Sigma \mapsto \mathcal { T } \{ \Sigma \} \mapsto G ( \mathcal { T } \{ \Sigma \} ) G$ , restricted to $\Sigma \in \mathcal { H } _ { B } ^ { G }$ . Then $G ^ { \otimes 2 } \circ \mathcal { T } \ \ u { \mid } \ \mathcal { H } _ { B } ^ { G }$ has the following eigendecomposition. + +1. Eigenspace $\mathbb { R } G$ with eigenvalue $\lambda _ { G } ^ { G , 7 } \ : = \ B ^ { - 1 } ( ( B - 1 ) ( \alpha _ { 1 1 } - \alpha _ { 2 1 } ) - ( \alpha _ { 1 2 } - \alpha _ { 2 2 } ) ) ,$ where as in Thm E.61, + +$$ +\begin{array} { l } { \alpha _ { 1 1 } = [ 1 1 | 1 1 ] + [ 1 1 | 2 2 ] ( B - 1 ) } \\ { \alpha _ { 1 2 } = 2 ( B - 1 ) [ 1 1 | 1 2 ] + ( B - 2 ) ( B - 1 ) [ 1 1 | 2 3 ] } \\ { \alpha _ { 2 1 } = 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] } \\ { \alpha _ { 2 2 } = [ 1 2 | 1 2 ] + 4 ( B - 2 ) [ 1 2 | 1 3 ] + [ 1 2 | 2 1 ] + ( B - 2 ) ( B - 3 ) [ 1 2 | 3 4 ] } \end{array} +$$ + +2. Eigenspace L with eigenvalue $\lambda _ { \mathbb { L } } ^ { G , \mathcal { T } } : = B ^ { - 1 } ( ( B - 2 ) \beta _ { 1 1 } + \beta _ { 1 2 } + 2 ( B - 2 ) \beta _ { 2 1 } + 2 \beta _ { 2 2 } ) ,$ where as in Thm E.61, + +$$ +\begin{array} { r l } & { \beta _ { 1 1 } = [ 1 1 | 1 1 ] - [ 1 1 | 2 2 ] } \\ & { \beta _ { 1 2 } = 2 ( B - 2 ) ( [ 1 1 | 2 3 ] - [ 1 1 | 1 2 ] ) } \\ & { \beta _ { 2 1 } = - [ 1 2 | 1 1 ] + [ 1 2 | 3 3 ] } \\ & { \beta _ { 2 2 } = [ 1 2 | 2 1 ] + [ 1 2 | 1 2 ] + 2 ( B - 4 ) [ 1 2 | 1 3 ] - 2 ( B - 3 ) [ 1 2 | 3 4 ] . } \end{array} +$$ + +3. Eigenspace M with eigenvalue $\lambda _ { \mathbb { M } } ^ { G , \mathcal { T } } : = \lambda _ { \mathbb { M } } ^ { \mathcal { T } } = [ 1 2 | 1 2 ] + [ 1 2 | 2 1 ] - 4 [ 1 2 | 1 3 ] + 2 [ 1 2 | 3 4 ] .$ + +Proof. Item 1 As in the proof of $\mathrm { T h m } \mathrm { E } . 6 1 $ , we find + +$$ +\mathcal { T } \{ \mathrm { B S B 1 } ( a , b ) \} = \mathrm { B S B 1 } \left( \left( \begin{array} { l l } { \alpha _ { 1 1 } } & { \alpha _ { 1 2 } } \\ { \alpha _ { 2 1 } } & { \alpha _ { 2 2 } } \end{array} \right) \left( \begin{array} { l } { a } \\ { b } \end{array} \right) \right) +$$ + +where + +$$ +\begin{array} { l } { \alpha _ { 1 1 } = [ 1 1 | 1 1 ] + [ 1 1 | 2 2 ] ( B - 1 ) } \\ { \alpha _ { 1 2 } = 2 ( B - 1 ) [ 1 1 | 1 2 ] + ( B - 2 ) ( B - 1 ) [ 1 1 | 2 3 ] } \\ { \alpha _ { 2 1 } = 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] } \\ { \alpha _ { 2 2 } = [ 1 2 | 1 2 ] + 4 ( B - 2 ) [ 1 2 | 1 3 ] + [ 1 2 | 2 1 ] + ( B - 2 ) ( B - 3 ) [ 1 2 | 3 4 ] . } \end{array} +$$ + +For $a = B - 1 , b = - 1$ so that $\mathrm { B S B } 1 ( B - 1 , - 1 ) = B G$ , we get + +$$ +\begin{array} { r l } & { \qquad \mathcal { T } \{ \mathrm { B S B } 1 ( B - 1 , - 1 ) \} = \mathrm { B S B } 1 ( ( B - 1 ) \alpha _ { 1 1 } - \alpha _ { 1 2 } , ( B - 1 ) \alpha _ { 2 1 } - \alpha _ { 2 2 } ) } \\ & { \qquad \boldsymbol { G } ^ { \otimes 2 } \circ \mathcal { T } \{ \mathrm { B S B } 1 ( B - 1 , - 1 ) \} = G \mathrm { B S B } 1 \left( ( B - 1 ) \alpha _ { 1 1 } - \alpha _ { 1 2 } , ( B - 1 ) \alpha _ { 2 1 } - \alpha _ { 2 2 } \right) \ G } \\ & { \qquad = \left( ( B - 1 ) ( \alpha _ { 1 1 } - \alpha _ { 2 1 } ) - ( \alpha _ { 1 2 } - \alpha _ { 2 2 } ) \right) G } \\ & { \qquad = B ^ { - 1 } \left( ( B - 1 ) ( \alpha _ { 1 1 } - \alpha _ { 2 1 } ) - ( \alpha _ { 1 2 } - \alpha _ { 2 2 } ) \right) \mathrm { B S B } 1 ( B - 1 , - 1 ) } \\ & { \qquad = \lambda _ { G } ^ { G } \mathcal { T } \mathrm { B S B } 1 ( B - 1 , - 1 ) } \end{array} +$$ + +by Lemma E.52. + +Item 2. It suffices to show that $\mathrm { L } ( B - 2 , 1 )$ is an eigenmatrix with the eigenvalue $\lambda _ { \mathbb { L } } ^ { G , T }$ + +As in the proof of Thm E.61, we find + +$$ +{ \mathcal { T } } \{ \mathrm { L } ( a , b ) \} = \mathrm { L } \left( \left( { \begin{array} { c c } { \beta _ { 1 1 } } & { \beta _ { 1 2 } } \\ { \beta _ { 2 1 } } & { \beta _ { 2 2 } } \end{array} } \right) { \binom { a } { b } } \right) +$$ + +where + +$$ +\begin{array} { r l } & { \beta _ { 1 1 } = [ 1 1 | 1 1 ] - [ 1 1 | 2 2 ] } \\ & { \beta _ { 1 2 } = 2 ( B - 2 ) ( [ 1 1 | 2 3 ] - [ 1 1 | 1 2 ] ) } \\ & { \beta _ { 2 1 } = - [ 1 2 | 1 1 ] + [ 1 2 | 3 3 ] } \\ & { \beta _ { 2 2 } = [ 1 2 | 2 1 ] + [ 1 2 | 1 2 ] + 2 ( B - 4 ) [ 1 2 | 1 3 ] - 2 ( B - 3 ) [ 1 2 | 3 4 ] . } \end{array} +$$ + +So with $a = B - 2 , b = 1$ , we have + +$$ +\begin{array} { r l } & { \quad \mathcal { T } \{ \mathrm { L } ( B - 2 , 1 ) \} = \mathrm { L } ( ( B - 2 ) \beta _ { 1 1 } + \beta _ { 1 2 } , ( B - 2 ) \beta _ { 2 1 } + \beta _ { 2 2 } ) } \\ & { \quad G ^ { \otimes 2 } \circ \mathcal { T } \{ \mathrm { L } ( B - 2 , 1 ) \} = G \mathrm { L } ( ( B - 2 ) \beta _ { 1 1 } + \beta _ { 1 2 } , ( B - 2 ) \beta _ { 2 1 } + \beta _ { 2 2 } ) G } \\ & { \quad \quad \quad \quad \quad = B ^ { - 1 } ( ( B - 2 ) \beta _ { 1 1 } + \beta _ { 1 2 } + 2 ( ( B - 2 ) \beta _ { 2 1 } + \beta _ { 2 2 } ) ) \mathrm { L } ( B - 2 , 1 ) } \\ & { \quad \quad \quad \quad = \lambda _ { \mathrm { L } } ^ { G , \top } \mathrm { L } ( B - 2 , 1 ) } \end{array} +$$ + +by Lemma E.60. + +Item 3. The proof is exactly the same as in that of Thm E.61. + +Noting that the eigenspaces of $\cdot$ are orthogonal, we have + +Proposition E.63. $G ^ { \otimes 2 } \circ \mathcal { T } \uparrow \mathcal { H } ^ { G }$ for any ultrasymmetric operator $\tau$ is self-adjoint. + +DOS Operators. In some cases, such as the original study of vanilla tanh networks by Schoenholz et al. (2016), the ultrasymmetric operator involved has a much simpler form: + +Definition E.64. Let $\mathcal { T } : \mathcal { H } _ { B } \mathcal { H } _ { B }$ be such that for any $\Sigma \in \mathcal { H } _ { B }$ and any $i \neq j \in [ B ]$ , + +$$ +\begin{array} { l } { { \mathcal { T } \{ \Sigma \} _ { i i } = u \Sigma _ { i i } } } \\ { { \mathcal { T } \{ \Sigma \} _ { i j } = v \Sigma _ { i i } + v \Sigma _ { j j } + w \Sigma _ { i j } } } \end{array} +$$ + +Then we say that $\tau$ is diagonal-off-diagonal semidirect, or $D O S$ for short. We write more specifically $\mathcal { T } = \dot { \mathrm { D O S } } _ { B } ( u , v , w )$ . + +$\cdot$ and $\mathrm { T h m ~ E . 6 1 }$ still hold for DOS operators, but we can simplify the results and reason about them in a more direct way. + +Lemma E.65. Let $\mathcal { T } : = \mathrm { D O S } _ { B } ( u , v , w )$ . Then ${ \mathcal { T } } \{ \mathrm { L } _ { B } ( a , b ) \} = \mathrm { L } _ { B } ( u a , w b - v a ) .$ . + +Proof. Let $L : = \operatorname { L } _ { B } ( a , b )$ . It suffices to verify the following, each of which is a direct computation. + +1. $\gamma \{ L \} _ { 3 : B , 3 : B } = 0 .$ . +2. $\mathcal { T } \{ L \} _ { 1 , 2 } = \mathcal { T } \{ L \} _ { 2 , 1 } = 0$ +3. $\mathcal { T } \{ L \} _ { 1 , 1 } = - \mathcal { T } \{ L \} _ { 2 , 2 } = u a$ +4. $\mathcal { T } \{ L \} _ { 1 , i } = - \mathcal { T } \{ L \} _ { 2 , i } = \mathcal { T } \{ L \} _ { i , 1 } = - \mathcal { T } \{ L \} _ { i , 2 } = v a - w b .$ + +Lemma E.66. Let $\mathcal { T } : = \mathrm { D O S } _ { B } ( u , v , w )$ . Then $\mathrm { L } _ { B } ( w - u , v )$ and $\mathrm { L } _ { B } ( 0 , 1 )$ are its eigenvectors: + +$$ +\begin{array} { r } { \mathcal { T } \{ \mathrm { L } _ { B } ( w - u , v ) \} = u \mathrm { L } _ { B } ( w - u , v ) } \\ { \mathcal { T } \{ \mathrm { L } _ { B } ( 0 , 1 ) \} = w \mathrm { L } _ { B } ( 0 , 1 ) \qquad } \end{array} +$$ + +Proof. The map $( a , b ) \mapsto ( u a , w b - v a )$ has eigenvalues $u$ and $w$ with corresponding eigenvectors $( w - u , v )$ and $( 0 , 1 )$ . By Lemma E.65, this implies the desired results. + +Lemma E.67. Let $\mathcal { T } : = \mathrm { D O S } _ { B } ( u , v , w )$ . Then for any $L \in \mathbb { L } _ { B }$ $\begin{array} { r } { { _ 3 } , G ^ { \otimes 2 } \circ \mathcal { T } \{ L \} = \frac { ( u - 2 v ) ( B - 2 ) + 2 w } { B } L . } \end{array}$ + +Proof. By Lemma E.60 and Lemma E.65, ${ \cal G } ^ { \otimes 2 } \circ { \mathcal T } \{ \mathrm { L } ( B - 2 , 1 ) \} = { \cal G } ^ { \otimes 2 } \{ \mathrm { L } ( u ( B - 1 ) , w - v ( B -$ $\begin{array} { r } { 1 ) ) \} \} = { \frac { ( u - 2 v ) ( B - 2 ) + 2 w } { B } } \mathrm { L } _ { B } ( B - 2 , 1 ) } \end{array}$ . By permutation. Since they span mmetry, we also have the general res, this gives the conclusion we want. $\mathrm { L } _ { B } ( B - 2 , 1 )$ $L$ $\mathbb { L } _ { B }$ $\boxed { \begin{array} { r l } \end{array} }$ + +Lemma E.68. Let $\mathcal { T } : = \mathrm { D O S } _ { B } ( u , v , w )$ . Then $\mathcal { T } \{ \mathrm { B S B } 1 _ { B } ( a , b ) \} = \mathrm { B S B } 1 _ { B } ( u a , w b + 2 v a ) .$ + +Proof. Direct computation. + +Lemma E.69. Let $\mathcal { T } : = \mathrm { D O S } _ { B } ( u , v , w )$ . Then $\mathrm { B S B } 1 _ { B } ( w - u , w v )$ and $\mathrm { B S B 1 } _ { B } ( 0 , 1 )$ are eigenvectors of $\tau$ . + +$$ +\begin{array} { c } { { \mathcal { T } \{ \mathrm { B S B } 1 _ { B } ( u - w , 2 v ) \} = u \mathrm { B S B } 1 _ { B } ( u - w , 2 v ) } } \\ { { \mathcal { T } \{ \mathrm { B S B } 1 _ { B } ( 0 , 1 ) \} = w \mathrm { B S B } 1 _ { B } ( 0 , 1 ) } } \end{array} +$$ + +Proof. The linear map $( a , b ) \mapsto ( u a , w b + 2 v a )$ has eigenvalues $u$ and $w$ with corresponding eigenvectors $( u - w , 2 v )$ and $( 0 , 1 )$ . The result then immediately follows from Lemma E.68. □ + +Lemma E.70. Let $\mathcal { T } : = \mathrm { D O S } _ { B } ( u , v , w )$ . Then $\begin{array} { r } { G ^ { \otimes 2 } \circ \mathcal { T } \{ G \} = \frac { ( B - 1 ) ( u - 2 v ) + w } { B } G . } \end{array}$ + +Proof. Direct computation with Lemma E.68 and Lemma E.52 + +Definition E.71. Define ${ \overline { { \mathbb { M } } } } _ { B } : = \{ \Sigma \in { \cal S } _ { B } : \mathrm { D i a g } \Sigma = 0 \}$ (so compared to $\mathbb { M }$ , matrices in $\overline { { \mathbb { M } } }$ do not need to have zero row sums). In addition, for any $a , b \in \mathbb { R }$ , set + +$$ +\begin{array} { r } { \overline { { \mathrm { L } } } _ { B } ( a , b ) : = \left( \begin{array} { l l l l } { a } & { - b } & { - b } & { \cdots } \\ { - b } & { 0 } & { 0 } & { \cdots } \\ { - b } & { 0 } & { 0 } & { \cdots } \\ { \vdots } & { \vdots } & { \vdots } & { \ddots } \end{array} \right) \in \mathcal { H } _ { B } . } \end{array} +$$ + +Define $\overline { { \mathbb { L } } } _ { B } ( a , b ) : = \mathrm { s p a n } ( P _ { 1 i } ^ { T } \overline { { \mathrm { L } } } _ { B } ( a , b ) P _ { 1 i } : i \in [ B ] )$ where $P _ { 1 i }$ is the permutation matrix that swap the first entry with the ith entry, i.e. $\overline { { \mathbb { L } } } _ { B } ( a , b )$ is the span of the orbit of $\textstyle { \overline { { \operatorname { L } } } } _ { B } ( a , b )$ under permuting rows and columns simultaneously. + +Note that + +$$ +\begin{array} { c } { \mathrm { L } _ { B } ( a , b ) = \overline { { \mathrm { L } } } _ { B } ( a , b ) - P _ { 1 2 } ^ { T } \overline { { \mathrm { L } } } _ { B } ( a , b ) P _ { 1 2 } } \\ { \mathrm { B S B 1 } _ { B } ( a , - 2 b ) = \displaystyle \sum _ { i = 1 } ^ { B } P _ { 1 i } ^ { T } \overline { { \mathrm { L } } } _ { B } ( a , b ) P _ { 1 i } } \end{array} +$$ + +So $\mathrm { L } _ { B } ( a , b ) , \mathrm { B S B 1 } _ { B } ( a , - 2 b ) \in \overline { { \mathbb { L } } } _ { B } ( a , b )$ + +Theorem E.72. Let $\mathcal { T } : = \mathrm { D O S } _ { B } ( u , v , w )$ . Suppose $w \ne u$ . Then $\mathcal { T } \ \lceil \ \mathcal { H } _ { B }$ has the following eigendecomposition: + +• ${ \overline { { \mathbb { M } } } } _ { B }$ has eigenvalue $w$ $( \dim \overline { { \mathbb { M } } } _ { B } = B ( B - 1 ) / 2 )$ • $\overline { { \mathbb { L } } } _ { B } ( w - u , v )$ has eigenvalue of $u$ $\iota ( \dim \mathbb { L } _ { B } ( w - u , v ) = B )$ + +If $w = u$ , then $\mathbb { L } _ { B } ( w - u , v ) \subseteq \overline { { \mathbb { M } } } _ { B }$ + +Proof. The case of ${ \overline { { \mathbb { M } } } } _ { B }$ is obvious. We will show that $\overline { { \mathbb { L } } } _ { B } ( w - u , v )$ is an eigenspace with eigenvalue $u$ . Then by dimensionality consideration they are all of the eigenspaces of $\tau$ . + +Let $L : = \overline { { \mathrm { L } } } _ { B } ( w - u , v )$ . Then it’s not hard to see ${ \mathcal { T } } \{ L \} = { \overline { { \mathrm { L } } } } _ { B } ( a , b )$ for some $a , b \in \mathbb { R }$ . It follows that $a$ has to be $u ( w - u )$ and $b$ has to be $- ( v ( w - u ) - w v ) = u v$ , which yields what we want. + +Theorem E.73. Let $\mathcal { T } : = \mathrm { D O S } _ { B } ( u , v , w )$ . Then $G ^ { \otimes 2 } \circ \mathcal { T } \mid \mathcal { H } _ { B } ^ { G } : \mathcal { H } _ { B } ^ { G } \mathcal { H } _ { B } ^ { G }$ has the following eigendecomposition: + +RG has eigenvalu e (B−1)(u−2v)+w (dim RG = 1) • $\mathbb { M } _ { B }$ has eigenvalue w $( \dim \mathbb { M } _ { B } = B ( B - 3 ) / 2 )$ • $\mathbb { L } _ { B }$ has eigenvalue $\frac { ( B - 2 ) ( u - 2 v ) + 2 w } { B } ( \dim \mathbb { L } _ { B } = B - 1 )$ + +Proof. The case of ${ \mathbb { M } } _ { B }$ is obvious. The case for $\mathbb { R } G$ follows from Lemma E.70. The case for $\mathbb { L } _ { B }$ follows from Lemma E.67. By dimensionality considerations these are all of the eigenspaces. □ + +# F FORWARD DYNAMICS + +In this section we will be interested in studying the dynamics on PSD matrices of the form + +$$ +\Sigma ^ { l } = { \bf V } _ { B _ { \phi } } ( \Sigma ^ { l - 1 } ) = \mathbb { E } [ \phi ( \sqrt { B } G h / \| G h \| ) ^ { \otimes 2 } : h \sim \mathcal { N } ( 0 , \Sigma ^ { l - 1 } ) ] +$$ + +where $\Sigma ^ { l } \in S _ { B }$ and $\phi : \mathbb { R } \mathbb { R }$ . + +# F.1 GLOBAL CONVERGENCE + +Basic questions regarding the dynamics Eq. (43) are 1) does it converge? 2) What are the limit points? 3) How fast does it converge? Here we answer these questions definitively when $\phi = \mathrm { i d }$ . + +The following is the key lemma in our analysis. + +Lemma F.1. Consider the dynamics $\Sigma ^ { l } = \mathbb { E } [ ( h / \| h \| ) ^ { \otimes 2 } : h \sim \mathcal { N } ( 0 , \Sigma ^ { l - 1 } ) ]$ on $\Sigma ^ { l } \in { \mathcal { S } } _ { A }$ . Suppose $\Sigma ^ { 0 }$ is full rank. Then + +$$ +\begin{array} { r } { I . \ \operatorname* { l i m } _ { l \infty } \Sigma ^ { l } = \frac { 1 } { A } I . } \end{array} +$$ + +2. This convergence is exponential in the sense that, for any full rank $\Sigma ^ { 0 }$ , there is a constant $K < 1$ such that $\lambda _ { 1 } ( \bar { \Sigma } ^ { l } ) - \lambda _ { A } ( \Sigma ^ { l } ) < K ( \lambda _ { 1 } ( \Sigma ^ { l - 1 } ) - \bar { \lambda _ { A } } ( \Sigma ^ { l - 1 } ) )$ for all $l \geq 2$ . Here $\lambda _ { 1 }$ (resp. $\lambda _ { A }$ ) denotes that largest (resp. smallest) eigenvalue. + +3. Asymptotically, $\begin{array} { r } { \lambda _ { 1 } ( \Sigma ^ { l } ) - \lambda _ { A } ( \Sigma ^ { l } ) = O ( ( 1 - \frac { 2 } { A + 2 } ) ^ { l } ) . } \end{array}$ + +Proof. Let ${ \lambda _ { 1 } } ^ { l } \ge { \lambda _ { 2 } } ^ { l } \ge \dots \ge { \lambda _ { A } } ^ { l }$ be the eigenvalues of $\Sigma ^ { l }$ . It’s easy to see that $\begin{array} { r } { \sum _ { i } \lambda _ { i } ^ { l } = 1 } \end{array}$ for all $l \geq 1$ . So WLOG we assume $l \geq 1$ and this equality holds. + +We will show the “exponential convergence” statement; that $\begin{array} { r } { \operatorname* { l i m } _ { l \to \infty } \Sigma ^ { l } = \frac { 1 } { A } I } \end{array}$ then follows from the trace condition above. + +Proof of Item 2. For brevity, we write suppress the superscript index $l$ , and use · to denote ·l. We will now compute the eigenvalues $\overline { { \lambda } } _ { 1 } \geq \cdots \geq \overline { { \lambda } } _ { A }$ of $\overline { { \Sigma } }$ . + +First, notice that $\overline { { \Sigma } }$ and $\Sigma$ can be simultaneously diagonalized, and by induction all of $\{ \Sigma ^ { l } \} _ { l \geq 0 }$ can be simultaneous diagonalized. Thus we will WLOG assume that $\Sigma$ is diagonal, $\Sigma = { \mathrm { D i a g } } ( \lambda _ { 1 } , \bar { . } . . , \lambda _ { A } )$ , so that $\overline { { \Sigma } } = \mathrm { D i a g } ( \gamma _ { 1 } , \dots , \gamma _ { A } )$ for some $\{ \gamma _ { i } \} _ { i }$ . These $\{ \gamma _ { i } \} _ { i }$ form the eigenvalues of $\overline { { \Sigma } }$ but a priori we don’t know whether they fall in decreasing order; in fact we will soon see that they do, and $\gamma _ { i } = \overline { { \lambda } } _ { i }$ . + +We have + +$$ +\begin{array} { r l } & { \gamma _ { 3 } = \binom { \pi } { 2 } ^ { - \frac { 1 } { 2 } - \theta / 2 } \int _ { 0 } ^ { \infty } \frac { \lambda \sin ^ { 2 } \theta } { \sum _ { j } \lambda _ { j } ^ { 2 } + \theta ^ { 2 } } \mathrm { e } ^ { - \lambda _ { j } \theta / 2 } } \\ & { \quad - ( \frac { \pi } { 2 } ) ^ { - \theta / 2 } \int _ { 0 } ^ { \infty } \frac { \lambda \sin ^ { 2 } \theta } { \lambda _ { j } \sin ^ { 2 } \theta } \mathrm { e } ^ { - \lambda _ { j } \theta / 2 } d \mathrm { e } ^ { \theta } \mathrm { e } ^ { - \gamma _ { 3 } \sum \theta / 2 } d \mathrm { e } } \\ & { \quad - ( \frac { \pi } { 2 } ) ^ { - \theta / 2 } \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \lambda _ { j } \cos ^ { 2 } \theta \mathrm { e } ^ { - \lambda _ { j } \theta / 2 } d \mathrm { e } ^ { \theta } \mathrm { e } ^ { \theta } \mathrm { e } ^ { \theta } \mathrm { e } ^ { \theta } \mathrm { e } ^ { \lambda _ { j } \theta / 2 } d \mathrm { e } } \\ & { \quad - ( \frac { \pi } { 2 } ) ^ { - \theta / 2 } \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \lambda _ { j } \sin ^ { 2 } \theta \mathrm { e } ^ { - \lambda _ { j } \theta / 2 } d \mathrm { e } ^ { \theta } \mathrm { e } ^ { \theta } \mathrm { e } ^ { \lambda _ { j } \theta / 2 } \mathrm { e } \mathrm { d } \mathrm { e } } \\ & { \quad - ( \frac { \pi } { 2 } ) ^ { - \theta / 2 } \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \lambda _ { j } \cos ^ { 2 } \theta \mathrm { e } ^ { - \lambda _ { j } \theta / 2 } d \mathrm { e } ^ { \theta } \mathrm { e } ^ { \theta } \mathrm { e } ^ { \lambda _ { j } \theta / 2 } \prod _ { j } ^ { ( 1 ) } [ \frac { \pi } { 2 } ( 1 + 2 \alpha \lambda _ { j } ) ^ { - \theta / 2 } \mathrm { e } ^ { \lambda _ { j } \theta / 2 } } \\ & \quad - ( \frac { \pi } { 2 } ) ^ { - \theta / 2 } \int _ { 0 } ^ { \infty } \lambda _ { j } ( 1 + 2 \end{array} +$$ + +Therefore, + +$$ +\begin{array} { r } { \gamma _ { i } - \gamma _ { k } = \displaystyle \int _ { 0 } ^ { \infty } \prod _ { j = 1 } ^ { A } ( 1 + 2 s \lambda _ { j } ) ^ { - 1 / 2 } ( \lambda _ { i } ( 1 + 2 s \lambda _ { i } ) ^ { - 1 } - \lambda _ { k } ( 1 + 2 s \lambda _ { k } ) ^ { - 1 } ) \mathrm { d } s } \\ { = ( \lambda _ { i } - \lambda _ { k } ) \displaystyle \int _ { 0 } ^ { \infty } \prod _ { j = 1 } ^ { A } ( 1 + 2 s \lambda _ { j } ) ^ { - 1 / 2 } ( 1 + 2 s \lambda _ { i } ) ^ { - 1 } ( 1 + 2 s \lambda _ { k } ) ^ { - 1 } \mathrm { d } s } \end{array} +$$ + +Since the RHS integral is always positive, $\lambda _ { i } \geq \lambda _ { k } \implies \gamma _ { i } \geq \gamma _ { k }$ and thus $\gamma _ { i } = \overline { { \lambda } } _ { i }$ for each $i$ . + +Define $\begin{array} { r } { T ( \lambda ) : = \int _ { 0 } ^ { \infty } \prod _ { j = 1 } ^ { A } ( 1 + 2 s \lambda _ { 1 } ) ^ { - 1 / 2 } ( 1 + 2 s \lambda _ { 1 } ) ^ { - 1 } ( 1 + 2 s \lambda _ { A } ) ^ { - 1 } \mathrm { d } s , } \end{array}$ , so that $\overline { { { \lambda } } } _ { 1 } \mathrm { ~ - ~ } \overline { { { \lambda } } } _ { A } \mathrm { ~ = ~ }$ $( \lambda _ { 1 } - \lambda _ { A } ) T ( \lambda )$ . + +Note first that $\begin{array} { r } { \prod _ { j = 1 } ^ { A } ( 1 + 2 s \lambda _ { j } ) ^ { - 1 / 2 } ( 1 + 2 s \lambda _ { i } ) ^ { - 1 } ( 1 + 2 s \lambda _ { k } ) ^ { - 1 } } \end{array}$ is (strictly) log-convex and hence (strictly) convex in $\lambda$ . Furthermore, $T ( \lambda )$ is (strictly) convex because it is an integral of (strictly) convex functions. Thus $T$ is maximized over any convex region by its extremal points, and only by its extremal points because of strict convexity. + +The convex region we are interested in is given by + +$$ +{ \mathcal { A } } : = \{ ( \lambda _ { i } ) _ { i } : \lambda _ { 1 } \geq \lambda _ { 2 } \geq \cdots \geq \lambda _ { A } \geq 0 \ \& \ \sum _ { i } \lambda _ { i } = 1 \} . +$$ + +The unit sum condition follows from the normalization $h / \| h \|$ of the iteration map. The extremal $k$ times points of $\mathcal { A }$ are $\{ \omega ^ { k } : = ( \widehat { 1 / k , 1 / k , \cdot \cdot \cdot , 1 / k } , 0 , 0 , \cdot \cdot \cdot , 0 ) \} _ { k = 1 } ^ { A }$ . For $k = 1 , \ldots , A - 1$ , we have + +$$ +\begin{array} { l } { \displaystyle T ( \omega ^ { k } ) = \int _ { 0 } ^ { \infty } ( 1 + 2 s / k ) ^ { - k / 2 - 1 } \mathrm { d } s } \\ { \displaystyle \qquad = \frac { 2 / k } { - k / 2 } ( 1 + 2 s / k ) ^ { - k / 2 } \bigg \rvert _ { 0 } ^ { \infty } } \\ { \displaystyle \qquad = 1 . } \end{array} +$$ + +But for $k = A$ , + +$$ +\begin{array} { l } { \displaystyle T ( \omega ^ { A } ) = \int _ { 0 } ^ { \infty } ( 1 + 2 s / A ) ^ { - A / 2 - 2 } \mathrm { d } s } \\ { \displaystyle \quad = \frac { 2 / A } { - A / 2 - 1 } ( 1 + 2 s / A ) ^ { - k / 2 } \bigg \vert _ { 0 } ^ { \infty } } \\ { \displaystyle \quad = \frac { A } { A + 2 } } \end{array} +$$ + +This shows that $T ( \lambda ) \leq 1$ , with equality iff $\lambda = \omega ^ { k }$ for $k = 1 , \ldots , A - 1$ . In fact, because every point $\lambda \in { \mathcal { A } }$ is a convex combination of $\omega ^ { k }$ for $k = 1 , \dots , A$ , $\textstyle \lambda = \sum _ { k = 1 } ^ { A } a _ { k } \omega ^ { k }$ , by convexity of $T$ we must have + +$$ +\begin{array} { r } { T ( \lambda ) \leq \displaystyle \sum _ { k = 1 } ^ { A } a _ { k } T ( \omega ^ { k } ) } \\ { = 1 - a _ { A } \displaystyle \frac { 2 } { A + 2 } } \\ { = 1 - \lambda _ { A } \displaystyle \frac { 2 } { A + 2 } } \end{array} +$$ + +where the last line follows because $\omega ^ { A }$ is the only point with last coordinate nonzero so that $a _ { A } =$ $\lambda _ { A }$ . + +We now show that the gap ${ \lambda _ { 1 } } ^ { l } - { \lambda _ { A } } ^ { l } 0$ as $l \infty$ . There are two cases: if ${ \lambda _ { A } } ^ { l }$ is bounded away from 0 infinitely often, then $T ( \lambda ) < 1 - \epsilon$ infinitely often for a fixed $\epsilon > 0$ so that the gap indeed vanishes with $l$ . Now suppose otherwise, that ${ \lambda _ { A } } ^ { l }$ converges to 0; we will show this leads to a contradiction. Notice that + +$$ +\begin{array} { c } { { \overline { { { \lambda } } } _ { A } = \displaystyle \int _ { 0 } \displaystyle \prod _ { j = 1 } ^ { \infty } ( 1 + 2 s \lambda _ { j } ) ^ { - 1 / 2 } \lambda _ { A } ( 1 + 2 s \lambda _ { A } ) ^ { - 1 } \mathrm { d } s } } \\ { { \overline { { { \lambda } } } _ { A } / \lambda _ { A } \geq \displaystyle \int _ { 0 } \displaystyle \prod _ { j = 1 } ^ { \infty } ( 1 + 2 s ( 1 - \lambda _ { A } ) / ( A - 1 ) ) ^ { - ( A - 1 ) / 2 } ( 1 + 2 s \lambda _ { A } ) ^ { - 3 / 2 } \mathrm { d } s } } \end{array} +$$ + +where the first lines is Eq. (44) and the 2nd line follows from the convexity of $\prod _ { j = 1 } ^ { A - 1 } ( 1 + 2 s \lambda _ { j } ) ^ { - 1 / 2 }$ as a function of $\left( \lambda _ { 1 } , \ldots , \lambda _ { A - 1 } \right)$ . By a simple application of dominated convergence, as $\lambda _ { A } \to 0$ , this integral converges to a particular simple form, + +$$ +\begin{array} { l } { \displaystyle \overline { { \lambda } } _ { A } / \lambda _ { A } \geq \int _ { 0 } ^ { \infty } \prod _ { j = 1 } ^ { A } ( 1 + 2 s / ( A - 1 ) ) ^ { - ( A - 1 ) / 2 } \mathrm { d } s } \\ { \displaystyle \qquad = 1 + \frac { 2 } { A - 3 } } \end{array} +$$ + +Thus for large enough $l$ , ${ \lambda _ { A } } ^ { l + 1 } / { \lambda _ { A } } ^ { l }$ is at least $1 + \epsilon$ for some $\epsilon > 0$ , but this contradicts the convergence of $\lambda _ { A }$ to 0. + +Altogether, this proves that ${ \lambda _ { 1 } } ^ { l } - { \lambda _ { A } } ^ { l } 0$ and therefore $\lambda ^ { l } \to \omega ^ { A }$ as $l \infty$ . Consequently, ${ \lambda _ { A } } ^ { l }$ is bounded from below, say by $K ^ { \prime }$ (where $K ^ { \prime } > 0$ because by Eq. (44), ${ \lambda _ { A } } ^ { l }$ is never 0), for all $l$ , and by Eq. (45), we prove Item 2 by taking $\begin{array} { r } { K = 1 - K ^ { \prime } \frac { 2 } { A + 2 } } \end{array}$ . In addition, asymptotically, the gap decreases exponentially as $\begin{array} { r } { T ( \omega ^ { A } ) ^ { l } = \left( \frac { A } { A + 2 } \right) ^ { l } } \end{array}$ , proving Item 3. + +Theorem F.2. Consider the dynamics $\Sigma ^ { l } = \mathbb { E } [ ( h / \| h \| ) ^ { \otimes 2 } : h \sim \mathcal { N } ( 0 , \Sigma ^ { l - 1 } ) ]$ on $\Sigma ^ { l } \in { \cal S } _ { A }$ . Suppose $\Sigma ^ { 0 } = M ^ { T } D M$ where $M$ is orthogonal and $D$ is a diagonal matrix. If $D$ has rank $C$ and $D _ { i i } \neq $ $0 , \forall 1 \leq i \leq C$ , then + +1. $\begin{array} { r } { \operatorname* { l i m } _ { l \to \infty } \Sigma ^ { l } = \frac { 1 } { C } M ^ { T } D ^ { \prime } M } \end{array}$ where $D ^ { \prime }$ is the diagonal matrix with $D _ { i i } ^ { \prime } = \mathbb { I } ( D _ { i i } \neq 0 )$ . + +2. This convergence is exponential in the sense that, for any $\Sigma ^ { 0 }$ of rank $C$ , there is a constant $K < 1$ such that $\lambda _ { 1 } ( \dot { \Sigma } ^ { l } ) - \lambda _ { C } ( \Sigma ^ { l } ) < K ( \lambda _ { 1 } ( \Sigma ^ { l \bar { - } 1 } ) - \dot { \lambda } _ { C } ( \Sigma ^ { \bar { l } - 1 } ) )$ for all $l \geq 2$ . Here $\lambda _ { i }$ denotes the ith largest eigenvalue. + +3. Asymptotically, $\begin{array} { r } { \lambda _ { 1 } ( \Sigma ^ { l } ) - \lambda _ { C } ( \Sigma ^ { l } ) = O ( ( 1 - \frac { 2 } { C + 2 } ) ^ { l } ) . } \end{array}$ . + +Proof. Note that $\Sigma ^ { l }$ can always be simultaneously diagonalized with $\Sigma ^ { 0 }$ , so that $\Sigma ^ { l } = { M } ^ { T } D ^ { l } { M }$ for some diagonal $D ^ { l }$ which has $D _ { i i } ^ { l } = 0 , \forall 0 \leq \overline { { i } } \leq \mathbf { \bar { C } }$ . Then we have $( D ^ { l } ) ^ { \prime } = \mathbb { E } [ ( h / \| h \| ) ^ { \otimes 2 } :$ $h \sim \mathcal { N } ( 0 , ( D ^ { l - 1 } ) ^ { \prime } ) ]$ , where $( D ^ { l } ) ^ { \prime }$ means the diagonal matrix obtained from $D ^ { l }$ by deleting the dimensions with zero eigenvalues. The proof then finishes by Lemma F.1. □ + +From this it easily follows the following characterization of the convergence behavior. + +Corollary F.3. Consider the dynamics of Eq. (43) for $\phi = \mathrm { i d }$ : $\Sigma ^ { l } = B \mathbb { E } [ ( G h / \| G h \| ) ^ { \otimes 2 } : h \sim$ $\mathcal { N } ( 0 , \Sigma ^ { l - 1 } ) ]$ on $\Sigma ^ { l } \in S _ { B }$ . Suppose $G \Sigma G$ has rank $C < B$ and factors as $\hat { \Phi } \bar { D \Phi } ^ { T }$ where $\mathbf { \varPhi } ^ { \prime \prime } \mathbf { \varTheta } ^ { \prime \prime } \in \mathbb { R } ^ { C \times C }$ is a diagonal matrix with no zero diagonal entries and $\hat { \Phi }$ is an $B \times C$ matrix whose columns form an orthonormal basis of a subspace of $\operatorname { i m } G$ . Then + +$$ +\begin{array} { r } { I . \ \operatorname* { l i m } _ { l \infty } \Sigma ^ { l } = \frac { B } { C } \hat { \mathbb { e } } I _ { C } \hat { \mathbb { e } } ^ { T } . } \end{array} +$$ + +2. This convergence is exponential in the sense that, for any $G \Sigma ^ { 0 } G$ of rank $C$ , there is $a$ constant $K < 1$ such that $\lambda _ { 1 } ( \Sigma ^ { l } ) - \lambda _ { C } ( \Sigma ^ { l } ) < K ( \bar { \lambda } _ { 1 } ( \Sigma ^ { l \bar { - } 1 } ) - \lambda _ { C } ( \bar { \Sigma } ^ { l - 1 } ) )$ for all $l \geq 2$ . Here $\lambda _ { i }$ denotes the ith largest eigenvalue. + +3. Asymptotically, $\begin{array} { r } { \lambda _ { 1 } ( \Sigma ^ { l } ) - \lambda _ { C } ( \Sigma ^ { l } ) = O ( ( 1 - \frac { 2 } { C + 2 } ) ^ { l } ) . } \end{array}$ + +General nonlinearity. We don’t (currently) have a proof of any characterization of the basin of attraction for Eq. (43) for general $\phi$ . Thus we are forced to resort to finding its fixed points manually and characterize their local convergence properties. + +# F.2 LIMIT POINTS + +Batchnorm $B _ { \phi }$ is permutation-equivariant, in the sense that $\boldsymbol { B } _ { \phi } ( \pi h ) = \pi \boldsymbol { B } _ { \phi } ( h )$ for any permtuation matrix $\pi$ . Along with the case of $\phi = \mathrm { i d }$ studied above, this suggests that we look into BSB1 fixed points $\Sigma ^ { * }$ . What are the BSB1 fixed points of $E q$ . (43)? + +# F.2.1 SPHERICAL INTEGRATION + +The main result $\begin{array} { r } { \left( \begin{array} { c c c c c c c } { \rho } & { \rho } & { \rho } & { \rho } & { \rho } & { \rho } \\ { \vdots } & { \vdots } & { \vdots } & { \vdots } & { \vdots } & { \vdots } \\ { \rho } & { \rho } & { \rho } & { \rho } & { \rho } & { \rho } \end{array} \right) } \end{array}$ ) of this section is an expression of the BSB1 fixed point diagonal and offdiagonal entries in terms of 1- and 2-dimensional integrals. This allows one to numerically compute such fixed points. + +By a simple symmetry argument, we have the following + +Lemma F.4. Suppose $X$ is a random vector in $\mathbb { R } ^ { B }$ , symmetric in the sense that for any permutation matrix $\pi$ and any subset $U$ of $\mathbb { R } ^ { B }$ measurable with respect to the distribution of $X$ , $P ( X \in U ) =$ $P ( X \in \pi ( U ) )$ . Let $\Phi : \mathbb { R } ^ { B } \mathbb { R } ^ { B }$ be a symmetric function in the sense that $\Phi ( \pi x ) = \pi \Phi ( x )$ for any permutation matrix $\pi$ . Then $\mathbb { E } \Phi ( X ) \dot { \Phi ( X ) } ^ { T } = \dot { \mu } I + \nu \mathbf { 1 } \mathbf { 1 } ^ { T }$ for some $\mu$ and $\nu$ . + +This lemma implies that $\mathrm { V } _ { B _ { \phi } } ( \Sigma )$ is BSB1 whenever $\Sigma$ is BSB1. In the following, We in fact show that for any BSB1 $\Sigma$ , $\mathrm { V } _ { B _ { \phi } } ( \Sigma )$ is the same, and this is the unique BSB1 fixed point of Eq. (43). + +Theorem F.5. The BSB1 fixed point $\Sigma ^ { * } = \mu ^ { * } I + \nu ^ { * } \mathbf { 1 1 } ^ { T }$ to Eq. (43) is unique and satisfies + +$$ +\begin{array} { r l } & { \mu ^ { * } + \nu ^ { * } = \frac { \Gamma \left( \frac { B - 1 } { 2 } \right) } { \Gamma \left( \frac { B - 2 } { 2 } \right) \sqrt { \pi } } \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } \sin ^ { B - 3 } \theta _ { 1 } \phi ( \sqrt { B } \zeta _ { 1 } ( \theta _ { 1 } ) ) ^ { 2 } } \\ & { \qquad \nu ^ { * } = \left\{ \begin{array} { l l } { \frac { B - 3 } { 2 \pi } \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 2 } \sin ^ { B - 3 } \theta _ { 1 } \sin ^ { B - 4 } \theta _ { 2 } \phi ( \sqrt { B } \zeta _ { 1 } ( \theta _ { 1 } ) ) \phi ( \sqrt { B } \zeta _ { 2 } ( \theta _ { 1 } , \theta _ { 2 } ) ) , } & { i f B \geq 4 } \\ { \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \mathrm { d } \theta \phi ( - \sqrt { 2 } \sin \theta ) \phi ( \sin \theta / \sqrt { 2 } - \cos \theta \sqrt { 3 } / \sqrt { 2 } ) , } & { i f B = 3 } \end{array} \right. } \end{array} +$$ + +where + +$$ +\begin{array} { l } { \displaystyle \zeta _ { 1 } ( \theta ) = - \sqrt { \frac { B - 1 } { B } } \cos \theta } \\ { \displaystyle \zeta _ { 2 } ( \theta _ { 1 } , \theta _ { 2 } ) = \frac { 1 } { \sqrt { B ( B - 1 ) } } \cos \theta _ { 1 } - \sqrt { \frac { B - 2 } { B - 1 } } \sin \theta _ { 1 } \cos \theta _ { 2 } } \end{array} +$$ + +Proof. For any BSB1 $\Sigma$ , $\mathbb { e } ^ { T } \Sigma \mathbb { e } = \mu I _ { B - 1 }$ for some $\mu$ . Thus we can apply Proposition E.33 to $G \Sigma G$ We observe that ${ \mathcal { K } } ( v ; G \Sigma G ) = 1$ , so that + +$$ +\begin{array} { r } { \mathrm { V } _ { \mathcal { B } _ { \phi } } ( \Sigma ) = \mathrm { V } _ { \mathcal { B } _ { \phi } } ( G \Sigma G ) = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathbb { e } v ) ^ { \otimes 2 } . } \end{array} +$$ + +Note that this is now independent of the specific values of $\Sigma$ . Now, using the specific form of $\mathbb { P }$ given by Eq. (33), we see that $\mathrm { V } _ { B _ { \phi } } ( \Sigma ) _ { 1 1 }$ only depends on $\theta _ { 1 }$ and $\mathrm { V } _ { B _ { \phi } } ( \Sigma ) _ { 1 2 }$ only depends on $\theta _ { 1 }$ and $\theta _ { 2 }$ . Then applying Proposition E.33 followed by Lemma E.36 and Lemma E.37 yield the desired result. □ + +# F.2.2 LAPLACE METHOD + +In the case that $\phi$ is positive-homogeneous, we can apply Laplace’s method Appendix E.1 and obtain the BSB1 fixed point in a much nicer form. + +Definition F.6. For any $\alpha \geq 0$ , define $\begin{array} { r } { K _ { \alpha , B } : = \mathsf { c } _ { \alpha } \mathrm { P } \left( \frac { B - 1 } { 2 } , \alpha \right) ^ { - 1 } \left( \frac { B - 1 } { 2 } \right) ^ { \alpha } } \end{array}$ where $\mathrm { P } \left( a , b \right) : = \Gamma ( a +$ $b ) / \Gamma ( a )$ is the Pochhammer symbol. + +Note that + +Proposition F.7. $\begin{array} { r } { \operatorname* { l i m } _ { B \to \infty } K _ { \alpha , B } = \mathsf { c } _ { \alpha } } \end{array}$ . + +We give a closed form description of the BSB1 fixed point for a positive homogeneous $\phi$ in terms of its J function. + +Theorem F.8. Suppose $\phi : \mathbb { R } \mathbb { R }$ is degree $\alpha$ positive-homogeneous. For any BSB1 $\Sigma \in \Sigma$ $\boldsymbol { S } _ { B }$ , $\mathrm { V } _ { B _ { \phi } } ( \Sigma )$ is BSB1. The diagonal entries are $K _ { \alpha , B } \mathbf { J } _ { \phi } ( 1 )$ and the off-diagonal entries are $\begin{array} { r } { K _ { \alpha , B } \mathbf { J } _ { \phi } \left( \frac { - 1 } { B - 1 } \right) } \end{array}$ . Here $\mathsf { C } _ { \alpha }$ is as defined in Defn E.7 and $\mathrm { J } _ { \phi }$ is as defined in Defn E.20. Thus a BSB1 fixed point of Eq. (43) exists and is unique. + +Proof. Let $\mathbb { P }$ be an $B \times ( B - 1 )$ matrix whose columns form an orthonormal basis of $\mathrm { i m } G : = \{ G v :$ $v \in \mathbb { R } ^ { B } \} = \{ w \in \mathbb { R } ^ { B } : \sum _ { i } w _ { i } = 0 \}$ . Let $\mathrm { B S B 1 } ( a , b )$ denote a matrix with $a$ on the diagonal and $b$ on the off-diagnals. Note that $\mathrm { V } _ { \phi }$ is positive homogeneous of degree $\alpha$ , so for any $\mu$ , + +$$ +\begin{array} { r l } & { \quad { \nabla _ { \phi } } ( \phi \mu I _ { B - 1 } ( I _ { B - 1 } + 2 s \mu I _ { B - 1 } ) ^ { - 1 } \mathfrak { g } ^ { T } ) } \\ & { = { \nabla _ { \phi } } ( \frac { \mu } { 1 + 2 s \mu } \frac { \beta \mathfrak { g } ^ { T } } { B } ) } \\ & { = \left( \frac { \mu } { 1 + 2 s \mu } \right) ^ { \alpha } \nabla _ { \phi } ( \mathfrak { g } \mathfrak { g } ^ { T } ) } \\ & { = \left( \frac { \mu } { 1 + 2 s \mu } \right) ^ { \alpha } \nabla _ { \phi } ( G ) } \\ & { = \left( \frac { \mu } { 1 + 2 s \mu } \right) ^ { \alpha } \nabla _ { \phi } ( G ) } \\ & { = \left( \frac { \mu } { 1 + 2 s \mu } \frac { B - 1 } { B } \right) ^ { \alpha } \nabla _ { \phi } \left( \mathtt { B S B } \left( 1 , \frac { - 1 } { B - 1 } \right) \right) } \\ & { = \left( \frac { \mu } { 1 + 2 s \mu } \frac { B - 1 } { B } \right) ^ { \alpha } \mathrm { c } _ { \phi } \mathrm { J } _ { \phi } \left( \mathtt { B S B } \left( 1 , \frac { - 1 } { B - 1 } \right) \right) } \end{array} +$$ + +So by Eq. (26), + +$$ +\begin{array} { l } { { \displaystyle \zeta _ { B _ { \phi } } ( \Phi \mu I \mathbb { P } ^ { T } ) = B ^ { \alpha } \Gamma ( \alpha ) ^ { - 1 } \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { \alpha - 1 } ( 1 + 2 s \mu ) ^ { - ( B - 1 ) / 2 } \left( \frac { \mu } { 1 + 2 s \mu } \frac { B - 1 } { B } \right) ^ { \alpha } \mathrm { c } _ { \alpha } \mathrm { J } _ { \phi } \left( \mathrm { B S B } 1 \left( 1 , \frac { \sqrt { \alpha - 1 } } { B } \right) \right) ^ { \alpha - 1 } \mathrm { d } s } } \\ { { \displaystyle \qquad = \Gamma ( \alpha ) ^ { - 1 } ( B - 1 ) ^ { \alpha } \mathrm { c } _ { \alpha } \mathrm { J } _ { \phi } \left( \mathrm { B S B } 1 \left( 1 , \frac { - 1 } { B - 1 } \right) \right) \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { \alpha - 1 } ( 1 + 2 s \mu ) ^ { - ( B - 1 ) / 2 - \alpha } \mu ^ { \alpha } } } \\ { { \displaystyle \qquad = \Gamma ( \alpha ) ^ { - 1 } ( B - 1 ) ^ { \alpha } \mathrm { c } _ { \alpha } \mathrm { J } _ { \phi } \left( \mathrm { B S B } 1 \left( 1 , \frac { - 1 } { B - 1 } \right) \right) \mathrm { B e t a } ( \alpha , ( B - 1 ) / 2 ) 2 ^ { - \alpha } } } \\ { { \displaystyle \qquad = \mathrm { c } _ { \alpha } \frac { \Gamma ( ( B - 1 ) / 2 ) } { \Gamma ( \alpha + ( B - 1 ) / 2 ) } \left( \frac { B - 1 } { 2 } \right) ^ { \alpha } \mathrm { J } _ { \phi } \left( \mathrm { B S B } 1 \left( 1 , \frac { - 1 } { B - 1 } \right) \right) } } \end{array} +$$ + +Corollary F.9. Suppose $\phi : \mathbb { R } \mathbb { R }$ is degree α positive-homogeneous. If $\Sigma ^ { * }$ is the BSB1 fixed point of Eq. (43) as given by Thm F.8, then $G \Sigma ^ { * } G = \mu ^ { * } I _ { B - 1 }$ where $\mu ^ { * } = \bar { K _ { \alpha , B } } \big ( \mathrm { J } _ { \phi } ( 1 ) - \mathrm { J } _ { \phi } \big ( - 1 / \bar { ( } B -$ 1)) + +By setting $\alpha = 1$ and $\phi =$ relu, we get + +Corollary F.10. For any BSB1 $\Sigma \in S _ { B }$ , $\mathrm { V } _ { B _ { \mathrm { r e l u } } } ( \Sigma )$ is BSB1 with diagonal entries $\frac { 1 } { 2 }$ and off-diagonal entries $\textstyle { \frac { 1 } { 2 } } \mathrm { J _ { 1 } } \left( { \frac { - 1 } { B - 1 } } \right)$ , so that $\begin{array} { r } { G ^ { \otimes 2 } \{ \mathsf { V } _ { \mathcal { B } _ { \mathrm { r e l u } } } ( \Sigma ) \} = G ( \mathsf { V } _ { \mathcal { B } _ { \mathrm { r e l u } } } ( \Sigma ) ) G = \left( \frac { 1 } { 2 } - \frac { 1 } { 2 } \mathsf { J } _ { 1 } \left( \frac { - 1 } { B - 1 } \right) \right) G . } \end{array}$ + +By setting $\alpha = 1$ and $\phi = \operatorname { i d } = x \mapsto \operatorname { r e l u } ( x ) - \operatorname { r e l u } ( - x )$ , we get + +Corollary F.11. For any BSB1 $\Sigma \in S _ { B }$ , $\mathrm { V } _ { B _ { \mathrm { i d } } } ( \Sigma )$ is BSB1 with diagonal entries 1 and off-diagonal entries $\frac { - 1 } { B - 1 }$ , so that $G ^ { \otimes 2 } \{ \mathrm { V } _ { { \cal B } _ { \mathrm { i d } } } ( \Sigma ) \} = B / ( \stackrel { } { B } - 1 ) G$ . + +Remark F.12. One might hope to tweak the Laplace method for computing the fixed point to work for the Fourier method, but because there is no nice relation between $\mathrm { V } _ { \phi } ( c \Sigma )$ and $\mathrm { V } _ { \phi } ( \bar { \Sigma } )$ in general, we cannot simplify Eq. (32) as we can Eq. (26) and Eq. (27). + +# F.2.3 GEGENBAUER EXPANSION + +It turns out that the BSB1 fixed point can be described very cleanly using the Gegenbauer coefficients of $\phi$ . + +When $\Sigma$ is BSB1, $\mathcal { K } ( v , \Sigma ) = 1$ for all $v \in S ^ { B - 2 }$ , so that by Proposition E.34, + +$$ +\begin{array} { r l } & { \Sigma ^ { * } = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathbb { e } v ) ^ { \otimes 2 } } \\ & { \Sigma _ { a b } ^ { * } = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathbb { e } _ { a , : } v ) \phi ( \sqrt { B } \mathbb { e } _ { b , : } v ) } \end{array} +$$ + +for any $a , b \in [ B ]$ , independent of the actual values of $\Sigma$ . Here $\Phi _ { a , \mathbf { \lambda } }$ : is the $a$ th row of $\mathbb { P }$ . + +Theorem F.13. If $\phi ( { \sqrt { B - 1 } } x ) \in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \frac { B - 3 } { 2 } } )$ has Gegenbauer expansion $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ , then $\Sigma ^ { * } = \mu ^ { * } I + \nu ^ { * } \mathbf { 1 1 } ^ { T }$ with + +$$ +\begin{array} { l } { { \displaystyle \mu ^ { * } = \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } } \\ { { \displaystyle \quad = \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } \left( \binom { B - 4 + l } { l } - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } } \\ { { \displaystyle \nu ^ { * } = \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) } } \end{array} +$$ + +Proof. By the reproducing property Fact E.46, we have + +$$ +\begin{array} { r l } { \Phi _ { \alpha \beta } ^ { \alpha } = } & { \langle \phi ( \sqrt { D } w _ { \alpha \beta } ; 0 ) \phi ( \sqrt { D } w _ { \alpha \beta } ) \rangle _ { \beta \alpha \beta } , } \\ & { = \langle \phi ( \sqrt { D } w - 1 ) \phi _ { \alpha \beta } , 1 \phi _ { \alpha \beta } \rangle _ { \beta \alpha } , } \\ & { = \langle \phi ( \sqrt { D } w _ { \alpha \beta } - 1 ) c _ { \alpha \beta } , 1 \phi _ { \alpha \beta } \rangle _ { \beta \alpha } , } \\ & { = \langle \displaystyle \sum _ { j = 0 } ^ { \infty } a _ { j } \frac { 1 } { c _ { \alpha \beta - 1 , j } } c _ { j } ^ { ( \alpha \beta \alpha ) } ( \phi _ { \alpha \beta } ) , \displaystyle \sum _ { j = 0 } ^ { \infty } a _ { j } \frac { 1 } { c _ { \alpha \beta - 1 , j } } c _ { j } ^ { ( \alpha \beta \alpha , 3 ) } ( \phi _ { \beta \beta } , + ) \rangle _ { \alpha \beta } , } \\ & { = \langle \displaystyle \sum _ { j = 0 } ^ { \infty } a _ { j } Z _ { \alpha \alpha \beta - 1 , j } ^ { \beta \alpha } , \displaystyle \sum _ { j = 0 } ^ { \infty } a _ { j } Z _ { \alpha \beta - 2 , j } ^ { \beta \alpha } \rangle _ { j \alpha \beta } \rangle _ { j \alpha \beta - 1 } \rangle _ { j \alpha \beta - 2 } \rangle } \\ & { = \langle \displaystyle \sum _ { j = 0 } ^ { \infty } a _ { j } Z _ { \alpha \beta - 2 , j } ^ { \beta \alpha \beta - 2 \beta } , \qquad \mathrm { f o r ~ a l l ~ } \qquad \mathrm { ~ a l l ~ } \qquad \mathrm { ~ f o r ~ a l l ~ } \qquad \mathrm { ~ a l l ~ } \qquad \mathrm { ~ f o r ~ a ~ \beta ~ } } \\ & { = \displaystyle \sum _ { j = 0 } ^ { \infty } a _ { j } ^ { 2 } Z _ { \alpha \beta - 2 } ^ { \beta \alpha - 2 \beta } ( \phi _ { \alpha \beta } ) , } \\ & { = \displaystyle \sum _ { j = 0 } ^ { \infty } a _ { j } ^ { 2 } \frac { 1 } { c _ { \alpha \beta - 1 , j } } c _ { j } ^ { ( \alpha \beta \alpha ) } ( \phi _ { \alpha \beta } , \phi _ { \beta \beta } ) , } \\ & = \displaystyle \sum _ { j = 0 } ^ { \infty } a _ { j } ^ { 2 } \frac { 1 } { c _ { \alpha \beta - 1 , j } } c _ { j } ^ { ( \alpha \beta \alpha ) } ( \ \end{array} +$$ + +This gives the third equation. Since $\mu ^ { * } = \Sigma _ { 1 1 } ^ { * } - \Sigma _ { 1 2 } ^ { * }$ , straightforward arithmetic yields the first equation. The second follows from Fact E.41. + +We will see later that the rate of gradient explosion also decomposes nicely in terms of Gegenbauer coefficients $\mathrm { \ T h m ~ G } . 5 )$ . However, this is not quite true for the eigenvalues of the forward convergence (Thms F.22 and F.24). + +# F.3 LOCAL CONVERGENCE + +In this section we consider linearization of the dynamics given in Eq. (43). Thus we must consider linear operators on the space of PSD linear operators $\boldsymbol { S } _ { B }$ . To avoid confusion, we use the following notation: If $\tau : S _ { B } S _ { B }$ (for example the Jacobian of $\mathrm { V } _ { B _ { \phi } }$ ) and $\Sigma \in S _ { B }$ , then write $\mathcal { T } \{ \Sigma \}$ for the image of $\Sigma$ under $\tau$ . + +A priori, the Jacobian of $\mathrm { V } _ { B _ { \phi } }$ at its BSB1 fixed point may seem like a very daunting object. But a moment of thought shows that + +Proposition F.14. The Jacobian $\frac { \mathrm { d } \mathrm { V } _ { B _ { \phi } } } { \mathrm { d } \Sigma } \big | _ { \Sigma } : \mathcal { H } _ { B } \mathcal { H } _ { B }$ is ultrasymmetric for any BSB1 $\Sigma$ + +Now we prepare several helper reults in order to make progress understanding batchnorm. + +Definition F.15. Define $\mathsf { n } ( x ) = \sqrt { B } x / \| x \|$ , i.e. division by sample standard deviation. + +Batchnorm $B _ { \phi }$ can be decomposed as the composition of three steps, $\phi \circ { \mathsf { n } } \circ G$ , where $G$ is meancentering, n is division by standard deviation, and $\phi$ is coordinate-wise application of nonlinearity. We have, as operators $\mathcal { H } \to \mathcal { H }$ , + +$$ +\begin{array} { r l } & { \frac { \mathrm { d } \mathrm { V } _ { \mathcal { B } _ { \phi } } ( \Sigma ) } { \mathrm { d } \Sigma } = \frac { \mathrm { d } \mathbb { E } [ \mathcal { B } _ { \phi } ( z ) ^ { \otimes 2 } : z \in \mathcal { N } ( 0 , \Sigma ) ] } { \mathrm { d } \Sigma } } \\ & { \quad \quad \quad = \frac { \mathrm { d } \mathbb { E } [ ( \phi \circ \mathfrak { n } ) ( x ) ^ { \otimes 2 } : x \in \mathcal { N } ( 0 , G \Sigma G ) ] } { \mathrm { d } \Sigma } } \\ & { \quad \quad \quad = \frac { \mathrm { d } \mathrm { V } _ { ( \phi \circ \mathfrak { n } ) } ( \Sigma ^ { G } ) } { \mathrm { d } \Sigma ^ { G } } \circ \frac { \mathrm { d } \Sigma ^ { G } } { \mathrm { d } \Sigma } } \\ & { \quad \quad \quad = \frac { \mathrm { d } \mathrm { V } _ { ( \phi \circ \mathfrak { n } ) } ( \Sigma ^ { G } ) } { \mathrm { d } \Sigma ^ { G } } \circ G ^ { \otimes 2 } } \end{array} +$$ + +Definition F.16. With $\Sigma ^ { * }$ being the unique BSB1 fixed point, write $\begin{array} { r l r } { \mathcal { U } } & { { } : = } & { G ^ { \otimes 2 } } \end{array}$ ◦ $\frac { \mathrm { d } \mathrm { V } _ { ( \phi \circ \mathfrak { n } ) } ( \Sigma ^ { G } ) } { \mathrm { d } \Sigma ^ { G } } \Big | _ { \Sigma ^ { G } = G \Sigma ^ { * } G } : \mathcal { H } ^ { G } \to \mathcal { H } ^ { G }$ + +It turns out to be advantageous to study $\mathcal { U }$ first and relate its eigendecomposition back to that of $\frac { \mathrm { d } \mathrm { V } _ { B _ { \phi } } ( \Sigma ) } { \mathrm { d } \Sigma } \big | _ { \Sigma = \Sigma ^ { * } }$ ) Σ=Σ∗ , where Σ∗ is the BSB1 fixed point, by applying Lemma F.17. + +Lemma F.17. Let $X$ and $Y$ be two vector spaces. Consider linear operators $A : X \to Y , B : Y \to$ $X$ . Then + +$$ +I . { \mathrm { ~ \ r a n k { \ } } } A B = { \mathrm { r a n k } } B A +$$ + +2. If $v \in Y$ is an eigenvector of $A B$ with nonzero eigenvalue, then $X \ni B v \neq 0$ and $B v$ is an eigenvector of $B A$ of the same eigenvalue + +3. Suppose $A B$ has $k = { \mathrm { r a n k } } A B$ linearly independent eigenvectors $\{ v _ { i } \} _ { i = 1 } ^ { k }$ of nonzero eigenvalues $\{ \lambda _ { i } \} _ { i = 1 } ^ { k }$ . Then $B A$ has $k$ linearly independent eigenvectors $\{ B v _ { i } \} _ { i = 1 } ^ { k }$ with the same eigenvalues $\{ \lambda _ { i } \} _ { i = 1 } ^ { k }$ which are all eigenvectors of $B A$ with nonzero eigenvalues, up to linear combinations within eigenvectors with the same eigenvalue. + +With $A = G ^ { \otimes 2 }$ and $\begin{array} { r } { B = \frac { \mathrm { d } \mathrm { V } _ { ( \phi \circ \mathsf { n } ) } \left( \Sigma ^ { G } \right) } { \mathrm { d } \Sigma ^ { G } } } \end{array}$ , $\mathrm { T h m } \mathrm { E } . 6 1 $ implies that $A B$ and $B A$ can both be diagonalized, and this lemma implies that all nonzero eigenvalues of $\frac { \mathrm { d } \mathrm { V } _ { B _ { \phi } } ( \Sigma ) } { \mathrm { d } \Sigma }$ can be recovered from those of $\mathcal { U }$ . + +Proof. (Item 1) Observe rank $A B = \operatorname { r a n k } A B A B \leq \operatorname { r a n k } B A$ . By symmetry the two sides are in fact equal. + +(Item 2) $B v$ cannot be zero or otherwise $A B v \ = \ A 0 \ = \ 0$ , contradicting the fact that $v$ is an eigenvector with nonzero eigenvalue. Suppose $\lambda$ is the eigenvalue associated to $v$ . Then $B A ( B v ) =$ $\bar { B ( A B v ) } = B ( \lambda v ) = \lambda B v$ , so $B v$ is an eigenvector of $B A$ with the same eigenvalue. + +(Item 3) Item 2 shows that $\{ B v _ { i } \} _ { i = 1 } ^ { k }$ are eigenvectors $B A$ with the same eigenvalues $\{ \lambda _ { i } \} _ { i = 1 } ^ { k }$ . The eigenspaces with different eigenvalues are linearly independent, so it suffices to show that if $\{ B v _ { i _ { j } } \} _ { j }$ are eigenvectors of the same eigenvalue $\lambda _ { s }$ , then they are linearly independent. But $\textstyle \sum _ { j } a _ { j } B v _ { i _ { j } } =$ $\begin{array} { r } { \mathrm { ~ 0 ~ \implies ~ } \sum _ { j } a _ { j } v _ { i _ { j } } = 0 } \end{array}$ because $B$ is injective on eigenvectors by Item 2, so that $a _ { j } = 0$ identically. Hence $\{ B v _ { j } \} _ { j }$ is linearly independent. + +Since rank $B A = k$ , these are all of the eigenvectors with nonzero eigenvalues of $B A$ up to linearly combinations. + +Lemma F.18. Let $f : \mathbb { R } ^ { B } \to \mathbb { R } ^ { A }$ be measurable, and $\Sigma \in { \cal S } _ { B }$ be invertible. Then for any $\Lambda \ \in$ $\mathbb { R } ^ { B \times B }$ , with $\langle \cdot , \cdot \rangle$ denoting trace inner product, + +$$ +{ \frac { \mathrm { d } } { \mathrm { d } \Sigma } } \mathbb { E } [ f ( z ) : z \sim { \mathcal { N } } ( 0 , \Sigma ) ] \{ \Lambda \} = { \frac { 1 } { 2 } } \mathbb { E } [ f ( z ) \left. \Sigma ^ { - 1 } z z ^ { T } \Sigma ^ { - 1 } - \Sigma ^ { - 1 } , \Lambda \right. : z \sim { \mathcal { N } } ( 0 , \Sigma ) ] +$$ + +If $f$ is in addition twice-differentiable, then + +$$ +{ \frac { \mathrm { d } } { \mathrm { d } \Sigma } } \mathbb { E } [ f ( z ) : z \sim { \mathcal { N } } ( 0 , \Sigma ) ] = { \frac { 1 } { 2 } } \mathbb { E } \left[ { \frac { \mathrm { d } ^ { 2 } f ( z ) } { \mathrm { d } z ^ { 2 } } } : z \sim { \mathcal { N } } ( 0 , \Sigma ) \right] +$$ + +whenever both sides exist. + +Proof. Let $\Sigma _ { t } , t \in ( - \epsilon , \epsilon )$ be a smooth path in $\boldsymbol { S } _ { B }$ , with $\Sigma _ { 0 } = \Sigma$ . Write $\begin{array} { r } { \frac { d } { d t } \Sigma _ { t } = \dot { \Sigma } _ { t } } \end{array}$ . Then + +$$ +\begin{array} { r l } & { \frac { d } { d t } \mathbb { E } [ f ( z ) : z \sim \mathcal { N } ( 0 , \Sigma _ { \varepsilon } ) ] } \\ & { = \frac { d } { d t } ( 2 \pi ) ^ { - \delta / 2 } \mathrm { d } \mathrm { e t } \Sigma _ { \varepsilon } ^ { - 1 / 2 } \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma _ { \varepsilon } \cdot 1 } { _ 1 } ^ { t } f ( z ) \Big | _ { \mathrm { L } = 0 } } \\ & { = ( 2 \pi ) ^ { - \delta / 2 } \frac { 1 } { 2 } \mathrm { d } \mathrm { e t } \Sigma _ { \varepsilon } ^ { - 1 / 2 } \mathrm { e t } \left( \sum _ { 0 } ^ { - 1 } \bar { \Sigma } _ { 0 } ^ { T } \Sigma _ { 0 } ^ { - 1 } \right) \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma _ { 0 } ^ { - 1 } z ^ { T } \int \bar { f } ( z ) } } \\ & { \qquad + ( 2 \pi ) ^ { - \delta / 2 } \mathrm { d } \mathrm { e t } \Sigma _ { 0 } ^ { - 1 / 2 } \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma _ { 0 } ^ { T } \Sigma _ { 0 } ^ { - 1 } \cdot 2 } \frac { 1 } { 2 } z ^ { T } ( - \Sigma _ { 0 } ^ { - 1 } \Sigma _ { 0 } ^ { - 1 } ) z ( z ) ^ { - 1 } \mathrm { e / \Sigma } } \\ & { = \frac { - 1 } { 2 } ( 2 \pi ) ^ { - \delta / 2 } \mathrm { d } \mathrm { e t } \Sigma _ { 0 } ^ { - 1 / 2 } \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma _ { 0 } ^ { - 1 } z } f ( z ) ( ( \mathrm { r } \Sigma _ { 0 } ^ { - 1 } \Sigma _ { 0 } - z ^ { T } \Sigma _ { 0 } ^ { - 1 } \Sigma _ { 0 } ^ { - 1 } \Sigma _ { 0 } ^ { - 1 } \Sigma _ { 0 } ^ { - 1 } \Sigma _ { 0 } ^ { - 1 } ) } \\ & = \frac { 1 } { 2 } ( 2 \pi ) ^ { - \delta / 2 } \mathrm { d } \mathrm { e t } \Sigma _ { 0 } ^ { - 1 / 2 } \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma _ { 0 } ^ { - 1 } z } f ( z ) ( \Sigma ^ { - 1 } z ^ { T } \Sigma ^ { - 1 } - \Sigma ^ \end{array} +$$ + +Note that + +$$ +\begin{array} { c } { { v ^ { T } \displaystyle \frac { \mathrm { d } } { \mathrm { d } z } e ^ { \frac { - 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } = - v ^ { T } \Sigma ^ { - 1 } z e ^ { \frac { - 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } } } \\ { { w ^ { T } \left( \displaystyle \frac { \mathrm { d } ^ { 2 } } { \mathrm { d } z ^ { 2 } } e ^ { \frac { - 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \right) v = \left( w ^ { T } \Sigma ^ { - 1 } z z ^ { T } \Sigma ^ { - 1 } v - w ^ { T } \Sigma ^ { - 1 } v \right) e ^ { \frac { - 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } } } \\ { { { } } } \\ { { { } = \left. \Sigma ^ { - 1 } z z ^ { T } \Sigma ^ { - 1 } - \Sigma ^ { - 1 } , v w ^ { T } \right. e ^ { \frac { - 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } } } \end{array} +$$ + +so that as a cotensor, + +$$ +{ \frac { \mathrm { d } ^ { 2 } } { \mathrm { d } z ^ { 2 } } } e ^ { { \frac { - 1 } { 2 } } z ^ { T } \Sigma ^ { - 1 } z } \{ \Lambda \} = \left. \Sigma ^ { - 1 } z z ^ { T } \Sigma ^ { - 1 } - \Sigma ^ { - 1 } , \Lambda \right. e ^ { { \frac { - 1 } { 2 } } z ^ { T } \Sigma ^ { - 1 } z } +$$ + +for any Λ RB×B . + +Therefore, + +$$ +\begin{array} { r l r } & { } & { \frac { \displaystyle \mathrm { d } \mathbb { E } [ f ( z ) : z \sim { \mathcal { N } ( 0 , \Sigma ) } ] } { \mathrm { d } \Sigma } \{ \Lambda \} = \frac { 1 } { 2 } ( 2 \pi ) ^ { - B / 2 } \operatorname* { d e t } \Sigma ^ { - 1 / 2 } \int \mathrm { d } z f ( z ) \frac { \displaystyle \mathrm { d } ^ { 2 } e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } } { \mathrm { d } ^ { 2 } z } \{ \Lambda \} } \\ & { } & { \quad \quad \quad \quad = \frac { 1 } { 2 } ( 2 \pi ) ^ { - B / 2 } \operatorname* { d e t } \Sigma ^ { - 1 / 2 } \int \mathrm { d } z e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \frac { \displaystyle \mathrm { d } ^ { 2 } f ( z ) } { \displaystyle \mathrm { d } ^ { 2 } z } \{ \Lambda \} } \\ & { } & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ( \mathrm { b y ~ i n t e g r a t i o n ~ b y ~ p a r t s } ) } \\ & { } & { \quad \quad \quad \quad = \frac { 1 } { 2 } \mathbb { E } \left[ \frac { \mathrm { d } ^ { 2 } f ( z ) } { \displaystyle \mathrm { d } ^ { 2 } z } : z \sim \mathcal { N } ( 0 , \Sigma ) \right] \{ \Lambda \} } \end{array} +$$ + +Note that for any $\Lambda \in \mathcal { H } _ { B }$ with $\| \boldsymbol { \Lambda } \| _ { o p } < \mu ^ { * }$ , + +$$ +\begin{array} { r l } & { \mathbb { E } [ \phi \circ \mathsf { n } ( z ) ^ { \otimes 2 } : z \in \mathcal { N } ( 0 , \mu ^ { * } G + \Lambda ) ] = \mathbb { E } [ \phi \circ \mathsf { n } ( G z ) ^ { \otimes 2 } : z \in \mathcal { N } ( 0 , \mu ^ { * } I + \Lambda ) ] } \\ & { \qquad = \mathbb { E } [ \mathcal { B } _ { \phi } ( z ) ^ { \otimes 2 } : z \in \mathcal { N } ( 0 , \mu ^ { * } I + \Lambda ) ] } \end{array} +$$ + +so that we have, for any $\Lambda \in \mathcal { H } _ { B }$ , + +$$ +\begin{array} { r l } { u \{ \Delta \} } & { = \frac { \mathrm { d } \mathrm { V } _ { \mathrm { S } } \alpha ( \Sigma ) } { \mathrm { d } \Sigma } \Big | _ { \Sigma = \mu ^ { * } } \{ \Delta \} } \\ & { \phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\ & { = \frac { 1 } { 2 } \mathbb { E } [ B _ { \delta } ( z ) ^ { \otimes 2 } \langle a ^ { \mu - 1 } z z ^ { T } - \mu ^ { \mu - 1 } I , \Delta \rangle : z \sim \mathcal { N } ( 0 , \mu ^ { * } I ) ] } \\ & { \phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\ & { \phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\ & { = \frac { 1 } { 2 } \mu ^ { \mu _ { \mathrm { S } } \alpha - 2 } \mathbb { E } [ B _ { \delta } ( z ) ^ { \otimes 2 } \langle z z ^ { T } , \Delta \rangle : z \sim \mathcal { N } ( 0 , \mu ^ { * } I ) ] - \frac { 1 } { 2 } \mu ^ { \mu _ { \mathrm { S } } - 1 } \Sigma ^ { * } \langle I , \Delta \rangle } \\ & { \phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\ & { \phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\ & { = ( 2 \mu ^ { * } ) ^ { - 1 } \left( \mathbb { E } [ B _ { \delta } ( z ) ^ { \otimes 2 } \langle z z ^ { T } G , \Delta \rangle : z \sim \mathcal { N } ( 0 , I ) ] - \mathbb { E } ^ { * } \langle I , \Delta \rangle \right) } \\ & { \phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \end{array} +$$ + +Let’s extend to all matrices by this formula: + +Definition F.19. Define + +$$ +\begin{array} { r l } & { \tilde { \mathcal { U } } : \mathbb { R } ^ { B \times B } S _ { B } , } \\ & { \qquad \Lambda \mapsto ( 2 \mu ^ { * } ) ^ { - 1 } ( \mathbb { E } [ \mathcal { B } _ { \phi } ( z ) ^ { \otimes 2 } \langle G z z ^ { T } G , \Lambda \rangle : z \sim \mathcal { N } ( 0 , I ) ] - \Sigma ^ { * } \langle I , \Lambda \rangle ) } \end{array} +$$ + +# F.3.1 SPHERICAL INTEGRATION + +$\mathbf { S o } \tilde { \mathcal { U } } \mid \mathcal { H } _ { B } = \mathcal { U } \mid \mathcal { H } _ { B }$ . Ultimately we will apply $\cdot$ to $G ^ { \otimes 2 } \circ \tilde { \mathcal { U } } \mid \mathcal { H } _ { B } = G ^ { \otimes 2 } \circ \mathcal { U }$ Definition F.20. Write $\begin{array} { r } { \tau _ { i j } = \frac { 1 } { 2 } \left( \delta _ { i } \delta _ { j } ^ { T } + \delta _ { j } \delta _ { i } ^ { T } \right) } \end{array}$ . + +Then + +$$ +\begin{array} { r l } & { \tilde { \mathcal { U } } \{ \tau _ { i j } \} _ { k l } = ( 2 \mu ^ { * } ) ^ { - 1 } \left( \mathbb { E } [ \mathcal { B } _ { \phi } ( z ) _ { k } \mathcal { B } _ { \phi } ( z ) _ { l } ( G z ) _ { i } ( G z ) _ { j } : z \sim \mathcal { N } ( 0 , I ) ] - \Sigma _ { k l } ^ { * } \mathbb { I } ( i = j ) \right) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \mathrm { f o r d e d } \ M [ i = j ] , } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{array} +$$ + +Here we will realize $\mathbb { P }$ as the matrix in Eq. (33). + +Definition F.21. Define $W _ { i j | k l } : = \mathbb { E } [ \phi ( \mathsf { n } ( \mathtt { e } x ) ) _ { k } \phi ( \mathsf { n } ( \mathtt { e } x ) ) _ { l } ( \mathtt { e } x ) _ { i } ( \mathtt { e } x ) _ { j } : x \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] .$ + +Then $\tilde { \mathcal { U } } \{ \tau _ { i j } \} _ { k l } = ( 2 \mu ^ { * } ) ^ { - 1 } ( W _ { i j | k l } - \Sigma _ { k l } ^ { * } \mathbb { I } ( i = j ) )$ . If we can evaluate $W _ { i j | k l }$ then we can use Thm E.61 to compute the eigenvalues of $G ^ { \otimes 2 } \circ u$ . It’s easy to see that $W _ { i j | k l }$ is ultrasymmetric Thus WLOG we can take $i , j , k , l$ from $\{ 1 , 2 , 3 , 4 \}$ . + +By Lemma E.35, and the fact that $x \mapsto \Phi x$ is an isometry, + +$$ +\begin{array} { l } { { W _ { i j | k l } = \mathbb E [ r ^ { 2 } \phi ( \sqrt { B } \mathfrak e v ) _ { k } \phi ( \sqrt { B } \mathfrak e v ) _ { l } ( \mathfrak e v ) _ { i } ( \mathfrak e v ) _ { j } : x \sim \mathcal N ( 0 , I _ { B - 1 } ) ] } } \\ { { \ = ( B - 5 ) ( B - 3 ) ( B - 1 ) ( 2 \pi ) ^ { - 2 } \times } } \\ { { \displaystyle \qquad \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } \cdot \cdot \cdot \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 4 } \phi ( \sqrt { B } \mathfrak e v ) _ { k } \phi ( \sqrt { B } \mathfrak e v ) _ { l } ( \mathfrak e v ) _ { i } ( \mathfrak e v ) _ { j } \sin ^ { B - 3 } \theta _ { 1 } \cdot \cdot \cdot \sin ^ { B - 6 } \theta _ { 4 } } } \end{array} +$$ + +If WLOG we further assume that $k , l \in \{ 1 , 2 \}$ (by ultrasymmetry), then there is no dependence on $\theta _ { 3 }$ and $\theta _ { 4 }$ inside $\phi$ . So we can expand $( \mathbb { e } v ) _ { i }$ and $( \mathbb { e } v ) _ { j }$ in trigonometric expressions as in Eq. (34) and integrate out $\theta _ { 3 }$ and $\theta _ { 4 }$ via Lemma E.29. We will not write out this integral explicitly but instead focus on other techniques for evaluating the eigenvalues. + +# F.3.2 GEGENBAUER EXPANSION + +Now let’s compute the local convergence rate via Gegenbauer expansion. By differentiating Proposition E.34 through a path $\Sigma _ { t } \in { \cal S } _ { B } , t \in ( - \epsilon , \epsilon )$ , we get + +$$ +\begin{array} { l } { \displaystyle { { ^ \prime } } _ { B _ { \phi } } ( { \Sigma } _ { t } ) = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathfrak { e } v ) ^ { \otimes 2 } \frac { d } { d t } K ( v ; { \Sigma } _ { t } ) } \\ { = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathfrak { e } v ) ^ { \otimes 2 } K ( v ; { \Sigma } _ { t } ) \left( \frac { B - 1 } { 2 } ( v ^ { T } { \Sigma } _ { \boxdot } { v } ^ { - 1 } v ) ^ { - 1 } v ^ { T } { \Sigma } _ { \boxdot } { } ^ { - 1 } \frac { d { \Sigma } _ { \boxdot } { \Sigma } } { d t } { \Sigma } _ { \boxdot } { } ^ { - 1 } v - \frac { 1 } { 2 } \operatorname { t r } ( { \Sigma } _ { \boxdot } { \dot } ^ { - 1 } \frac { d { \Sigma } _ { \boxdot } { \Sigma } } { d t } ) ^ { \otimes 2 } \right) , } \end{array} +$$ + +where $\Sigma _ { \Pi } = \Sigma _ { \Pi t } = \Phi ^ { T } \Sigma _ { t ^ { \mathbb { C } } } \in S _ { B - 1 }$ (as introduced below Defn E.3). At $\Sigma _ { 0 } = \Sigma ^ { * }$ , the BSB1 fixed point, we have $\Sigma _ { \perp 0 } = \mu ^ { * } I$ , and + +$$ +\begin{array} { r l } & { \frac { d } { d t } \nabla _ { B _ { \phi } } ( \Sigma _ { t } ) \bigg | _ { t = 0 } = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathfrak { e } v ) ^ { \otimes 2 } K ( v ; \Sigma ^ { * } ) \mu ^ { * - 1 } \left( \frac { B - 1 } { 2 } v ^ { T } \frac { d \Sigma _ { \boxplus } } { d t } v - \frac { 1 } { 2 } \operatorname { t r } ( \frac { d \Sigma _ { \boxplus } } { d t } ) \right) } \\ & { \qquad = \mu ^ { * - 1 } \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathfrak { e } v ) ^ { \otimes 2 } \left( \frac { B - 1 } { 2 } v ^ { T } \frac { d \Sigma _ { \boxplus } } { d t } v - \frac { 1 } { 2 } \operatorname { t r } ( \frac { d \Sigma _ { \boxplus } } { d t } ) \right) } \end{array} +$$ + +If $\textstyle { \frac { d } { d t } } \Sigma | _ { t = 0 } = G$ , then the term in the parenthesis vanishes. This shows that $\left. \frac { \mathrm { d } \mathrm { V } \mathfrak { s } _ { \phi } } { \mathrm { d } \Sigma } \right| _ { \Sigma = \Sigma ^ { * } } \left\{ G \right\} = 0 .$ + +Eigenvalue for L. If $\begin{array} { r } { \frac { d } { d t } \Sigma | _ { t = 0 } = \operatorname { L } ( B - 2 , 1 ) = B G ( \delta _ { 1 } ^ { \otimes 2 } - \delta _ { 2 } ^ { \otimes 2 } ) G } \end{array}$ where $\delta _ { 1 } = ( 1 , 0 , \ldots , 0 ) , \delta _ { 2 } =$ $( 0 , 1 , 0 , \ldots , 0 )$ , both in $\mathbb { R } ^ { B }$ (Proposition E.59), then we know by Thm E.62 that $G ^ { \otimes 2 } \circ$ $\begin{array} { r } { \frac { d } { d t } \mathrm { V } _ { B _ { \phi } } ( \Sigma _ { t } ) \big | _ { t = 0 } } \end{array}$ is a multiple of $\mathrm { L } ( B - 2 , 1 )$ . We compute + +$$ +\left( \frac { d } { d t } \nabla _ { B _ { \phi } } ( \Sigma _ { t } ) \bigg | _ { t = 0 } \right) _ { a b } = B \frac { B - 1 } { 2 } \mu ^ { * - 1 } \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathbb { e } _ { a , : } v ) \phi ( \sqrt { B } \mathbb { e } _ { b , : } v ) ( ( \mathbb { e } _ { 1 , : } v ) ^ { 2 } - ( \mathbb { e } _ { 2 , : } v ) ^ { 2 } ) . +$$ + +Clearly, $\begin{array} { r } { \frac { d } { d t } \mathrm { V } _ { B _ { \phi } } ( \Sigma _ { t } ) \big | _ { t = 0 } } \end{array}$ is L-shaped. + +Now define the quantity $\begin{array} { r l r } { A ( { a } , { b } ; { c } ) } & { : = } & { \mathbb { E } _ { v \sim S ^ { B - 2 } } \phi ( \sqrt { B } \mathbb { e } _ { a , : } v ) \phi ( \sqrt { B } \mathbb { e } _ { b , : } v ) ( \mathbb { e } _ { c , : } v ) ^ { 2 } } \end{array}$ , so that $\begin{array} { r l r } { \left( \frac { d } { d t } \nabla _ { \mathcal { B } _ { \phi } } ( \Sigma _ { t } ) \big | _ { t = 0 } \right) _ { a b } } & { { } = } & { \frac { B ( B - 1 ) } { 2 } \mu ^ { * - 1 } ( \mathcal { A } ( a , b ; 1 ) - \mathcal { A } ( a , b ; 2 ) ) } \end{array}$ B(B−1)2 µ∗−1(A(a, b; 1) − A(a, b; 2)). Because e is an isometry, $\begin{array} { r } { \sum _ { b } ( \Phi _ { b , : } v ) ^ { 2 } = \| v \| ^ { 2 } = 1 } \end{array}$ for all $v \in S ^ { B - 2 }$ . Thus + +$$ +\sum _ { c = 1 } ^ { B } A ( a , b ; c ) = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathbb { e } _ { a , : } v ) \phi ( \sqrt { B } \mathbb { e } _ { b , : } v ) = \Sigma _ { a b } ^ { * } . +$$ + +By symmetry, $\mathcal { A } ( a , b ; a ) = \mathcal { A } ( a , b ; b )$ and for any $c , c ^ { \prime } \notin \{ a , b \} , A ( a , b ; c ) = A ( a , b ; c ^ { \prime } )$ . So we have, for any $a , b , c$ not equal, + +$$ +\begin{array} { r } { A ( a , a ; a ) + ( B - 1 ) A ( a , a ; c ) = \Sigma _ { a a } ^ { * } } \\ { 2 A ( a , b ; b ) + ( B - 2 ) A ( a , b ; c ) = \Sigma _ { a b } ^ { * } } \end{array} +$$ + +So the eigenvalue associated to $\mathrm { L } ( B - 2 , 1 )$ is, So by Lemma E.60, ${ \cal G } ^ { \otimes 2 } \circ \mathcal { U } \{ \mathrm { L } ( B - 2 , 1 ) \} =$ $\lambda _ { \mathbb { L } } ^ { \uparrow } \mathrm { L } ( B - 2 , 1 )$ , where + +$$ +\begin{array} { r l } & { \overset { \uparrow } { \underset { \mathrm { L } } { \mathrm { T } } } = B \frac { B - 1 } { 2 } \mu ^ { * - 1 } \frac { ( \frac { d } { d t } \mathrm { V } _ { \mathcal { B } _ { \phi } } ( \Sigma _ { t } ) | _ { t = 0 } ) _ { 1 1 } - 2 ( \frac { d } { d t } \mathrm { V } _ { \mathcal { B } _ { \phi } } ( \Sigma _ { t } ) \big | _ { t = 0 } ) _ { 1 3 } } { B } } \\ & { \quad = B \frac { B - 1 } { 2 } \mu ^ { * - 1 } \frac { A ( 1 , 1 ; 1 ) - A ( 1 , 1 ; 2 ) - 2 ( A ( 1 , 3 ; 1 ) - A ( 1 , 3 ; 2 ) ) } { B } } \\ & { \quad = \frac { B - 1 } { 2 } \mu ^ { * - 1 } ( A ( 1 , 1 ; 1 ) - \frac { 1 } { B - 1 } ( \Sigma _ { a a } ^ { * } - A ( 1 , 1 ; 1 ) ) - 2 ( A ( 1 , 3 ; 1 ) - \frac { 1 } { B - 2 } ( \Sigma _ { a b } ^ { * } - 2 A ( 1 , 3 ; 1 ) - \frac { 1 } { B - 2 } ( \Sigma _ { a b } ^ { * } - 2 A ( 1 , 3 ; 1 ) - \frac { 1 } { B - 2 } \Sigma _ { a b } ^ { * } ) ) } \\ & { \quad = \frac { B - 1 } { 2 } \mu ^ { * - 1 } ( \frac { B } { B - 1 } A ( 1 , 1 ; 1 ) - \frac { 1 } { B - 1 } \Sigma _ { a a } ^ { * } - 2 \frac { B } { B - 2 } A ( 1 , 3 ; 1 ) + \frac { 2 } { B - 2 } \Sigma _ { a b } ^ { * } ) \qquad ( 4 8 ) } \end{array} +$$ + +Thus, if $\phi ( \sqrt { B - 1 } x ) \in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \frac { B - 3 } { 2 } } )$ has Gegenbauer expansion $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ then by Proposition E.43, + +$$ +\begin{array} { r l } { { } } & { { \ x ^ { 2 } \phi ( \sqrt { B - 1 } x ) } } \\ { { } } & { { = \displaystyle \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } x ^ { 2 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } } \\ { { } } & { { = \displaystyle \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } \biggl ( \kappa _ { 2 } ( l , \frac { B - 3 } { 2 } ) C _ { l + 2 } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } } \\ { { } } & { { \qquad + \kappa _ { 0 } ( l , \frac { B - 3 } { 2 } ) C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) + \kappa _ { - 2 } ( l , \frac { B - 3 } { 2 } ) C _ { l - 2 } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) \biggr ) } } \end{array} +$$ + +where $C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x )$ is understood to be 0 for $l < 0$ + +$$ +\begin{array} { l } { \displaystyle = \sum _ { l = 0 } ^ { \infty } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \\ { \displaystyle \quad \quad \quad \left( \frac { a _ { l - 2 } } { c _ { B - 1 , l - 2 } } \kappa _ { 2 } ( l - 2 , \frac { B - 3 } { 2 } ) + \frac { a _ { l } } { c _ { B - 1 , l } } \kappa _ { 0 } ( l , \frac { B - 3 } { 2 } ) + \frac { a _ { l + 2 } } { c _ { B - 1 , l + 2 } } \kappa _ { - 2 } ( l + 2 , \frac { B - 3 } { 2 } ) \right) } \end{array} +$$ + +$$ +\begin{array} { l } { \displaystyle = \sum _ { l = 0 } ^ { \infty } c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \\ { \displaystyle \quad \quad \left( \frac { a _ { l - 2 } c _ { B - 1 , l } } { c _ { B - 1 , l - 2 } } \kappa _ { 2 } ( l - 2 , \frac { B - 3 } { 2 } ) + a _ { l } \kappa _ { 0 } ( l , \frac { B - 3 } { 2 } ) + \frac { a _ { l + 2 } c _ { B - 1 , l } } { c _ { B - 1 , l + 2 } } \kappa _ { - 2 } ( l + 2 , \frac { B - 3 } { 2 } ) \right) } \end{array} +$$ + +We can evaluate, for any $a , b$ (possibly equal), + +$$ +\begin{array} { l } { { 4 ( a , b ; b ) = \displaystyle \operatorname * { \mathbb { E } } _ { v \sim S ^ { B - 2 } } \phi ( \sqrt { B } \mathfrak { S } a , v ) \phi ( \sqrt { B } \mathfrak { e } _ { b , : } v ) ( \mathfrak { e } _ { b , : } v ) ^ { 2 } } } \\ { { \displaystyle \quad = \operatorname * { \mathbb { E } } _ { v \sim S ^ { B - 2 } } \phi ( \sqrt { B - 1 } \hat { \mathfrak { e } } a , v ) \phi ( \sqrt { B - 1 } \hat { \mathfrak { e } } _ { b , : } v ) ( \hat { \mathfrak { e } } _ { b , : } v ) ^ { 2 } \frac { B - 1 } { B } } } \\ { { \displaystyle \quad = \frac { B - 1 } { B } \sum _ { l = 0 } ^ { \infty } c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( \langle \hat { \mathfrak { e } } _ { a , : } , \hat { \mathfrak { e } } _ { b , : } \rangle ) } } \\ { { \displaystyle a _ { l } \left( \frac { a _ { l - 2 } c _ { B - 1 , l } } { c _ { B - 1 , l - 2 } } \kappa _ { 2 } ( l - 2 , \frac { B - 3 } { 2 } ) + a _ { l } \kappa _ { 0 } ( l , \frac { B - 3 } { 2 } ) + \frac { a _ { l + 2 } c _ { B - 1 , l } } { c _ { B - 1 , l + 2 } } \kappa _ { - 2 } ( l + 2 , \frac { B - 3 } { 2 } ) \right) } } \end{array} +$$ + +By Eq. (48), the eigenvalue associated with $\mathrm { L } ( B - 2 , 1 )$ is + +$$ +\begin{array} { c } { { \displaystyle \frac { B - 1 } { 2 } \mu ^ { * - 1 } ( \frac { B } { B - 1 } A ( 1 , 1 ; 1 ) - \frac { 1 } { B - 1 } \Sigma _ { a a } ^ { * } - 2 \frac { B } { B - 2 } A ( 1 , 3 ; 1 ) + \frac { 2 } { B - 2 } \Sigma _ { a b } ^ { * } ) } } \\ { { = \displaystyle \frac { B - 1 } { 2 } \mu ^ { * - 1 } \sum _ { l = 0 } ^ { \infty } c _ { B - 1 , l } ^ { - 1 } ( \gamma _ { l } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) - \tau _ { l } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( \frac { - 1 } { B - 1 } ) ) } } \end{array} +$$ + +where + +$$ +\begin{array} { l } { \displaystyle { \ u _ { \iota } : = a _ { l } \left( \frac { a _ { l - 2 } c _ { B - 1 , l } } { c _ { B - 1 , l - 2 } } \kappa _ { 2 } ( l - 2 , \frac { B - 3 } { 2 } ) + a _ { l } \kappa _ { 0 } ( l , \frac { B - 3 } { 2 } ) + \frac { a _ { l + 2 } c _ { B - 1 , l } } { c _ { B - 1 , l + 2 } } \kappa _ { - 2 } ( l + 2 , \frac { B - 3 } { 2 } ) \right) - \frac { 1 } { B - 1 } \mathrm { ~ c ~ o ~ n ~ s ~ t ~ } , } } \\ { \displaystyle { \ u _ { \tilde { \iota } } : = \frac { 2 ( B - 1 ) } { B - 2 } \gamma \ u _ { l } . } } \end{array} +$$ + +Thus, + +Theorem F.22. Suppose φ(√B − 1x) has Gegenbauer expansion P∞l=0 al 1cB−1,l $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ . Then the eigenvalue of $\mathcal { U }$ with respect to the eigenspace $\mathbb { L }$ is a ratio of quadratic forms + +$$ +\lambda _ { \mathbb { L } } ^ { \uparrow } = \frac { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } w _ { B - 1 , l } + a _ { l } a _ { l + 2 } u _ { B - 1 , l } } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } v _ { B - 1 , l } } +$$ + +where + +$$ +\begin{array} { r l } & { r _ { ( k - 1 ) , ( k ) } = \varepsilon _ { ( k - 1 ) , ( k ) } ^ { - 1 } \left( C _ { i } ^ { ( k ^ { 2 } - 2 ) } ( 1 ) - C _ { i } ^ { ( k ^ { 2 } - 2 ) } ( \frac { 1 } { B } ) \right) } \\ & { r _ { ( k - 1 ) , ( k ^ { 2 } - 1 ) } ^ { - 1 } \left( r _ { ( k ) } \left( \frac { 1 } { B } - \frac { 1 } { 2 } \right) - \frac { 1 } { 1 - 1 } \right) e _ { ( k - 1 ) } ^ { r _ { ( k ) } } \left( C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( 1 ) - \frac { 2 ( B - 1 ) } { B - 2 } C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( \frac { 1 } { B - 1 } ) \right) } \\ & { \quad \quad \quad - \frac { 1 } { ( B } - 3 + 1 ) ( B - 3 ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { = \frac { 1 } { ( B - 5 ) + ( B ) ( B - 1 + 3 ) ( B - 3 + 2 ) ) } e _ { ( k - 1 ) } ^ { r _ { ( k ) } } \left( C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( 1 ) - \frac { 2 ( B - 1 ) } { B - B } C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( \frac { 1 } { B - 1 } ) \right) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { = \frac { 1 } { ( B - 1 ) ( B - 3 + 2 ) ( B - 3 + 2 ) ) } \left( C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( 1 ) - \frac { 2 ( B - 1 ) } { B } C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( \frac { 1 } { B - 1 } ) \right) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{array} +$$ + +Note that $v _ { B - 1 , 0 } = w _ { B - 1 , 0 } = u _ { B - 1 , 0 } = 0$ , so that there is in fact no dependence on $a _ { 0 }$ , as expected since batchnorm is invariant under additive shifts of $\phi$ . + +We see that $\lambda _ { \mathbb { L } } ^ { \uparrow } \geq 1$ iff + +$$ +0 \leq \sum _ { l = 1 } ^ { \infty } a _ { l } ^ { 2 } ( w _ { B - 1 , l } - v _ { B - 1 , l } ) + a _ { l } a _ { l + 2 } u _ { B - 1 , l } . +$$ + +This is a quadratic form on the coefficients $\{ a _ { l } \} _ { l }$ . We now analyze it heuristically and argue that the eigenvalue is $\geq 1$ typically when $\phi ( \sqrt { B - 1 } x )$ explodes sufficiently as $x \to 1$ or $x \to - 1$ ; in other words, the more explosive $\phi$ is, the less likely it is to induce Eq. (43) to converge to a BSB1 fixed point. + +Heuristically, $C _ { l + 2 } ^ { ( \frac { B - 3 } { 2 } ) } \big ( \frac { - 1 } { B - 1 } \big )$ is negligible compared to $C _ { l + 2 } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 )$ for sufficiently large $B$ and $l$ , so that $\begin{array} { r l r } { w _ { B - 1 , l } ~ - ~ v _ { B - 1 , l } } & { { } \approx } & { \left( \frac { l ( B - 3 + l ) ( B - 3 ) } { ( B - 5 + 2 l ) ( B - 1 + 2 l ) } - 1 \right) c _ { B - 1 , l } ^ { - 1 } C _ { l + 2 } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) \quad = } \end{array}$ $\begin{array} { r } { \left( \frac { B - 7 } { 4 } + O ( B ^ { - 2 } ) + O ( l ^ { - 2 } ) \right) c _ { B - 1 , l } ^ { - 1 } C _ { l + 2 } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) } \end{array}$ and is positive. For small $l$ , we can calculate + +$$ +\begin{array} { r l } & { w _ { B - 1 , 1 } - v _ { B - 1 , 1 } = \displaystyle \frac { - 2 B } { 1 + B } < 0 } \\ & { w _ { B - 1 , 2 } - v _ { B - 1 , 2 } = \displaystyle \frac { B ( B + 1 ) ( B ^ { 2 } - 9 B + 1 2 ) } { 2 ( B - 1 ) ( B + 3 ) } \geq 0 , \forall B \geq 1 0 } \\ & { w _ { B - 1 , 3 } - v _ { B - 1 , 3 } = \displaystyle \frac { ( B - 3 ) ( B + 3 ) B ^ { 2 } ( 2 B ^ { 3 } - 1 9 B ^ { 2 } + 1 6 B + 1 3 ) } { 6 ( B - 1 ) ^ { 2 } ( B + 1 ) ( B + 5 ) } \geq 0 , \forall B \geq 1 0 . } \end{array} +$$ + +In fact, plotting $w _ { B - 1 , l } - v _ { B - 1 , l }$ for various values of $l$ suggests that for $B \geq 1 0$ , $w _ { B - 1 , l } - v _ { B - 1 , l } \geq$ 0 for all $l \geq 1$ (Fig. 10). + +Thus, the more “linear” $\phi$ is, the larger $a _ { 1 } ^ { 2 }$ is compared to the rest of $\{ a _ { l } \} _ { l }$ , and the more likely that the eigenvalue is $< ~ 1$ . In the case that $a _ { l } a _ { l + 2 } = 0 \forall l$ , then indeed the eigenvalue is $< ~ 1$ precisely when $a _ { 1 } ^ { 2 }$ is sufficiently large. Because higher degree Gegenbauer polynomials explodes more violently as $x \pm 1$ , this is consistent with our claim. + +![](images/ea53df2404b9d1ec200fce8d9dce20ea8f850633079e937330d90fb992d6c9ad.jpg) +Figure 10: Plots of $w _ { B - 1 , l } - v _ { B - 1 , l }$ over $B$ for various $l$ + +Eigenvalue for M We use a similar technique shows the Gegenbauer expansion of the eigenvalue for $\mathbb { M }$ . + +Define $\tilde { \mathcal { A } } _ { a b } ( c , d ) : = \mathbb { E } _ { v \sim S ^ { B - 2 } } \phi ( \sqrt { B } \mathbb { e } _ { a , : } v ) \phi ( \sqrt { B } \mathbb { e } _ { b , : } v ) ( \mathbb { e } _ { c , : } v ) ( \mathbb { e } _ { d , : } v )$ . Then $\tilde { \mathcal { A } } _ { a b } ( c , c ) = \mathcal { A } ( a , b ; c )$ . Note the ultrasymmetries $\begin{array} { l l l } { \tilde { \mathcal { A } } _ { a b } ( c , d ) } & { = } & { \tilde { \mathcal { A } } _ { \pi ( a ) \pi ( b ) } ( \pi ( c ) , \pi ( d ) ) } \end{array}$ for any permutation $\pi$ , and $\tilde { \mathcal { A } } _ { a b } ( c , d ) = \tilde { \mathcal { A } } _ { a b } ( d , c ) = \tilde { \mathcal { A } } _ { b a } ( c , d ) = \tilde { \mathcal { A } } _ { b a } ( d , c )$ . + +By Eq. (46), we have + +$$ +\begin{array} { r l } & { \quad \tilde { \mathcal { U } } \{ \delta _ { c } \delta _ { d } ^ { T } \} _ { a b } } \\ & { = \mu ^ { * - 1 } \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathfrak { e } _ { a , : } v ) \phi ( \sqrt { B } \mathfrak { e } _ { b , : } v ) \left( \frac { B - 1 } { 2 } ( \mathfrak { e } _ { c , : } v ) ( \mathfrak { e } _ { d , : } v ) - \frac { 1 } { 2 } \left( \mathbb { I } ( c = d ) - \frac { 1 } { B } \right) \right) } \\ & { = \mu ^ { * - 1 } \frac { B - 1 } { 2 } \tilde { A } ( a , b ; c , d ) - \mu ^ { * - 1 } \frac { 1 } { 2 } \Sigma _ { a b } ^ { * } \left( \mathbb { I } ( c = d ) - \frac { 1 } { B } \right) . } \end{array} +$$ + +By $\cdot$ , $G ^ { \otimes 2 } \circ u$ has eigenspace $\mathbb { M }$ with eigenvalue + +$$ +\begin{array} { r l } & { \lambda _ { \mathbb { M } } ^ { \uparrow } = \tilde { \mathcal { U } } \{ \delta _ { 1 } \delta _ { 2 } ^ { T } \} _ { 1 2 } + \tilde { \mathcal { U } } \{ \delta _ { 2 } \delta _ { 1 } ^ { T } \} _ { 1 2 } - 4 \tilde { \mathcal { U } } \{ \delta _ { 1 } \delta _ { 3 } ^ { T } \} _ { 1 2 } + 2 \tilde { \mathcal { U } } \{ \delta _ { 3 } \delta _ { 4 } ^ { T } \} _ { 1 2 } } \\ & { \quad = 2 ( \tilde { \mathcal { U } } \{ \delta _ { 1 } \delta _ { 2 } ^ { T } \} _ { 1 2 } - 2 \tilde { \mathcal { U } } \{ \delta _ { 1 } \delta _ { 3 } ^ { T } \} _ { 1 2 } + \tilde { \mathcal { U } } \{ \delta _ { 3 } \delta _ { 4 } ^ { T } \} _ { 1 2 } ) } \\ & { \quad = \frac { B - 1 } { \mu ^ { * } } ( \tilde { A } _ { 1 2 } ( 1 2 ) - \tilde { A } _ { 1 2 } ( 1 3 ) + \tilde { A } _ { 1 2 } ( 3 4 ) ) . } \end{array} +$$ + +Thus we need to evaluate $\tilde { A } _ { 1 2 } ( 1 2 ) , \tilde { A } _ { 1 2 } ( 1 3 ) , \tilde { A } _ { 1 2 } ( 3 4 )$ . We will do so by exploiting linear dependences between different values of $\tilde { A } _ { a b } ( c , d )$ , the computed values of $\tilde { \mathcal { A } } _ { a b } ( c , c ) = \mathcal { A } ( a , b ; c )$ , and the value of $\tilde { A } _ { a b } ( a , b )$ computed below in Lemma F.23. Indeed, we have + +$$ +\sum _ { d = 1 } ^ { B } \tilde { \mathcal { A } } _ { a b } ( c , d ) = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathbb { e } _ { a , : } v ) \phi ( \sqrt { B } \mathbb { e } _ { b , : } v ) ( \mathbb { e } _ { c , : } v ) \sum _ { d = 1 } ^ { B } ( \mathbb { e } _ { d , : } v ) = 0 . +$$ + +Leveraging the symmmetries of $\tilde { \mathcal { A } }$ , we get + +$$ +\begin{array} { r } { ( B - 3 ) \tilde { A } _ { 1 2 } ( 3 4 ) + 2 \tilde { A } _ { 1 2 } ( 1 3 ) + \tilde { A } _ { 1 2 } ( 3 3 ) = 0 } \\ { ( B - 2 ) \tilde { A } _ { 1 2 } ( 1 3 ) + \tilde { A } _ { 1 2 } ( 1 1 ) + \tilde { A } _ { 1 2 } ( 1 2 ) = 0 . } \end{array} +$$ + +Expressing in terms of $\tilde { \mathcal { A } } _ { 1 2 } ( 1 1 )$ and $\tilde { A } _ { 1 2 } ( 1 2 )$ , we get + +$$ +\begin{array} { r l r } { { \tilde { \mathcal { A } } _ { 1 2 } ( 1 3 ) = \frac { - 1 } { B - 2 } ( \tilde { \mathcal { A } } _ { 1 2 } ( 1 1 ) + \tilde { \mathcal { A } } _ { 1 2 } ( 1 2 ) ) } } \\ & { } & { \tilde { \mathcal { A } } _ { 1 2 } ( 3 4 ) = \frac { - 1 } { B - 3 } ( 2 \tilde { \mathcal { A } } _ { 1 2 } ( 1 3 ) + \tilde { \mathcal { A } } _ { 1 2 } ( 3 3 ) ) } \\ & { } & { = \frac { - 1 } { B - 3 } ( \frac { - 2 } { B - 2 } ( \tilde { \mathcal { A } } _ { 1 2 } ( 1 1 ) + \tilde { \mathcal { A } } _ { 1 2 } ( 1 2 ) ) + \frac { 1 } { B - 2 } ( \Sigma _ { 1 2 } ^ { * } - 2 \tilde { \mathcal { A } } _ { 1 2 } ( 2 2 ) ) ) } \\ & { } & { \mathbf { B } \mathbf { y } \operatorname { E q . } ( 4 7 ) } \\ & { } & { = \frac { - 1 } { ( B - 3 ) ( B - 2 ) } ( \Sigma _ { 1 2 } ^ { * } - 4 \tilde { \mathcal { A } } _ { 1 2 } ( 1 1 ) - 2 \tilde { \mathcal { A } } _ { 1 2 } ( 1 2 ) ) . } \end{array} +$$ + +These relations allow us to simplify + +$$ +\lambda _ { \mathtt { M } } ^ { \uparrow } = \frac { B - 1 } { ( B - 3 ) \mu ^ { * } } \left( \frac { - 1 } { B - 2 } \Sigma _ { 1 2 } ^ { * } + \frac { 2 ( B - 1 ) } { B - 2 } \tilde { \mathcal { A } } _ { 1 2 } ( 1 1 ) + ( B - 1 ) \tilde { \mathcal { A } } _ { 1 2 } ( 1 2 ) \right) . +$$ + +By Eq. (49), we know the Gegenbauer expansion of $\tilde { \mathcal { A } } _ { 1 2 } ( 1 1 )$ . The following shows the Gegenbauer expansion of $\tilde { \mathcal { A } } _ { 1 2 } ( 1 2 )$ . + +Lemma F.23. Suppose $\phi ( \sqrt { B - 1 } x ) , x \phi ( \sqrt { B - 1 } x ) \in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \frac { B - 3 } { 2 } } )$ , and $\phi ( \sqrt { B - 1 } x )$ has Gegenbauer expansion P∞l=0 al 1cB−1,l $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ . Then + +$$ +\begin{array} { l } { \displaystyle \tilde { \mathcal { A } } _ { 1 2 } ( 1 2 ) = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B } \mathbb { e } _ { 1 , : } v ) \phi ( \sqrt { B } \mathbb { e } _ { 2 , : } v ) ( \mathbb { e } _ { 1 , : } v ) ( \mathbb { e } _ { 2 , : } v ) } \\ { \displaystyle = \sum _ { l = 0 } ^ { \infty } \alpha _ { l } a _ { l } ^ { 2 } + \beta _ { l } a _ { l } a _ { l + 2 } } \end{array} +$$ + +where + +$$ +\begin{array} { r l } & { v _ { l } = \frac { B - 1 } { B } \times \left\{ \begin{array} { l l } { \left( \frac { l + 1 } { B - 1 + 2 l } \right) ^ { 2 } c _ { B - 1 , l + 1 } ^ { - 1 } C _ { l + 1 } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) + \left( \frac { B + l - 4 } { B + 2 l - 5 } \right) ^ { 2 } c _ { B - 1 , l - 1 } ^ { - 1 } C _ { l - 1 } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) } & { i f l \sin ( j ) } \\ { \left( \frac { 1 } { B - 1 } \right) ^ { 2 } c _ { B - 1 , 1 } ^ { - 1 } C _ { 1 } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) } & { o t h } \end{array} \right. } \\ & { 3 _ { l } = \frac { B - 1 } { B } \times \frac { 2 ( l + 1 ) ( B + l - 2 ) } { ( B + 2 l - 1 ) ^ { 2 } } c _ { B - 1 , l + 1 } ^ { - 1 } C _ { l + 1 } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) . } \end{array} +$$ + +erwise + +Proof. Let $\psi ( x ) = x \phi ( { \sqrt { B - 1 } } x )$ . Then by Proposition E.43, we have + +$$ +\begin{array} { l } { { \displaystyle \psi ( x ) = \sum _ { l = 0 } ^ { \infty } a _ { l } c _ { B - 1 , l } ^ { - 1 } \frac { 1 } { B - 3 + 2 l } \left( ( l + 1 ) C _ { l + 1 } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) + ( B + l - 4 ) C _ { l - 1 } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) \right) } } \\ { { \displaystyle \quad = \sum _ { l = 0 } ^ { \infty } \left( a _ { l - 1 } \frac { l } { B + 2 l - 3 } + a _ { l + 1 } \frac { B + l - 3 } { B + 2 l - 3 } \right) c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) . } } \end{array} +$$ + +Then + +$$ +\begin{array} { l } { \displaystyle \tilde { A } _ { 1 2 } ( 1 2 ) = \frac { B - 1 } { B } \sum _ { v \sim S ^ { B - 2 } } \psi ( \hat { \bf e } _ { 1 , : } v ) \psi ( \hat { \bf e } _ { 2 , : } v ) } \\ { \displaystyle = \frac { B - 1 } { B } \sum _ { l = 0 } ^ { \infty } \left( a _ { l - 1 } \frac { l } { B + 2 l - 3 } + a _ { l + 1 } \frac { B + l - 3 } { B + 2 l - 3 } \right) ^ { 2 } c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) . } \end{array} +$$ + +Rearranging the sum in terms of $a _ { l }$ gives the result. + +Combining all of the above, we obtain the following + +Theorem F.24. Suppose φ(√B − 1x) has Gegenbauer expansion P∞l=0 al 1cB−1,l C $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ ( B−32 )l (x). Then the eigenvalue of $\mathcal { U }$ with respect to the eigenspace M is a ratio of quadratic forms + +$$ +\lambda _ { \mathbb { M } } ^ { \uparrow } = \frac { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \tilde { w } _ { B - 1 , l } + a _ { l } a _ { l + 2 } \tilde { u } _ { B - 1 , l } } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } v _ { B - 1 , l } } +$$ + +where + +$$ +\begin{array} { r l } & { \quad \frac { 2 ( B - 1 ) ^ { 5 } } { ( B - 1 ) ^ { 5 / 2 } } \textstyle ( \frac { ( B - 3 - l ) ( B - 2 + l ) } { ( B - 1 ) ^ { 6 } } \textstyle ( \frac { ( B - 3 - l ) ( B - 2 + 1 ) } { B - 1 + 2 l } C _ { l } ^ { ( \frac { 2 3 - 2 } { 2 } ) } \textstyle ( \frac { - 1 } { B - 1 } ) } \\ & { \qquad + \frac { ( B - 2 ) ( l + 1 ) ( B - 2 + 1 ) } { B - 1 + 2 l + 3 } C _ { l } ^ { ( \frac { 2 4 - 3 } { 2 } ) } \textstyle ( \frac { - 1 } { B - 1 } ) } \\ & { \qquad + \frac { ( 4 - 3 ) ( B - 1 + 2 + 1 ) ( B - 3 + 2 ) } { ( B + 1 + 1 ) ( B - 3 + 2 l ) } C _ { l } ^ { ( \frac { 2 3 - 2 } { 2 } ) } \textstyle ( \frac { - 1 } { B - 1 } ) ) } \\ & { \qquad + \frac { ( 4 - 3 ) ( B - 1 ) } { ( B + 2 l + 1 ) ( B - 3 + 2 l ) } \textstyle ( \frac { - 1 } { B + 2 l } ) ) } \\ & { \qquad \frac { B - 1 } { ( B - 3 ) ^ { 2 } ( B - 2 ) } \textstyle \bigg [ } \\ & { \qquad ( \frac { 2 ( B - 1 ) ^ { 2 } ( B - 2 ) ( B + 2 l - 3 ) ( 2 B + 3 + 2 l ^ { 2 } - 6 l - 5 ) } { B ( B + 2 l - 5 ) ( B + 2 l - 5 ) ( B + 2 l - 1 ) } - ( B - 3 + 2 l ) ) C _ { l } ^ { ( \frac { 2 3 - 2 } { 2 } ) } ( \frac { - 1 } { B } ) } \\ & { \qquad + \frac { ( B - 2 ) ( B - 1 ) ^ { 2 } ( l + 1 ) ^ { 2 } } { B ( B + 2 l - 1 ) ( B - 2 l + 1 ) ^ { 2 } } C _ { l } ^ { ( \frac { 2 3 - 2 } { 2 } ) } ( \frac { - 1 } { B - 1 } ) } \\ & \qquad + \frac { ( B - 2 ) ( B - 1 ) ^ { 2 } ( l + 1 ) ^ { 2 } } { B ( B + 2 l - 1 ) } C _ { l } ^ { ( \frac { 2 3 - 2 } { 2 } ) } ( \frac { - 1 } { B - 1 } ) \end{array} +$$ + +where $C _ { - 1 } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) = 0$ by convention. + +Note that $\tilde { u } _ { B - 1 , l }$ and $\tilde { w } _ { B - 1 , l }$ depends only on $C _ { \bullet } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right)$ which is much smaller than $C _ { \bullet } ^ { ( \frac { B - 3 } { 2 } ) }$ (1) for degree larger than 0. Thus the eigenvalue for $\mathbb { M }$ is typically much smaller than the eigenvalue for $\mathbb { L }$ (though there are counterexamples, like sin). + +Differentiating Eq. (26) In what follows, let $\phi$ be a positive-homogeneous function of degree $\alpha$ . We begin by studying $\frac { \mathrm { d } \mathrm { V } _ { ( \phi \circ \mathsf { n } ) } ( \Sigma ^ { G } ) } { \mathrm { d } \Sigma ^ { G } }$ . We will differentiate Eq. (26) directly at $G ^ { \otimes 2 } \{ \Sigma ^ { * } \} = G \Sigma ^ { * } G$ where $\Sigma ^ { * }$ is the BSB1 fixed point given by Thm F.8. To that end, consider a smooth path $\Sigma _ { t } \ \in$ ${ \cal S } _ { B } ^ { G } , t \in ( - \epsilon , \epsilon )$ for some $\epsilon > 0$ , with $\Sigma _ { 0 } = G \Sigma ^ { * } G$ . Set $\Sigma _ { \perp t } = \Phi ^ { T } \Sigma _ { t ^ { \mathbb { F } } } \in { \mathcal S } _ { B - 1 }$ , so that $\Sigma _ { t } =$ $\mathbb { e } \sum _ { \mathbf { \Pi } } { _ { t } } \mathbb { e } ^ { T }$ and $\Sigma _ { \perp 0 } = \mu ^ { * } I _ { B - 1 }$ where $\begin{array} { r } { \mu ^ { * } = K _ { \alpha , B } ( \mathrm { J } _ { \phi } ( 1 ) - \mathrm { J } _ { \phi } ( \frac { - 1 } { B - 1 } ) ) } \end{array}$ as in Thm F.8. If we write $\dot { \Sigma } _ { \perp }$ and $\dot { \Sigma }$ for the time derivatives, we have + +$$ +\begin{array} { r l } & { \quad \left. B ^ { - \alpha } \Gamma ( \alpha ) \frac { 1 } { d t } { \cal M } _ { \mathrm { v } } ^ { \mathrm { i } } V ( \phi _ { \mathrm { c o s s } } ^ { \mathrm { o u p } } ( x ^ { \mathrm { i } } \mathrm { f r a l } ^ { \mathrm { o } ^ { T } } ) \biggr | _ { \mathrm { l o s s } } ^ { 2 } \right. } \\ & { = \left. \overline { { B ^ { - \alpha } \Gamma ( \alpha ) } } \frac { d } { d t } { \cal V } _ { \mathrm { a } } \varepsilon ( \mathrm { P } _ { \mathrm { A L } } ^ { \mathrm { i } } \mathrm { P } _ { \mathrm { E } } ^ { \mathrm { o u p } } ) \right| _ { \mathrm { t e s o } } } \\ & { = \int _ { 0 } ^ { \infty } \mathrm { d } s s ^ { - 1 } \left[ \frac { d } { d t } \mathrm { d } { \bf f } \left( I + 2 s \Sigma _ { 3 2 } \right) ^ { - 1 / 2 } \right] _ { \mathrm { c o s s } } ^ { - 1 / 2 } \biggr | _ { \mathrm { v e l o } } \times { \cal V } _ { \mathrm { s g } } \left( \beta \Sigma _ { 2 0 } \left( I + 2 s \Sigma _ { 2 0 } \right) ^ { - 1 } \mathrm { e } ^ { T } \right) } \\ & { \qquad + \operatorname { d } \mathrm { e t } ( I + 2 s \Sigma _ { 3 2 } ) ^ { - 1 / 2 } \frac { d } { d t } { \cal V } _ { \mathrm { v e l o } } \left( \phi \Sigma _ { 3 2 } \left( I + 2 s \Sigma _ { 3 2 } \right) ^ { - 1 } \mathrm { e } ^ { T } \right) \biggr | _ { \mathrm { l o s s } } \biggr ] } \\ & { = \int _ { 0 } ^ { \infty } \mathrm { d } s s ^ { - 1 } \left[ - \frac { 3 } { 1 + 2 s \beta \beta \sigma ^ { 4 } } \operatorname { d i f } \left( I + 2 s \Sigma _ { 3 2 } \right) ^ { - 1 / 2 } \operatorname { d i f } \left( \tilde { \Sigma } _ { \mathrm { f o l } } \right) \times \left( \frac { \beta ^ { \prime } } { 1 + 2 s \beta \sigma ^ { 4 } } \right) ^ { \alpha } \nabla _ { \mathrm { s g } } \left( G \right) \right. } \\ & { \qquad \left. + \operatorname { d i f } \left( I + 2 s \Sigma _ { 1 0 } \right) ^ { - 1 / 2 } \frac { d } { \mathrm { d } s \sigma } \mathrm { d } t \beta \left( \Sigma \right) \right] } \end{array} +$$ + +(apply Lemma F.26 and Lemma F.27) + +$$ +\begin{array} { l } { { \displaystyle = \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { \alpha - 1 } \bigg [ - \frac { s } { 1 + 2 s \mu ^ { * } } ( 1 + 2 s \mu ^ { * } ) ^ { - ( B - 1 ) / 2 } \mathrm { t r } \left( \dot { \Sigma } _ { \bigstar \mathrm { D 0 } } \right) \times \left( \frac { \mu ^ { * } } { 1 + 2 s \mu ^ { * } } \right) ^ { \alpha } \mathrm { V } _ { \phi } ( G ) } } \\ { { \displaystyle \qquad + \left( 1 + 2 s \mu ^ { * } \right) ^ { - ( B - 1 ) / 2 } \left( \frac { \mu ^ { * } } { 1 + 2 s \mu ^ { * } } \right) ^ { \alpha - 1 } \frac { \mathrm { d } \mathrm { V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } \bigg \vert _ { \Sigma = G } \left\{ ( 1 + 2 s \mu ^ { * } ) ^ { - 2 } \mathrm { e } \dot { \Sigma } _ { \big \mathrm { D 0 } } \mathrm { e } ^ { T } \right\} \bigg ] } } \end{array} +$$ + +(using fact that $\mathrm { d } \mathrm { V } _ { \phi } / \mathrm { d } \Sigma$ is degree $( \alpha - 1 )$ -positive homogeneous) + +$$ +\begin{array} { r l } & { = - \left( \displaystyle \int _ { 0 } ^ { \infty } ( 1 + 2 s \mu ^ { * } ) ^ { - \frac { B - 1 } { 2 } - 1 - \alpha } \mu ^ { * \alpha } s ^ { \alpha } \right) \mathrm { t r } \left( \dot { \Sigma } _ { \bigtriangledown \mathrm { D } 0 } \right) { \bf V } _ { \phi } ( G ) } \\ & { \quad + \left( \displaystyle \int _ { 0 } ^ { \infty } ( 1 + 2 s \mu ^ { * } ) ^ { - \frac { B - 1 } { 2 } - 1 - \alpha } \mu ^ { * \alpha - 1 } s ^ { \alpha - 1 } \right) \frac { \mathrm { d } { \bf V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } \biggr | _ { \Sigma = G } \left\{ \mathrm { e } \dot { \Sigma } _ { \bigtriangledown \mathrm { D } 0 } \mathrm { e } ^ { T } \right\} } \\ & { = - \mu ^ { * - 1 } 2 ^ { - 1 - \alpha } \mathrm { B e t a } \left( \frac { B - 1 } { 2 } , \alpha + 1 \right) \mathrm { t r } \left( \dot { \Sigma } _ { \bigtriangledown 0 } \right) { \bf V } _ { \phi } ( G ) } \\ & { \quad + \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { B e t a } \left( \frac { B + 1 } { 2 } , \alpha \right) \frac { \mathrm { d } { \bf V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } \biggr | _ { \Sigma = G } \left\{ \dot { \Sigma } _ { 0 } \right\} } \end{array} +$$ + +(apply Lemma F.28) + +if $\frac { B - 1 } { 2 } + 2 > \alpha$ (precondition for Lemma F.28). + +With some trivial simplifications, we obtain the following + +Lemma F.25. Let $\phi$ be positive-homogeneous of degree α. Consider a smooth path $\Sigma _ { t } \in S _ { B } ^ { G }$ with $\Sigma _ { 0 } = \mu ^ { * } G$ . If $\frac { B - 1 } { 2 } + 2 > \alpha$ , then + +$$ +\begin{array} { r l } & { \quad \frac { d } { d t } \nabla _ { ( \phi \circ \mathbf { n } ) } ( \Sigma _ { t } ) \bigg | _ { t = 0 } } \\ & { = \frac { \mathrm { d } \mathrm { V } _ { ( \phi \circ \mathbf { n } ) } ( \Sigma ) } { \mathrm { d } \Sigma } \bigg | _ { \Sigma = \mu ^ { * } G } \{ \dot { \Sigma } _ { 0 } \} } \\ & { = - \alpha B ^ { \alpha } \mu ^ { * - 1 } 2 ^ { - 1 - \alpha } \mathrm { P } \left( \frac { B - 1 } { 2 } , \alpha + 1 \right) ^ { - 1 } \mathrm { t r } \left( \dot { \Sigma } _ { \Sigma 0 } \right) \mathrm { V } _ { \phi } ( G ) } \\ & { \quad + B ^ { \alpha } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } \frac { \mathrm { d } \mathrm { V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } \bigg | _ { \Sigma = G } \left\{ \dot { \Sigma } _ { 0 } \right\} } \end{array} +$$ + +where $\mathrm { P } \left( a , b \right) = \Gamma ( a + b ) / \Gamma ( a )$ is the Pochhammer symbol. + +Lemma F.26. For any $\begin{array} { r } { s \in \mathbb { R } , \frac { d } { d t } \operatorname* { d e t } ( I + 2 s \Sigma _ { \Sigma } ) ^ { - 1 / 2 } = - s \operatorname* { d e t } ( I + 2 s \Sigma _ { \Sigma } ) ^ { - 1 / 2 } \operatorname { t r } ( ( I + 2 s \Sigma _ { \Sigma } ) ^ { - 1 / 2 } = 0 , 1 ) } \end{array}$ $2 s \Sigma _ { \perp } ) ^ { - 1 } d \Sigma _ { \perp } / d t )$ . + +For Σ0 = µI, this is also equal to − s1+2sµ $\begin{array} { r l r } { \mathrm { ~ } } & { { } } & \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm { ~ } \mathrm ~ \end{array}$ at $t = 0$ . + +Proof. Straightforward computation. + +Lemma F.27. For any $s \in \mathbb { R }$ , $\begin{array} { r } { \frac { d } { d t } \Sigma _ { \bigtriangledown } ( I + 2 s \Sigma _ { \bigtriangledown } ) ^ { - 1 } = ( I + 2 s \Sigma _ { \bigtriangledown } ) \frac { d \Sigma _ { \bigtriangledown } } { d t } ( I + 2 s \Sigma _ { \bigtriangledown } ) ^ { - 1 } . } \end{array}$ + +Proof. Straightforward computation. + +Lemma F.28. For $a > b + 1$ , $\begin{array} { r } { \int _ { 0 } ^ { \infty } ( 1 + 2 s \mu ) ^ { - a } s ^ { b } \mathrm { d } s = ( 2 \mu ) ^ { - 1 - b } \operatorname { B e t a } ( b + 1 , a - 1 - b ) . } \end{array}$ + +Proof. Apply change of variables $\begin{array} { r } { x = \frac { 2 \mu s } { 1 + 2 \mu s } } \end{array}$ . + +This immediately gives the following consequence. + +Theorem F.29. Let $\phi : \mathbb { R } \mathbb { R }$ be any function with finite first and second Gaussian moments. Then φ Σ=BSB1(a,b) : HGB → HGB has the following eigendecomposition: + +· $\mathbb { R } G$ has eigenvalu e (B−1)(u−2v)+w (dim RG = 1) • $\mathbb { M } _ { B }$ has eigenvalue $w$ $( \dim \mathbb { M } _ { B } = B ( B - 3 ) / 2 )$ • $\mathbb { L } _ { B }$ has eigenvalue $\frac { ( B - 2 ) ( u - 2 v ) + 2 w } { B } ( \dim \mathbb { L } _ { B } = B - 1 )$ + +where + +$$ +u = \left. \frac { \partial \mathrm { V } _ { \phi } ( \Sigma ) _ { 1 1 } } { \partial \Sigma _ { 1 1 } } \right. _ { \Sigma = \mathrm { B S B } 1 ( a , b ) } , \quad v = \left. \frac { \partial \mathrm { V } _ { \phi } ( \Sigma ) _ { 1 2 } } { \partial \Sigma _ { 1 1 } } \right. _ { \Sigma = \mathrm { B S B } 1 ( a , b ) } , \quad w = \left. \frac { \partial \mathrm { V } _ { \phi } ( \Sigma ) _ { 1 2 } } { \partial \Sigma _ { 1 2 } } \right. _ { \Sigma = \mathrm { B S B } 1 ( a , b ) } . +$$ + +Theorem F.30. Let $\phi : \mathbb { R } \mathbb { R }$ be positive-homogeneous of degree $\alpha$ . Then for any $p \neq 0 , c \in \mathbb { R }$ , Σ=BSB1(p,cp) : HGB → HGB has the following eigendecomposition: + +$$ +\begin{array} { c } { { \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \frac 1 B \left( ( B - 2 ) \alpha \left[ { \bf J } _ { \phi } ( 1 ) - { \bf J } _ { \phi } \left( c \right) \right] + \left( 2 + c ( B - 2 ) \right) { \bf J } _ { \phi } ^ { \prime } \left( c \right) \right) } } \\ { { \mathrm { , ~ } } } \\ { { \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \frac 1 B \left( ( B - 1 ) \alpha \left[ { \bf J } _ { \phi } ( 1 ) - { \bf J } _ { \phi } \left( c \right) \right] + \left( 1 + c ( B - 1 ) \right) { \bf J } _ { \phi } ^ { \prime } \left( c \right) \right) } } \end{array} +$$ + +Proof. By Proposition E.22, $\frac { \mathrm { d } \mathrm { V } _ { \phi } } { \mathrm { d } \Sigma } \Big | _ { \Sigma = \mathrm { B S B } 1 ( p , c p ) }$ is $\mathrm { D O S } ( u , v , w )$ with + +$$ +\begin{array} { r l } & { \boldsymbol { u } = \frac { \partial N _ { \phi } ( E ( \Sigma ) ) _ { 1 1 } } { \partial \Sigma _ { 1 1 } } \Bigg | _ { \{ \Sigma = \mathrm { R } \ S \} \cup \{ \rho _ { \phi } \} } } \\ & { \quad = c _ { \phi , \Phi } p ^ { i - 1 } \dot { \Sigma } _ { \phi , \Phi } ( 1 ) } \\ & { \quad = \frac { \partial N _ { \phi } ( \Sigma ) _ { 1 2 } } { \partial \Sigma _ { 1 1 } } \Bigg | _ { \{ \Sigma = \mathrm { R } \ S \mathrm { M } \{ \rho _ { \phi } \} } } \\ & { \quad = \frac { 1 } { 2 } c _ { \phi , \Phi } \Sigma _ { i \alpha } ^ { ( i \alpha - 2 ) } \Sigma _ { \Delta \phi } ^ { \mathrm { t } _ { \alpha } } ( \Omega _ { \phi } \langle c _ { \phi } \rangle - c _ { \phi , \Phi } J _ { \phi } ^ { \prime } ( c _ { \phi } ) \Bigg ) } \\ & { \quad = \frac { 1 } { 2 } c _ { \phi , \Phi } p ^ { i - 1 } \left( \Omega _ { \phi } c _ { \phi } ( \Sigma ) - c _ { \phi } ^ { \dagger } \right) ^ { \alpha } } \\ & { \quad = \frac { \partial N _ { \phi } ( \Sigma ) _ { 1 2 } } { \partial \Sigma _ { 1 2 } } \Bigg | _ { \{ \Sigma = \mathrm { R } \ S \mathrm { M } \{ \rho _ { \phi } \} } } \\ & { \quad = c _ { \phi , \Phi } \Sigma _ { i \alpha } ^ { ( i \alpha - 1 ) / 2 } \Sigma _ { \mathrm { a r } } ^ { ( i \alpha - 1 ) / 2 } \Sigma _ { \phi } ^ { \mathrm { t } _ { \alpha } } ( c _ { \phi } ) } \\ & { \quad = c _ { \phi , \Phi } \pi ^ { ( i \alpha - 1 ) / 2 } \Sigma _ { 1 \phi } ^ { ( i \alpha - 1 ) / 2 } \Sigma _ { \phi } ^ { \mathrm { t } _ { \alpha } } ( c _ { \phi } ) } \\ & { \quad = c _ { \phi , \Phi } \pi ^ { ( i \alpha - 1 ) / 2 } \Sigma _ { 1 \phi } ^ { ( i \alpha - 1 ) / 2 } \Gamma _ { \phi } ^ { ( i \alpha ) } ( c _ { \phi } ) } \end{array} +$$ + +With $\cdot$ , we can do the computation: + +• $\mathbb { M } _ { B }$ has eigenvalue $w = \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \mathsf { J } _ { \phi } ^ { \prime } \left( c \right)$ + +• $\mathbb { L } _ { B }$ has eigenvalue + +$$ +\begin{array} { l } { { \displaystyle \frac { ( u - 2 v ) ( B - 2 ) + 2 w } { B } } \ ~ } \\ { { \displaystyle = \frac { 1 } { B } \left( ( B - 2 ) \left( \mathsf { c } _ { \alpha } \alpha p ^ { \alpha - 1 } \mathbf { J } _ { \phi } ( 1 ) - \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \left( \alpha \mathbf { J } _ { \phi } \left( c \right) - c \mathbf { J } _ { \phi } ^ { \prime } \left( c \right) \right) \right) + 2 \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \mathbf { J } _ { \phi } ^ { \prime } \left( c \right) \right) } \ ~ } \\ { { \displaystyle = \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \frac { 1 } { B } \left( ( B - 2 ) \alpha \left[ \mathsf { J } _ { \phi } ( 1 ) - \mathsf { J } _ { \phi } \left( c \right) \right] + ( 2 + c ( B - 2 ) ) \mathbf { J } _ { \phi } ^ { \prime } \left( c \right) \right) } } \end{array} +$$ + +• $\mathbb { R } G$ has eigenvalue + +$$ +\begin{array} { l } { { \displaystyle \frac { ( B - 1 ) ( u - 2 v ) + w } { B } } \ ~ } \\ { { \displaystyle = \frac { 1 } { B } \left( ( B - 1 ) \left( \mathsf { c } _ { \alpha } \alpha p ^ { \alpha - 1 } \mathbf { J } _ { \phi } ( 1 ) - \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \left( \alpha \mathbf { J } _ { \phi } \left( c \right) - c \mathbf { J } _ { \phi } ^ { \prime } \left( c \right) \right) \right) + \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \mathbf { J } _ { \phi } ^ { \prime } \left( c \right) \right) } \ ~ } \\ { { \displaystyle = \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \frac { 1 } { B } \left( ( B - 1 ) \alpha \left[ \mathbf { J } _ { \phi } ( 1 ) - \mathbf { J } _ { \phi } \left( c \right) \right] + \left( 1 + c ( B - 1 ) \right) \mathbf { J } _ { \phi } ^ { \prime } \left( c \right) \right) } } \end{array} +$$ + +We record the following consequence which will be used frequently in the sequel. + +Theorem F.31. Let $\phi : \mathbb { R } \mathbb { R }$ be positive-homogeneous of degree $\alpha$ . Then $G ^ { \otimes 2 } \circ \left. \frac { \mathrm { d } \mathrm { V } _ { \phi } } { \mathrm { d } \Sigma } \right| _ { \Sigma = G }$ $\mathcal { H } _ { B } ^ { G } \to \mathcal { H } _ { B } ^ { G }$ has the following eigendecomposition: + +$$ +\begin{array} { r l } & { \iota e \lambda _ { \mathbb { M } } ^ { G , \phi } ( B , \alpha ) : = \mathsf { c } _ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha - 1 } \operatorname { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) } \\ & { } \\ & { \mathrm { \ } ^ { \beta \lambda _ { \mathbb { L } } ^ { G , \phi } } ( B , \alpha ) : = \mathsf { c } _ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha - 1 } \frac { 1 } { B } \left( ( B - 2 ) \alpha \left[ \operatorname { J } _ { \phi } ( 1 ) - \operatorname { J } _ { \phi } \left( \frac { - 1 } { B - 1 } \right) \right] + \frac { B } { B - 1 } \operatorname { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) \right) } \\ & { } \\ & { e \lambda _ { G } ^ { G , \phi } ( B , \alpha ) : = \mathsf { c } _ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha } \alpha \left[ \mathsf { J } _ { \phi } ( 1 ) - \operatorname { J } _ { \phi } \left( \frac { - 1 } { B - 1 } \right) \right] } \end{array} +$$ + +Proof. Plug in $\begin{array} { r } { p = \frac { B - 1 } { B } } \end{array}$ and $c = - 1 / ( B - 1 )$ for Thm F.30. + +Theorem F.32. Let $\phi : \mathbb { R } \mathbb { R }$ be a degree $\alpha$ positive-homogeneous function. Then for $p \neq 0 , c \in$ $\begin{array} { r } { \mathbb { R } , \ \frac { \mathrm { d } \mathrm { V } _ { \phi } } { \mathrm { d } \Sigma } \Big | _ { \Sigma = \mathrm { B S B } 1 ( p , c p ) } : \mathcal { H } _ { B } \to \mathcal { H } _ { B } } \end{array}$ has the following eigendecomposition: + +• ${ \overline { { \mathbb { M } } } } _ { B }$ has eigenvalue ${ \mathsf { c } } _ { \alpha } p ^ { \alpha - 1 } \mathrm { J } _ { \phi } ^ { \prime } \left( c \right)$ +• $\overline { { \mathbb { L } } } _ { B } ( a , b )$ has eigenvalue $\mathsf { c } _ { \alpha } \alpha p ^ { \alpha - 1 } \mathrm { J } _ { \phi } ( 1 )$ where $a \ = \ \mathrm { 2 } \left( \mathrm { J } _ { \phi } ^ { \prime } \left( c \right) - \alpha \mathrm { J } _ { \phi } ( 1 ) \right)$ and $\begin{array} { r l } { b } & { { } = } \end{array}$ $\alpha \mathrm { J } _ { \phi } \left( c \right) - c \mathrm { J } _ { \phi } ^ { \prime } \left( c \right)$ + +Proof. Use Thm E.72 with the computations from the proof of $\mathrm { T h m } \ \mathrm { F } . 3 0$ as well as the following computation + +$$ +\begin{array} { c } { { w - u = \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \mathsf { J } _ { \phi } ^ { \prime } \left( c \right) - \mathsf { c } _ { \alpha } \alpha p ^ { \alpha - 1 } \mathsf { J } _ { \phi } ( 1 ) } } \\ { { \displaystyle = \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \left( \mathsf { J } _ { \phi } ^ { \prime } \left( c \right) - \alpha \mathsf { J } _ { \phi } ( 1 ) \right) } } \\ { { \displaystyle v = \frac { 1 } { 2 } \mathsf { c } _ { \alpha } p ^ { \alpha - 1 } \left( \alpha \mathsf { J } _ { \phi } \left( c \right) - c \mathsf { J } _ { \phi } ^ { \prime } \left( c \right) \right) } } \end{array} +$$ + +Theorem F.33. Let $\phi$ be positive-homogeneous with degree $\alpha$ . Assume $\textstyle { \frac { B - 1 } { 2 } } > \alpha$ . The operator $\begin{array} { r } { \mathcal { U } = G ^ { \otimes 2 } \circ \frac { \mathrm { d } \mathbf { V } _ { ( \phi \circ \mathfrak { n } ) } \left( \Sigma ^ { G } \right) } { \mathrm { d } \Sigma ^ { G } } \big | _ { \Sigma ^ { G } = \mu ^ { * } G } : \mathcal { H } _ { B } ^ { G } \to \mathcal { H } _ { B } ^ { G } } \end{array}$ has 3 distinct eigenvalues. They are as follows: + +1. $\lambda _ { G } ^ { \uparrow } ( B , \alpha ) : = 0$ with dimension 1 eigenspace $\mathbb { R } G$ . + +2. + +$$ +\begin{array} { r l } & { \lambda _ { \mathtt { M } } ^ { \uparrow } ( B , \alpha ) : = B ^ { \alpha } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } \lambda _ { \mathtt { M } } ^ { G , \phi } ( B , \alpha ) } \\ & { \quad \quad \quad = \mathtt { c } _ { \alpha } B ( B - 1 ) ^ { \alpha - 1 } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } \mathrm { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) } \end{array} +$$ + +with dimension $\frac { B ( B - 3 ) } { 2 }$ eigenspace M. + +3. + +$$ +\begin{array} { r l } & { \lambda _ { \mathbb { L } } ^ { \uparrow } ( B , \alpha ) : = B ^ { \alpha } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \displaystyle \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } \lambda _ { \mathbb { L } } ^ { G , \phi } ( B , \alpha ) } \\ & { \quad \quad \quad = \mathsf { c } _ { \alpha } ( B - 1 ) ^ { \alpha - 1 } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \displaystyle \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } } \\ & { \quad \quad \quad \quad \quad \left( ( B - 2 ) \alpha \left[ \mathrm { J } _ { \phi } ( 1 ) - \mathrm { J } _ { \phi } \left( \displaystyle \frac { - 1 } { B - 1 } \right) \right] + \displaystyle \frac { B } { B - 1 } \mathrm { J } _ { \phi } ^ { \prime } \left( \displaystyle \frac { - 1 } { B - 1 } \right) \right) } \end{array} +$$ + +with dimension $B - 1$ eigenspace $\mathbb { L }$ . + +Proof. Item 1. The case of $\lambda _ { G } ^ { \uparrow } ( B , \alpha )$ . Since $( \phi \circ \mathsf { n } ) ( \mu ^ { * } G ) = ( \phi \circ \mathsf { n } ) ( \mu ^ { * } G + \omega G )$ for any $\omega$ , the operator $\frac { \mathrm { d } \mathrm { V } _ { ( \phi \circ \mathsf { n } ) } \left( \Sigma ^ { G } \right) } { \mathrm { d } \Sigma ^ { G } } \Big | _ { \Sigma ^ { G } = \mu ^ { * } G }$ ΣG=µ∗G sends G to 0. + +Item 2. The case of $\lambda _ { \mathbb { M } } ^ { \uparrow } ( B , \alpha )$ . Assume $\dot { \Sigma } _ { 0 } \in \mathbb { M }$ , Thm F.31 gives (with $\odot$ denoting Hadamard product, i.e. entrywise multiplication) + +$$ +\begin{array} { l } { { \displaystyle \left. { \frac { \mathrm { d } \mathrm { V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } } \right| _ { \Sigma = G } \{ \dot { \Sigma } _ { 0 } \} = \lambda _ { \mathrm { M } } ^ { G , \phi } ( B , \alpha ) \dot { \Sigma } _ { 0 } } } \\ { { \displaystyle \qquad = \mathsf { c } _ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha - 1 } \mathrm { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) \dot { \Sigma } _ { 0 } } } \end{array} +$$ + +Since $\mathrm { t r } ( \dot { \Sigma } _ { 0 } ) = 0$ , Eq. (50) gives + +$$ +\begin{array} { r l } & { \quad G ^ { \otimes 2 } \simeq \frac { \mathrm { d } \mathrm { V } ( \phi \circ \mathrm { e n } ) ( \Sigma ^ { G } ) } { \mathrm { d } \Sigma ^ { G } } \Big | _ { \Sigma ^ { G } = \mu ^ { - } } \{ \dot { \Sigma } _ { 0 } \} } \\ & { = G ^ { \otimes 2 } \circ \frac { d } { d t } \mathrm { V } _ { ( \phi \circ \mathrm { e n } ) } ( \Sigma _ { t } ) \Big | _ { t = 0 } ^ { \Sigma } ; } \\ & { = B ^ { \alpha } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } G ^ { \otimes 2 } \circ \frac { \mathrm { d } \mathrm { V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } \Big | _ { \Sigma = G } \left\{ \dot { \Sigma } _ { 0 } \right\} } \\ & { = B ^ { \alpha } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } \mathrm { c } _ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha - 1 } \mathrm { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) G ^ { \otimes 2 } \{ \dot { \Sigma } _ { 0 } \} } \\ & { = c _ { \alpha } B ( B - 1 ) ^ { \alpha - 1 } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } \mathrm { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) \dot { \Sigma } _ { 0 } } \\ & { = c _ { \alpha } B ( B - 1 ) ^ { \alpha - 1 } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } \mathrm { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) \dot { \Sigma } _ { 0 } } \end{array} +$$ + +So the eigenvalue for $\mathbb { M }$ is $\begin{array} { r } { \lambda _ { \mathbb { L } } ^ { \uparrow } = \mathsf { c } _ { \alpha } B ( B - 1 ) ^ { \alpha - 1 } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } \mathrm { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) . } \end{array}$ + +Item 3. The case of $\lambda _ { \mathbb { L } } ^ { \uparrow } ( B , \alpha )$ . Let $\dot { \Sigma } _ { 0 } = \mathrm { L } ( B - 2 , 1 )$ . Thm F.31 gives + +$$ +G ^ { \otimes 2 } \circ \frac { \mathrm { d } \mathrm { V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } \bigg | _ { \Sigma = G } \{ \dot { \Sigma } _ { 0 } \} = \lambda _ { \mathbb { L } } ^ { G , \phi } ( B , \alpha ) \dot { \Sigma } _ { 0 } +$$ + +where + +$$ +\textsc { \det { \detokenize { \left( B , \alpha \right) } } } : = \frac { 1 } { B } \mathfrak { c } _ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha - 1 } \left( ( B - 2 ) \alpha \left[ \mathbf { J } _ { \phi } ( 1 ) - \mathbf { J } _ { \phi } \left( \frac { - 1 } { B - 1 } \right) \right] + \frac { B } { B - 1 } \mathbf { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) \right) , +$$ + +Since $\mathrm { t r } ( \dot { \Sigma } _ { 0 } ) = 0$ , Eq. (50) gives + +$$ +\begin{array} { r l } & { \bar { C } ^ { \mathrm { S } ^ { 2 } } \lesssim \frac { \mathrm { d } \nabla \langle \phi \phi \rangle \big ( \frac { 2 \xi } { 2 } \big ) } { \mathrm { d } \xi \mathrm { d } \xi ^ { \prime } } \Big | _ { \mathrm { c o l } \sim \phi } \xi \big | _ { \mathrm { c o l } \sim \phi } \xi \big | _ { \mathrm { c o l } } ^ { \xi } } \\ & { = G ^ { \mathrm { S } ^ { 2 } } \lesssim \frac { d } { d \xi } \sqrt { \kappa ( \mathrm { s m } / \xi ) } \Big | _ { \mathrm { r e c h } } } \\ & { = B ^ { \nu } \mu ^ { \epsilon - 1 } 2 ^ { - \alpha } \mathrm { P r } ( \frac { B + 1 } { 2 } , \alpha ) ^ { - 1 } G ^ { \mathrm { C } ^ { 2 } } \lesssim \frac { \mathrm { d } \nabla _ { \phi } ( \frac { 2 \xi } { 2 } ) } { \mathrm { d } \xi } \Big | _ { \mathrm { s } = \epsilon } \big \{ \dot { \Sigma } \big \} } \\ & { = \bar { B } ^ { \nu } \mu ^ { \epsilon - 1 } 2 ^ { - \alpha } \mathrm { P r } ( \frac { B + 1 } { 2 } , \alpha ) ^ { - 1 } \bar { \lambda } ^ { \frac { \epsilon } { 2 } } \mathrm { o } } \\ & { = ( B - 1 ) ^ { \alpha - 1 } \mathrm { c } _ { \phi } \mu ^ { \epsilon - 1 } 2 ^ { - \alpha } \mathrm { P r } ( \frac { B + 1 } { 2 } , \alpha ) ^ { - 1 } } \\ & { \quad \times ( ( B - 2 ) \alpha \Big [ \mathbb { L } _ { \phi } ( 1 ) - \mathrm { ] } _ { \phi } ( \frac { - 1 } { B - 1 } ) ] + \frac { B } { B - 1 } \big \} _ { \phi } ^ { \prime } ( \frac { - 1 } { B - 1 } ) ) } \\ & { \quad \times \hat { \Sigma } _ { \mathrm { s p } } \bigg [ \mathrm { c m } \Big [ \mathrm { L } _ { \phi } ( 1 ) - \mathrm { ] } _ { \phi } \Big ] } \\ & { \quad \times \hat { \Sigma } _ { \mathrm { s p } } ^ { \epsilon } } \end{array} +$$ + +$$ +\begin{array} { r } { \textnormal { o } \lambda _ { \mathrm { L } } ^ { \uparrow } = ( B - 1 ) ^ { \alpha - 1 } \mathfrak { c } _ { \alpha } \mu ^ { * - 1 } 2 ^ { - \alpha } \mathrm { P } \left( \frac { B + 1 } { 2 } , \alpha \right) ^ { - 1 } \left( ( B - 2 ) \alpha \left[ \mathrm { J } _ { \phi } ( 1 ) - \mathrm { J } _ { \phi } \left( \frac { - 1 } { B - 1 } \right) \right] + \frac { B } { B - 1 } \mathrm { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) \right) } \\ { \prod } \end{array} +$$ + +With some routine computation, we obtain + +Proposition F.34. With $\phi$ and $\alpha$ as above, as $B \infty$ , + +$$ +\begin{array} { r l } & { \lambda _ { \mathtt { M } } ^ { \mathtt { T } } ( B , \alpha ) \sim \alpha + B ^ { - 1 } ( 2 \alpha ^ { 3 } - 4 \alpha ^ { 2 } - \alpha + \frac { \mathtt { J } _ { \phi } ^ { \prime } ( 0 ) } { \mathtt { J } _ { \phi } ( 1 ) - \mathtt { J } _ { \phi } ( 0 ) } ) + O ( B ^ { - 2 } ) } \\ & { \overset { \mathtt { J } ^ { \prime } } { \underset { \mathrm { M } } { = } } ( B , \alpha ) \sim \frac { \mathtt { J } _ { \phi } ^ { \prime } ( 0 ) } { \mathtt { J } _ { \phi } ( 1 ) - \mathtt { J } _ { \phi } ( 0 ) } } \\ & { \quad \quad \ + B ^ { - 1 } \left( ( 2 \alpha ^ { 2 } - 4 \alpha + 1 ) \frac { \mathtt { J } _ { \phi } ^ { \prime } ( 0 ) } { \mathtt { J } _ { \phi } ( 1 ) - \mathtt { J } _ { \phi } ( 0 ) } - \frac { \mathtt { J } _ { \phi } ^ { \prime \prime } ( 0 ) } { \mathtt { J } _ { \phi } ( 1 ) - \mathtt { J } _ { \phi } ( 0 ) } - \left( \frac { \mathtt { J } _ { \phi } ^ { \prime } ( 0 ) } { \mathtt { J } _ { \phi } ( 1 ) - \mathtt { J } _ { \phi } ( 0 ) } \right) ^ { 2 } \right) } \\ & { \quad \ \quad + O \left( B ^ { - 2 } \right) } \end{array} +$$ + +We can compute, by Proposition E.23, + +$$ +\begin{array} { r l r } { { \frac { \mathrm { J } _ { \phi } ^ { \prime } ( 0 ) } { \mathrm { J } _ { \phi } ( 1 ) - \mathrm { J } _ { \phi } ( 0 ) } = \frac { ( a + b ) ^ { 2 } \frac { 1 } { \sqrt { \pi } } \frac { \Gamma ( \frac { \alpha } { 2 } + 1 ) ^ { 2 } } { \Gamma ( \alpha + \frac { 1 } { 2 } ) } } { ( a ^ { 2 } + b ^ { 2 } ) - ( a - b ) ^ { 2 } \frac { 1 } { 2 \sqrt { \pi } } \frac { \Gamma ( \frac { \alpha } { 2 } + \frac { 1 } { 2 } ) ^ { 2 } } { \Gamma ( \alpha + \frac { 1 } { 2 } ) } } } } \\ & { } & { \leq \frac { 2 \Gamma ( \frac { \alpha } { 2 } + 1 ) ^ { 2 } } { \sqrt { \pi } \Gamma ( \alpha + \frac { 1 } { 2 } ) } } \end{array} +$$ + +where the last part is obtained by optimizing over a and $b$ . On $\alpha \in ( - 1 / 2 , \infty )$ , this is greater than $\alpha$ iff $\alpha < 1$ . Thus for $\alpha \geq 1$ , the maximum eigenvalue of $\mathcal { U }$ is always achieved by eigenspace L for large enough $B$ . + +# G BACKWARD DYNAMICS + +Let’s extend the definition of the $\mathrm { V }$ operator: + +Definition G.1. Suppose $T _ { x } : \mathbb { R } ^ { n } \mathbb { R } ^ { m }$ is a linear operator parametrized by $\boldsymbol { x } \in \mathbb { R } ^ { k }$ . Then for a PSD matrix $\Sigma \in S _ { k }$ , define $\mathrm { V } _ { T } ( \Sigma ) : = \mathbb { E } [ T _ { x } \otimes T _ { x } : x \in \mathcal { N } ( 0 , \Sigma ) ] : \mathbb { R } ^ { n \times n } \to \mathbb { R } ^ { m \times m }$ , which acts on $n \times n$ matrices $\Pi$ by + +$$ +\mathrm { V } _ { T } ( \Sigma ) \{ \Pi \} = \mathbb { E } [ T _ { x } \Pi T _ { x } ^ { T } : x \in \mathcal { N } ( 0 , \Sigma ) ] \in \mathbb { R } ^ { m \times m } . +$$ + +Under this definition, $\mathrm { V } _ { B _ { \phi } ^ { \prime } } ( \Sigma )$ is a linear operator $\mathbb { R } ^ { B \times B } \xrightarrow { } \mathbb { R } ^ { B \times B }$ . Recall the notion of adjoint: + +Definition G.2. If $V$ is a (real) vector space, then $V ^ { \dagger }$ , its dual space, is defined as the space of linear functionals $f : V \to \mathbb { R }$ on $V$ . If $T : V \to W$ is a linear operator, then $T ^ { \dagger }$ , its adjoint, is a linear operator $W ^ { \dagger } V ^ { \dagger }$ , defined by $T ^ { \dagger } ( f ) = v \mapsto f ( T ( v ) )$ . + +If a linear operator is represented by a matrix, with function application represented by matrixvector multiplication (matrix on the left, vector on the right), then the adjoint is represented by matrix transpose. + +The backward equation. In this section we are interested in the backward dynamics, given by the following equation + +$$ +\Pi ^ { l } = \mathrm { V } _ { B _ { \phi } ^ { \prime } } ( \Sigma ^ { l } ) ^ { \dagger } \{ \Pi ^ { l + 1 } \} +$$ + +where $\Sigma ^ { l }$ is given by the forward dynamics. Particularly, we are interested in the specific case when we have exponential convergence of $\Sigma ^ { l }$ to a BSB1 fixed point. Thus we will study the asymptotic approximation of the above, namely the linear system. + +$$ +\Pi ^ { l } = \mathrm { V } _ { B _ { \phi } ^ { \prime } } ( \Sigma ^ { * } ) ^ { \dagger } \{ \Pi ^ { l + 1 } \} +$$ + +where $\Sigma ^ { * }$ is the BSB1 fixed point. Note that after one step of backprop, $\Pi ^ { L - 1 } = \mathrm { V } _ { \mathcal { B } _ { \phi } ^ { \prime } } ( \Sigma ^ { * } ) ^ { \dagger } \{ \Pi ^ { L } \}$ is in $\mathcal { H } _ { B } ^ { G }$ . Thus the large $L$ dynamics of Eq. (51) is given by the eigendecomposition of $\mathrm { V } _ { B _ { \phi } ^ { \prime } } ( \Sigma ^ { * } ) ^ { \dag } \ \mathinner { \uparrow }$ $\mathcal { H } _ { B } ^ { G } : \mathcal { H } _ { B } ^ { G } \to \mathcal { H } _ { B } ^ { G }$ . It turns out to be much more convenient to study its adjoint $G ^ { \otimes 2 } \circ \mathrm { V } _ { B _ { \phi } ^ { \prime } } ( \Sigma ^ { * } )$ , which has the same eigenvalues (in fact, it will turn out that it is self-adjoint). + +# G.1 EIGENDECOMPOSITION OF $G ^ { \otimes 2 } \circ \mathrm { V } _ { B _ { \phi } ^ { \prime } } ( \Sigma ^ { * } )$ + +By chain rule, $\begin{array} { r } { \mathcal { B } _ { \phi } ^ { \prime } ( x ) = \frac { \mathrm { d } \phi \mathsf { o n } ( z ) } { \mathrm { d } z } \big | _ { z = G x } \frac { \mathrm { d } G x } { \mathrm { d } x } = \frac { \mathrm { d } \phi \mathsf { o n } ( z ) } { \mathrm { d } z } \big | _ { z = G x } G , \mathcal { B } _ { \phi } ^ { \prime } ( x ) \Delta x = \frac { \mathrm { d } \phi \mathsf { o n } ( z ) } { \mathrm { d } z } \big | _ { z = G x } G \Delta x } \end{array}$ Thus + +$$ +\begin{array} { r l } & { \nabla _ { B _ { \phi } ^ { \prime } } ( \Sigma ) = \mathbb { E } \left[ B _ { \phi } ^ { \prime } ( x ) ^ { \otimes 2 } : x \sim \mathcal { N } ( 0 , \Sigma ) \right] } \\ & { \qquad = \mathbb { E } \left[ \left( \left. \frac { \mathrm { d } \phi \circ \mathrm { n } ( z ) } { \mathrm { d } z } \right| _ { z = G x } G \right) ^ { \otimes 2 } : x \sim \mathcal { N } ( 0 , \Sigma ) \right] } \\ & { \qquad = \mathbb { E } \left[ \left. \frac { \mathrm { d } \phi \circ \mathrm { n } ( z ) } { \mathrm { d } z } \right| _ { z = G x } ^ { \otimes 2 } \circ G ^ { \otimes 2 } : x \sim \mathcal { N } ( 0 , \Sigma ) \right] ^ { - 1 } } \\ & { \qquad = \mathbb { E } \left[ \left. \frac { \mathrm { d } \phi \circ \mathrm { n } ( z ) } { \mathrm { d } z } \right| _ { z = G x } ^ { \otimes 2 } : x \sim \mathcal { N } ( 0 , \Sigma ) \right] \circ G ^ { \otimes 2 } } \\ & { \qquad = \mathbb { V } \left[ \left. \frac { \mathrm { d } \phi \circ \mathrm { n } ( \Sigma ) } { \mathrm { d } z } \right| _ { z = G x } ^ { \otimes 2 } \right] \left( \Sigma \right) c G ^ { \otimes 2 } } \end{array} +$$ + +If we let $\mathcal { F } ( \Sigma ) : = \mathrm { V } _ { \left[ x \mapsto \frac { \mathtt { d } \phi \circ \mathsf { n } ( z ) } { \mathtt { d } z } \right| _ { z = G x } } \big ] ( \Sigma )$ , then $\mathrm { V } _ { \mathcal { B } _ { \phi } ^ { \prime } } ( \Sigma ) = \mathcal { F } ( \Sigma ) \circ G ^ { \otimes 2 }$ . As discussed above, we will seek the eigendecomposition of $G ^ { \otimes 2 } \circ { \mathcal { F } } ( \Sigma ) \mid { \mathcal { H } } _ { B } ^ { G } : { \mathcal { H } } _ { B } ^ { G } \to { \mathcal { H } } _ { B } ^ { G }$ . + +We first make some basic calculations. + +# Proposition G.3. + +$$ +\left. { \frac { \operatorname { d n } ( z ) } { \operatorname { d } z } } \right| _ { z = y } = { \sqrt { B } } r ^ { - 1 } ( I - v v ^ { T } ) , +$$ + +where $r = \| y \| , v = y / \| y \| = \mathsf { n } ( y ) / \sqrt { B }$ . + +Proof. We have $\begin{array} { r } { \left. \frac { \mathrm { d } { \bf n } ( z ) } { \mathrm { d } z } \right| _ { z = y } = \sqrt { B } \left. \frac { \mathrm { d } z / \| z \| } { \mathrm { d } z } \right| _ { z = y } , } \end{array}$ , and + +$$ +\begin{array} { r l } { \frac { \partial ( y _ { i } / \| y \| ) } { \partial y _ { j } } = \frac { \delta _ { i j } \| y \| - ( \partial \| y \| / \partial y _ { j } ) y _ { i } } { \| y \| ^ { 2 } } } & { } \\ & { \quad = \frac { \delta _ { i j } } { \| y \| } - \frac { y _ { i } } { \| y \| ^ { 2 } } \frac { \partial \sqrt { y \cdot y } } { \partial y _ { j } } } \\ & { \quad = \frac { \delta _ { i j } } { \| y \| } - \frac { y _ { i } } { \| y \| ^ { 2 } } \frac { y _ { j } } { \| y \| } } \\ & { \quad = \frac { \delta _ { i j } } { \| y \| } - \frac { y _ { i } y _ { j } } { \| y \| ^ { 3 } } } \end{array} +$$ + +so that + +$$ +\left. { \frac { \operatorname { d } { \boldsymbol { z } } / \| { \boldsymbol { z } } \| } { \operatorname { d } z } } \right| _ { z = y } = r ^ { - 1 } ( I - v v ^ { T } ) . +$$ + +By chain rule, this easily gives + +Proposition G.4. + +$$ +\begin{array} { c } { \displaystyle \left. \frac { \mathrm { d } \phi \circ \mathsf { n } ( z ) } { \mathrm { d } z } \right| _ { z = y } = \sqrt { B } D r ^ { - 1 } ( I - v v ^ { T } ) , } \\ { D = \mathrm { D i a g } ( \phi ^ { \prime } ( \mathsf { n } ( y ) ) ) , r = \| y \| , v = y / \| y \| = \mathsf { n } ( y ) / \sqrt { B } . } \end{array} +$$ + +With $v = y / \| y \|$ , $r = \| y \|$ , and $\tilde { D } = \mathrm { D i a g } ( \phi ^ { \prime } ( { \mathsf { n } } ( y ) ) )$ , we then have + +$$ +\begin{array} { r l } & { \mathcal { F } ( \Sigma ) = \mathbb { E } [ \tilde { D } ^ { \otimes 2 } \circ ( \sqrt { B } \frac { \mathrm { d } z / \| z \| } { \mathrm { d } z } | _ { z = G x } ) ^ { \otimes 2 } : x \sim \mathcal { N } ( 0 , \Sigma ) ] } \\ & { \quad = \mathbb { E } [ \tilde { D } ^ { \otimes 2 } \circ ( \sqrt { B } \frac { \mathrm { d } z / \| z \| } { \mathrm { d } z } | _ { z = y } ) ^ { \otimes 2 } : y \sim \mathcal { N } ( 0 , G \Sigma G ) ] } \\ & { \quad = B \mathbb { E } [ r ^ { - 2 } ( \tilde { D } ^ { \otimes 2 } + ( \tilde { D } v v ^ { T } ) ^ { \otimes 2 } - \tilde { D } \otimes ( \tilde { D } v v ^ { T } ) - ( \tilde { D } v v ^ { T } ) \otimes \tilde { D } ) : y \sim \mathcal { N } ( 0 , G \Sigma G ) ] } \end{array} +$$ + +$$ +\mathcal { F } ( \Sigma ) \{ \Lambda \} = B \mathbb { E } \left[ r ^ { - 2 } \left( \tilde { D } \Lambda \tilde { D } + \tilde { D } v v ^ { T } \Lambda v v ^ { T } \tilde { D } - \tilde { D } \Lambda v v ^ { T } \tilde { D } - \tilde { D } v v ^ { T } \Lambda \tilde { D } \right) : y \sim \mathcal { N } ( 0 , G \Sigma G ) \right] +$$ + +# G.1.1 SPHERICAL INTEGRATION + +When $\Sigma = \Sigma ^ { * }$ is the BSB1 fixed point, one can again easily see that $\mathcal { F } ( \Sigma ^ { * } )$ is ultrasymmetric. With $\begin{array} { r } { \tau _ { i j } = \frac { 1 } { 2 } \left( \delta _ { i } \delta _ { j } ^ { T } + \delta _ { j } \delta _ { i } ^ { T } \right) } \end{array}$ (Defn F.20), we have + +$$ +\begin{array} { r l } { \iota ( \Sigma ^ { * } ) \langle \lambda \rangle = B \mathbb { E } \Bigg [ \Gamma ^ { - 2 } \Big ( \tilde { D } \Lambda \tilde { D } + \tilde { D } w ^ { \top } \Delta w ^ { \top } \tilde { D } - \tilde { D } \tilde { D } + \Delta a w ^ { \top } \tilde { D } - \tilde { D } w w ^ { \top } \tilde { D } - \tilde { D } w ^ { \top } \tilde { A } \tilde { D } \Big ) : y \sim N ( 0 , p \tilde { \Sigma } \tilde { U } ) \Bigg ] } & { } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \quad \quad \quad \quad \\ & { \quad = B ^ { - 1 } B \mathbb { E } \Bigg [ - 2 \Big ( \tilde { D } \Lambda \tilde { D } + \tilde { D } w ^ { \top } \tilde { D } w \chi \chi w ^ { \top } \tilde { D } - \tilde { D } \Lambda w \pi ^ { \top } \tilde { T } - \tilde { D } w \pi ^ { \top } \tilde { D } - \tilde { D } w \pi ^ { \top } \tilde { A } \tilde { D } \Big ) \cdot \tilde { y } \times N ( 0 , \mathcal { A } ) \Bigg ] } & { } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & \quad \quad \quad \quad \quad \quad \quad \quad \end{array} +$$ + +by Lemma E.35, where we assume, WLOG by ultrasymmetry, $k , l \in \{ 1 , 2 \} ; i , j \in \{ 1 , \ldots , 4 \}$ , and + +$$ +\begin{array} { l } { { \mathbf { \dot { \tau } } ( v _ { 1 } , \dots , v _ { 4 } ) : = \displaystyle \frac { 1 } { 2 } \phi ^ { \prime } ( \sqrt { B } \mathfrak { v } v ) _ { k } \phi ^ { \prime } ( \sqrt { B } \mathfrak { e } v ) _ { l } \big ( } } \\ { { \mathbf { \dot { \tau } } ( k = i \ \& \ l = j ) + \mathbb { I } ( k = j \ \& \ l = i ) + 2 ( \mathfrak { e } v ) _ { i } ( \mathfrak { e } v ) _ { j } ( \mathfrak { e } v ) _ { k } ( \mathfrak { e } v ) _ { l } } } \\ { { \mathbf { \dot { \tau } } - ( \mathbb { I } ( i = k ) ( \mathfrak { e } v ) _ { j } ( \mathfrak { e } v ) _ { l } + \mathbb { I } ( j = k ) ( \mathfrak { e } v ) _ { i } ( \mathfrak { e } v ) _ { l } + \mathbb { I } ( i = l ) ( \mathfrak { e } v ) _ { j } ( \mathfrak { e } v ) _ { k } + \mathbb { I } ( j = l ) ( \mathfrak { e } v ) _ { i } ( \mathfrak { e } v ) _ { l } } } \\ { { \mathbf { \dot { \tau } } } } \\ { { \mathbf { \dot { \tau } } _ { 1 } ^ { \theta } = \cos { \theta _ { 1 } } } } \\ { { v _ { 2 } ^ { \theta } = \sin { \theta _ { 1 } } \cos { \theta _ { 2 } } } } \\ { { v _ { 3 } ^ { \theta } = \sin { \theta _ { 1 } } \sin { \theta _ { 2 } } \cos { \theta _ { 3 } } } } \\ { { v _ { 4 } ^ { \theta } = \sin { \theta _ { 1 } } \sin { \theta _ { 2 } } \sin { \theta _ { 3 } } \cos { \theta _ { 4 } } . } } \end{array} +$$ + +and $\mathbb { P }$ as in Eq. (33). + +We can integrate out $\theta _ { 3 }$ and $\theta _ { 4 }$ symbolically by Lemma E.29 since their dependence only appear outside of $\phi ^ { \prime }$ . This reduces each entry of $\mathcal { F } ( \Sigma ^ { * } ) \{ \tau _ { i j } \} _ { k l }$ to 2-dimensional integrals to evaluate numerically. The eigenvalues of $G ^ { \otimes 2 } \circ { \mathcal { F } } ( \Sigma ^ { * } )$ can then be obtained from Thm E.62. We omit details here and instead focus on results from other methods. + +# G.1.2 GEGENBAUER EXPANSION + +Notice that when $\Lambda = G$ , the expression for $\mathcal { F } ( \Sigma ^ { * } ) \{ \Lambda \}$ significantly simplifies because $G$ acts as identity on $v$ : + +$$ +\begin{array} { r l } & { \boldsymbol { \varepsilon } ( \Sigma ^ { * } ) \{ G \} = \mu ^ { * } \ ) ^ { 2 } \boldsymbol { B } \mathbb { E } \left[ r ^ { - 2 } \left( \boldsymbol { \tilde { D } } \boldsymbol { \Lambda } \boldsymbol { \tilde { D } } + \boldsymbol { \tilde { D } } \boldsymbol { v } \boldsymbol { v } ^ { T } \boldsymbol { G } \boldsymbol { v } \boldsymbol { v } ^ { T } \boldsymbol { \tilde { D } } - \boldsymbol { \tilde { D } } \boldsymbol { G } \boldsymbol { v } \boldsymbol { v } ^ { T } \boldsymbol { \tilde { D } } - \boldsymbol { \tilde { D } } \boldsymbol { v } \boldsymbol { v } ^ { T } \boldsymbol { G } \boldsymbol { \tilde { D } } \right) : \boldsymbol { y } \sim \mathcal { N } ( \boldsymbol { 0 } , \boldsymbol { G } ) \right] } \\ & { \quad \quad \quad = \mu ^ { * } - 1 \boldsymbol { B } \mathbb { E } \left[ r ^ { - 2 } \left( \boldsymbol { \tilde { D } } \boldsymbol { G } \boldsymbol { \tilde { D } } - \boldsymbol { \tilde { D } } \boldsymbol { v } \boldsymbol { v } ^ { T } \boldsymbol { \tilde { D } } \right) : \boldsymbol { y } \sim \mathcal { N } ( \boldsymbol { 0 } , \boldsymbol { G } ) \right] } \end{array} +$$ + +Then the diagonal of the image satisfies + +$$ +\begin{array} { r l } & { \quad \mathcal { F } ( \Sigma ^ { * } ) \} G \int _ { a } } \\ & { = \mu ^ { * - 1 } B \mathbb { E } \left[ r ^ { - 2 } \left( \frac { B - 1 } { B } \phi ^ { \prime } ( \sqrt { B } v _ { a } ) ^ { 2 } - v _ { a } ^ { 2 } \phi ^ { \prime } ( \sqrt { B } v _ { a } ) ^ { 2 } \right) : r v \sim \mathcal { N } ( 0 , G ) \right] } \\ & { = \mu ^ { * - 1 } B ( B - 3 ) ^ { - 1 } \underset { v \sim \mathbb { S } ^ { B } } { \mathbb { E } } \ \left( \frac { B - 1 } { B } \phi ^ { \prime } ( \sqrt { B } \mathbf { e } _ { a } , v ) ^ { 2 } - ( \mathbf { e } _ { a } , v ) ^ { 2 } \phi ^ { \prime } ( \sqrt { B } \mathbf { e } _ { a } , v ) ^ { 2 } \right) \mathcal { K } ( v ; G , 2 ) } \\ & { \qquad \mathrm { ( b y ~ P r o p o s i t i o n ~ E \lambda ^ { 3 , 3 } ) } } \\ & { = \mu ^ { * - 1 } B ( B - 3 ) ^ { - 1 } \underset { v \sim \mathbb { S } ^ { B - 1 } } { \underbrace { \mathbb { E } } } \left( 1 - ( \hat { \mathbf { e } } _ { a } , v ) ^ { 2 } \right) \frac { B - 1 } { B } \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathbf { e } } _ { a } , v ) ^ { 2 } } \\ & { \qquad \quad - \mu ^ { * - 1 } ( B - 1 ) ( B - 3 ) ^ { - 1 } \underset { v \sim \mathbb { S } ^ { B - 2 } } { \underbrace { \mathbb { E } } } \left( 1 - ( \hat { \mathbf { e } } _ { a } , v ) ^ { 2 } \right) \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathbf { e } } _ { a } , v ) ^ { 2 } } \end{array} +$$ + +And the off-diagonal entries satisfy + +$$ +\begin{array} { r l } & { \qquad \mathcal { F } ( \Sigma ^ { * } ) \{ G \} _ { a b } } \\ & { = \mu ^ { * - 1 } B \mathbb { E } [ r ^ { - 2 } ( \cfrac { - 1 } { B } \phi ^ { \prime } ( \sqrt { B } \mathfrak { e } _ { a , v } ) \phi ^ { \prime } ( \sqrt { B } \mathfrak { e } _ { b , v } ) } \\ & { \qquad - ( \mathfrak { e } _ { a , v } ) ( \mathfrak { e } _ { b , : } v ) \phi ^ { \prime } ( \sqrt { B } \mathfrak { e } _ { a , : } v ) \phi ^ { \prime } ( \mathfrak { e } _ { a , : } v ) ] : r v \sim \mathcal { N } ( 0 , G ) ] } \\ & { = \mu ^ { * - 1 } B ( B - 3 ) ^ { - 1 } \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } ( \cfrac { - 1 } { B } \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathfrak { e } } _ { a , v } ) \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathfrak { e } } _ { b , v } ) } \\ & { \qquad - \cfrac { B - 1 } { B } ( \hat { \mathfrak { e } } _ { a , v } ) ( \hat { \mathfrak { e } } _ { b , v } ) \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathfrak { e } } _ { a , : } v ) \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathfrak { e } } _ { b , : } v ) ) K ( v ; G , 2 ) } \end{array} +$$ + +(by Proposition E.33) + +$$ +\begin{array} { l } { { = \mu ^ { * - 1 } B ( B - 3 ) ^ { - 1 } \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \frac { - 1 } { B } ( 1 + ( B - 1 ) ( \hat { \mathfrak { e } } _ { a , v } ) ( \hat { \mathfrak { e } } _ { b , v } ) ) } } \\ { { \qquad \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathfrak { e } } _ { a , v } ) \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathfrak { e } } _ { b , v } , v ) } } \\ { { \qquad \quad } } \\ { { = - \mu ^ { * - 1 } ( B - 3 ) ^ { - 1 } \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } ( 1 + ( B - 1 ) ( \hat { \mathfrak { e } } _ { a , v } ) ( \hat { \mathfrak { e } } _ { b , v } ) ) \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathfrak { e } } _ { a , v } ) \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathfrak { e } } _ { b , v } , v ) } } \end{array} +$$ + +Lifting Spherical Expectation. We first show a way of expressing $\mathcal { F } ( \Sigma ^ { * } ) _ { a a }$ in terms of Gegenbauer basis by lifting the spherical integral over $S ^ { B - 2 }$ to spherical integral over $S ^ { B }$ . While this technique cannot be extended to $\mathcal { F } ( \Sigma ^ { * } ) _ { a b } , a \neq b$ , we believe it could be of use to future related problems. + +If $\phi ( { \sqrt { B - 1 } } x )$ has Gegenbauer expansion $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ , then $\phi ^ { \prime } ( \sqrt { B - 1 } x ) =$ $\begin{array} { r c l } { { \frac { 1 } { \sqrt { B - 1 } } \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) \prime } ( x ) } } & { { = } } & { { \frac { 1 } { \sqrt { B - 1 } } \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { B - 3 } { c _ { B - 1 , l } } C _ { l - 1 } ^ { ( \frac { B - 1 } { 2 } ) } ( x ) } } \end{array}$ $\begin{array} { r } { \frac { 1 } { \sqrt { B - 1 } } \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { B - 3 } { c _ { B - 1 , l } } C _ { l - 1 } ^ { ( \frac { B - 1 } { 2 } ) } ( \hat { \mathbb { e } } _ { a , : } v ) = \frac { 1 } { \sqrt { B - 1 } } \sum _ { l = 0 } ^ { \infty } a _ { l } ( B - 3 ) \frac { c _ { B + 1 , l - 1 } } { c _ { B - 1 , l } } Z _ { \hat { \mathbb { e } } _ { a , : } } ^ { B , ( l - 1 ) } ( v ) . } \end{array}$ , so that $\phi ^ { \prime } ( \sqrt { B } \mathbb { e } _ { a , : } v ) \ =$ . Here $\begin{array} { r } { \hat { \Phi } _ { a , } } \end{array}$ : and $v$ are treated as points on the $B$ -dimensional sphere with last 2 coordinates 0. + +Thus by Lemmas E.38 and E.39, + +$$ +\begin{array} { r l } & { \begin{array} { r l } & { \rho _ { \mathrm { e f f } } ^ { ( 1 ) } ( x , y ) } \\ & { = \rho _ { \mathrm { e f f } } ^ { ( 1 ) } ( x , y ) } \\ & { \rho _ { \mathrm { e f f } } ^ { ( 1 ) } ( x , y ) } \end{array} } \\ & \begin{array} { r l } & ( x - y ) ^ { 2 } ( x - 1 ) ^ { 2 } \left( x - 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ { 2 } \left( x + 1 \right) ^ 2 \end{array} \end{array} +$$ + +.45 + +Dirichlet-like form of $\phi$ . For ${ \mathcal { F } } ( \Sigma ^ { * } ) \{ G \} _ { a b }$ , there’s no obvious way of lifting the spherical expectation to a higher dimension with the weight $( 1 + ( B - 1 ) ( \hat { \mathbb { E } } _ { a , : } v ) ( \hat { \mathbb { e } } _ { b , : } v ) )$ . We will instead leverage $\cdot$ . In Thm E.47, setting $u _ { 1 } = \hat { \Phi } _ { a , : } = u _ { 2 }$ , we get + +$$ +\begin{array} { l } { \displaystyle \mathcal { F } ( \Sigma ^ { * } ) \{ G \} _ { a a } = \mu ^ { * - 1 } \frac { B - 1 } { B - 3 } \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } ( ( \hat { \mathbb { e } } _ { a , : } \cdot \hat { \mathbb { e } } _ { a , : } ) - ( \hat { \mathbb { e } } _ { a , : } v ) ^ { 2 } ) \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathbb { e } } _ { a , : } v ) ^ { 2 } } \\ { \displaystyle \quad \quad = \mu ^ { * - 1 } \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { l + B - 3 } { B - 3 } l c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( 1 \right) . } \end{array} +$$ + +Likewise, setting $u _ { 1 } = \widehat { \Phi } _ { a , : } , u _ { 2 } = \widehat { \Phi } _ { b , : }$ , we get + +$$ +\begin{array} { l } { \displaystyle { } ^ { r } ( \Sigma ^ { * } ) \{ G \} _ { a b } = \mu ^ { * - 1 } \displaystyle { \frac { B - 1 } { B - 3 } \displaystyle { \frac { \mathbb { E } } { v \sim S ^ { B - 2 } } \big ( \big ( \hat { \mathbf { e } } _ { a } , \cdot ~ \hat { \mathbf { e } } _ { b , : } \big ) - \big ( \hat { \mathbf { e } } _ { a , : } v \big ) \big ( \hat { \mathbf { e } } _ { b , : } v \big ) \big ) \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathbf { e } } _ { a , : } v ) \phi ^ { \prime } ( \sqrt { B - 1 } \hat { \mathbf { e } } _ { b , : } v ) } } } \\ { { \displaystyle \qquad = \mu ^ { * - 1 } \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \displaystyle { \frac { l + B - 3 } { B - 3 } l c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) } } } \end{array} +$$ + +Thus, + +Theorem G.5. SupposGegenbauer expansion $\phi ( \sqrt { B - 1 } x ) , \phi ^ { \prime } ( \sqrt { B - 1 } x ) \in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \frac { B - 3 } { 2 } } )$ and nique $\phi ( \sqrt { B - 1 } x )$ hasoint $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ $\Sigma ^ { * }$ + +of Eq. (43), the eigenvalue of $G ^ { \otimes 2 } \circ { \mathcal { F } } ( \Sigma ^ { * } )$ corresponding to eigenspace $\mathbb { R } G$ is + +$$ +\begin{array} { l } { { \displaystyle \lambda _ { G } ^ { \downarrow } = \mu ^ { * - 1 } \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { l + B - 3 } { B - 3 } l c _ { B - 1 , l } ^ { - 1 } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( 1 \right) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } } \\ { { \displaystyle \quad = \frac { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { l + B - 3 } { B - 3 } l c _ { B - 1 , l } ^ { - 1 } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( 1 \right) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } c _ { B - 1 , l } ^ { - 1 } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( 1 \right) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } } } \\ { { \displaystyle > 1 } } \end{array} +$$ + +This is minimized (over choices of $\phi$ ) iff $\phi$ is linear, in which case $\begin{array} { r } { \lambda _ { G } ^ { \downarrow } = \frac { B - 2 } { B - 3 } } \end{array}$ + +# G.2 LAPLACE METHOD + +In this section suppose that $\phi$ is degree $\alpha$ positive-homogeneous. Set $D = \operatorname { D i a g } ( \phi ^ { \prime } ( y ) )$ (and recall $v = y / \| y \| , r = \| y \| )$ . Then $D$ is degree $\alpha - 1$ positive-homogeneous in $x$ (because $\phi ^ { \prime }$ is). Consequently we can rewrite Eq. (52) as follows, + +$$ +\begin{array} { r l r } & { = B _ { \phantom { \alpha } y \sim N ( 0 , G \Sigma ^ { * } G ) } ^ { \alpha } \biggl [ r ^ { - 2 } \biggl ( r ^ { - 2 ( \alpha - 1 ) } D ^ { \otimes 2 } + r ^ { - 2 ( \alpha + 1 ) } ( D v v ^ { T } ) ^ { \otimes 2 } \phantom { x } } \\ & { \phantom { = } - r ^ { - 2 \alpha } D \otimes ( D v v ^ { T } ) - r ^ { - 2 \alpha } ( D v v ^ { T } ) \otimes D \biggr ) \biggr ] } \\ & { = B _ { \phantom { \alpha } y \sim N ( 0 , \mu \cdot \mathbf { G } ) } ^ { \alpha } \biggl [ r ^ { - 2 \alpha } D ^ { \otimes 2 } + r ^ { - 2 ( \alpha + 2 ) } ( D y ^ { T } ) ^ { \otimes 2 } - r ^ { - 2 ( \alpha + 1 ) } \left( D \otimes ( D y y ^ { T } ) + ( D y y ^ { T } ) \otimes D \right) } \\ & { = B _ { \phantom { \alpha } ( \mathbf { A } + \mathbf { B } - \mathbf { C } ) } ^ { \alpha } } & { ( 5 3 \qquad ( \mathbf { A } + \mathbf { B } - \mathbf { C } ) } \end{array} +$$ + +where $\Sigma ^ { * }$ is the BSB1 fixed point of Eq. (43), $G \Sigma ^ { * } G = \mu ^ { * } G$ , and + +$$ +\begin{array} { r l } & { \mathsf { A } = \mathbb { E } \left[ r ^ { - 2 \alpha } D ^ { \otimes 2 } : y \sim \mathcal { N } ( 0 , \mu ^ { * } G ) \right] } \\ & { \mathsf { B } = \mathbb { E } \left[ r ^ { - 2 ( \alpha + 2 ) } ( D y y ^ { T } ) ^ { \otimes 2 } : y \sim \mathcal { N } ( 0 , \mu ^ { * } G ) \right] } \\ & { \mathsf { C } = \mathbb { E } \left[ r ^ { - 2 ( \alpha + 1 ) } \left( D \otimes ( D y y ^ { T } ) + ( D y y ^ { T } ) \otimes D \right) : y \sim \mathcal { N } ( 0 , \mu ^ { * } G ) \right] } \end{array} +$$ + +Each term in the sum above is ultrasymmetric, so has the same eigenspaces $\mathbb { R } G , \mathbb { M } , \mathbb { L }$ (this will also become apparent in the computations below without resorting to Thm E.61). We can thus compute the eigenvalues for each of them in order. + +In all computation below, we apply Lemma E.2 first to relate the quantity in question to $\mathrm { V } _ { \phi }$ + +Computing A. For two matrices $\Sigma , \Lambda$ , write $\Sigma \odot \Lambda$ for entrywise multiplication. We have by Lemma E.2 + +$$ +\begin{array} { r l } { \Lambda \{ \Lambda \} } & { = \mathbb { E } [ r ^ { - 2 \alpha } D \Lambda D \cdot y \sim { \cal N } ( 0 , \mu ^ { * } Q ) ] } \\ & { = \mathbb { E } [ r ^ { - 2 \alpha } \Lambda \odot \psi ( \psi ) \psi ^ { T } \mid \psi ^ { T } : y \sim { \cal N } ( 0 , \mu ^ { * } C ) ] } \\ & { = \mathbb { E } [ \exp ^ { - \alpha } \int _ { 0 } ^ { \infty } \mathrm { d } s \ \mathrm { o } ^ { \alpha - 1 } \ \mathrm { d } \mathrm { e t } ( f + 2 s \mu ^ { * } G ) ^ { - 1 / 2 } \Lambda \odot \mathrm { V } _ { \psi } ( \frac { \mu ^ { * } } { 1 + 2 s \mu ^ { * } } G ) } \\ & { = \Gamma ( \alpha ) ^ { - 1 } \int _ { 0 } ^ { \infty } \mathrm { d } s \ { s } ^ { \alpha - 1 } ( 1 + 2 s \mu ^ { * } ) ^ { - ( N - 1 ) / 2 } \Lambda \odot ( \frac { \mu ^ { * } } { 1 + 2 s \mu ^ { * } } ) ^ { \alpha - 1 } \mathrm { V } _ { \psi } \langle G \rangle } \\ & { = \Gamma ( \alpha ) ^ { - 1 } \Lambda \odot \mathrm { V } _ { \psi } \langle G \rangle \int _ { 0 } ^ { \infty } \mathrm { d } s \ \mathrm { o } ( \mu ^ { * } s ) ^ { \alpha - 1 } ( 1 + 2 s \mu ^ { * } ) ^ { - ( N - 1 ) / 2 - \alpha + 1 } } \\ & { = \Gamma ( \alpha ) ^ { - 1 } \Lambda \odot \mathrm { V } _ { \psi } \langle G \rangle \mathrm { d } \mathrm { e t } ( \frac { B } { 2 } - 3 ) , } \\ & { = \Lambda \odot \mathsf { c } _ { \alpha - 1 } ( \frac { B - 1 } { B } ) ^ { \alpha - 1 } \mathrm { R S B } ( \mathrm { J } _ { \psi } ( 1 ) , \mathrm { J } _ { \psi } ( \frac { - 1 } { B - 1 } ) ) \mathrm { P } ( \frac { B - 3 } { 2 } , \alpha ) ^ { - 1 } \mid \mu ^ { * } ^ { - 1 } 2 ^ { - \alpha } } \end{array} +$$ + +Then $\cdot$ gives the eigendecomposition for $B ^ { \alpha } G ^ { \otimes 2 } \circ \mathsf { A } \mathrm { ~ } \uparrow \mathcal { H } _ { B } ^ { G }$ + +Theorem G.6. Let $\begin{array} { r } { R _ { B , \alpha } : = ( 2 \alpha - 1 ) \mathsf { c } _ { \alpha - 1 } B ( B - 1 ) ^ { \alpha - 1 } \mathrm { P } \left( \frac { B - 3 } { 2 } , \alpha \right) ^ { - 1 } 2 ^ { - \alpha } } \end{array}$ . Then $B ^ { \alpha } G ^ { \otimes 2 } \circ \mathsf { A } \mid \mathcal { H } _ { B } ^ { G }$ has the following eigendecomposition. + +$$ +\begin{array} { r l } & { \quad \lambda _ { \delta \_ { - } } ^ { \delta \_ { - } } - R _ { \beta , \omega \_ \mu ^ { - } } ^ { \delta \_ { - } - 1 } \left( \begin{array} { l l l } { R } & { 1 , } & { 1 , y _ { \delta } ^ { \delta } \left( 1 \right) } \\ { R } & { 1 , } & { 2 y _ { \delta } ^ { \delta } \left( \_ { S \_ { - } } - 1 \right) } \end{array} \right) , } \\ & { \quad \quad = \frac { \left( 2 \alpha \right) \cdot \left( R - 3 \right) \cdot \left( \alpha ^ { 2 } ( R - 1 ) y _ { \delta } ( y _ { \delta } ( 1 ) + ( 2 \alpha - 1 ) y _ { \delta } ( \frac { 1 } { 1 - \beta } ) \right) } { \left( 2 \alpha - 1 \right) \left( 3 R - 3 \right) \cdot \left( R - 1 \right) \left( \beta ( 1 ) - 1 \right) } } \\ & { \quad \quad \quad \quad \quad \lambda _ { \delta \_ { - } } ^ { \delta \_ { - } } = R _ { \beta , \omega \_ \mu ^ { - } } ^ { \delta \_ { - } - 1 } \left( \frac { R - 2 } { B } , y _ { \varepsilon } ^ { \varepsilon } ( 1 ) + \frac { 2 } { B } , y _ { \delta } ^ { \varepsilon } \left( \frac { - 1 } { B - 1 } \right) \right) , } \\ & { \quad \quad \quad - \frac { \left( 2 \alpha + B - 3 \right) \cdot \left( \alpha ^ { 2 } ( B - 2 ) y _ { \delta } ( x _ { \delta } ( 1 ) ) + 2 ( 2 \alpha - 1 ) y _ { \delta } ^ { \varepsilon } \left( \frac { - 1 } { 1 - B } \right) \right) } { \left( 2 \alpha - 1 \right) \left( B - 3 \right) \cdot \left( \beta ( 1 ) - 1 \right) \left( \beta ( 1 ) - 1 \right) } } \\ & { \quad \quad \quad \quad \lambda _ { \delta \_ { - } } ^ { \delta \_ { - } } = R _ { \beta , \omega ^ { \varepsilon - } } ^ { \delta \_ { - } - 1 } \gamma _ { \varepsilon } ^ { \delta } \left( \frac { - 1 } { B - 1 } \right) } \\ & { \quad \quad \quad - \frac { B \cdot \left( 2 \alpha + B - 3 \right) \cdot \varepsilon _ { 0 } ^ { \varepsilon } \left( \frac { - 1 } { 1 - \beta } \right) } { \left( B - 4 \right) \left( B - 1 \right) \cdot \left( 1 - \beta \right) } } \end{array} +$$ + +Proof. We have $\begin{array} { r } { B ^ { \alpha } \mathsf { A } = ( 2 \alpha - 1 ) ^ { - 1 } R _ { B , \alpha } \mu ^ { * } \overset { - 1 } { \underset { \ r { \infty } } { } } \mathrm { D O S } ( \mathrm { J } _ { \phi ^ { \prime } } ( 1 ) , 0 , \mathrm { J } _ { \phi ^ { \prime } } ( \frac { - 1 } { B - 1 } ) ) . } \end{array}$ . This allows us to apply $\cdot$ . We make a further simplification $\mathrm { J } _ { \phi ^ { \prime } } ( c ) = ( 2 \alpha - 1 ) \mathrm { J } _ { \phi } ^ { \prime } ( c )$ via Proposition E.22. + +Computing B. We simplify + +$$ +\begin{array} { r l } & { \mathsf { B } \{ \Lambda \} = \mathbb E \left[ r ^ { - 2 ( \alpha + 2 ) } D y y ^ { T } \Lambda y y ^ { T } D : y \sim \mathcal N ( 0 , \mu ^ { * } G ) \right] } \\ & { \qquad = \mathbb E \left[ r ^ { - 2 ( \alpha + 2 ) } \psi ( y ) y ^ { T } \Lambda y \psi ( y ) ^ { T } : y \sim \mathcal N ( 0 , \mu ^ { * } G ) \right] } \end{array} +$$ + +where $\psi ( y ) : = y \phi ^ { \prime } ( y )$ , which, in this case of $\phi$ being degree $\alpha$ positive-homogeneous, is also equal to $\alpha \phi ( y )$ . + +By Lemma E.2, $\mathsf { B } \{ \Lambda \}$ equals + +$$ +\Gamma ( \alpha + 2 ) ^ { - 1 } \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { \alpha + 1 } \operatorname * { d e t } ( I + 2 s \mu ^ { * } G ) ^ { - 1 / 2 } \mathbb { E } [ \psi ( y ) y ^ { T } \Lambda y \psi ( y ) ^ { T } : y \sim \mathcal { N } ( 0 , \frac { \mu ^ { * } } { 1 + 2 s \mu ^ { * } } G ) ] +$$ + +This expression naturally leads us to consider the following definition. + +Definition G.7. Let $\psi : \mathbb { R } \mathbb { R }$ be measurable and $\Sigma \in S _ { B }$ . Define $\mathrm { V } _ { \psi } ^ { ( 4 ) } ( \Sigma ) : { \mathcal { H } } _ { B } { \mathcal { H } } _ { B }$ by + +$$ +\nabla _ { \psi } ^ { ( 4 ) } ( \Sigma ) \{ \Lambda \} = \mathbb { E } [ \psi ( y ) y ^ { T } \Lambda y \psi ( y ) ^ { T } : y \sim \mathcal { N } ( 0 , \Sigma ) ] +$$ + +For two matrices $\Sigma , \Lambda$ , write $\langle \Sigma , \Lambda \rangle : = \mathrm { t r } ( \Sigma ^ { T } \Lambda )$ . We have the following identity. + +Proposition G.8. $\begin{array} { r } { \mathrm { V } _ { \phi } ^ { ( 4 ) } ( \Sigma ) \{ \Lambda \} = \mathrm { V } _ { \phi } ( \Sigma ) \langle \Sigma , \Lambda \rangle + 2 \frac { \mathrm { d } \mathrm { V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } \{ \Sigma \Lambda \Sigma \} . } \end{array}$ . + +Proof. We have + +$$ +\begin{array} { r l } & { \frac { \mathrm { I V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } \{ \Lambda \} = ( 2 \pi ) ^ { - B / 2 } \displaystyle \int \mathrm { d } z \phi ( z ) ^ { \otimes 2 } \left( \frac { \mathrm { d } } { \mathrm { d } \Sigma } \mathrm { d e t } ( \Sigma ) ^ { - 1 / 2 } e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \right) \{ \Lambda \} } \\ & { \quad \quad \quad = ( 2 \pi ) ^ { - B / 2 } \displaystyle \int \mathrm { d } z \phi ( z ) ^ { \otimes 2 } \left. \left[ \frac { - 1 } { 2 } \operatorname* { d e t } \Sigma ^ { - 1 / 2 } \Sigma ^ { - 1 } + \operatorname* { d e t } \Sigma ^ { - 1 / 2 } \frac { 1 } { 2 } \Sigma ^ { - 1 } z z z ^ { T } \Sigma ^ { - 1 } \right] e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } } \right. } \\ & { \quad \quad \quad = \displaystyle \frac { 1 } { 2 } ( 2 \pi ) ^ { - B / 2 } \mathrm { d e t } \Sigma ^ { - 1 / 2 } \displaystyle \int \mathrm { d } z \phi ( z ) ^ { \otimes 2 } e ^ { - \frac { 1 } { 2 } z ^ { T } \Sigma ^ { - 1 } z } \left[ \langle z z ^ { T } , \Sigma ^ { - 1 } \Lambda \Sigma ^ { - 1 } \rangle - \langle \Sigma ^ { - 1 } , \Lambda \rangle \right] } \\ & { \quad \quad \quad = \displaystyle \frac { 1 } { 2 } \mathrm { V } _ { \phi } ^ { ( 4 ) } ( \Sigma ) \{ \Sigma ^ { - 1 } \Lambda \Sigma ^ { - 1 } \} - \displaystyle \frac { 1 } { 2 } \mathrm { V } _ { \phi } ( \Sigma ) \langle \Sigma ^ { - 1 } , \Lambda \rangle . } \end{array} +$$ + +Making the substitution $\Lambda \Sigma \Lambda \Sigma$ , we get the desired result. + +If $\varphi$ is degree $\alpha$ positive-homogeneous, then $\mathrm { V } _ { \varphi } ^ { ( 4 ) }$ is degree $\alpha + 1$ positive-homogeneous. Thus, + +$$ +\begin{array} { r l } & { \mathbb { B } \{ \Lambda \} = \Gamma ( \alpha + 2 ) ^ { - 1 } \displaystyle \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { \alpha + 1 } \operatorname* { d e t } ( I + 2 s \mu ^ { * } G ) ^ { - 1 / 2 } \mathbb { V } _ { \psi } ^ { ( 4 ) } \left( \frac { \mu ^ { * } } { 1 + 2 s \mu ^ { * } } G \right) \{ \Lambda \} } \\ & { \quad \quad = \Gamma ( \alpha + 2 ) ^ { - 1 } \displaystyle \int _ { 0 } ^ { \infty } \mathrm { d } s \ s ^ { \alpha + 1 } ( 1 + 2 s \mu ^ { * } ) ^ { - ( B - 1 ) / 2 } \left( \frac { \mu ^ { * } } { 1 + 2 s \mu ^ { * } } \right) ^ { \alpha + 1 } \mathbb { V } _ { \psi } ^ { ( 4 ) } ( G ) \{ \Lambda \} } \\ & { \quad \quad = \Gamma ( \alpha + 2 ) ^ { - 1 } \mathbb { V } _ { \psi } ^ { ( 4 ) } ( G ) \{ \Lambda \} \displaystyle \int _ { 0 } ^ { \infty } \mathrm { d } s \ ( \mu ^ { * } s ) ^ { \alpha + 1 } ( 1 + 2 s \mu ^ { * } ) ^ { - ( B - 1 ) / 2 - \alpha - 1 } } \\ & { \quad \quad = \Gamma ( \alpha + 2 ) ^ { - 1 } \mathbb { V } _ { \psi } ^ { ( 4 ) } ( G ) \{ \Lambda \} \mathrm { B e t a } ( \frac { B - 3 } { 2 } , \alpha + 2 ) 2 ^ { - \alpha - 2 } \mu ^ { * - 1 } } \\ & { \quad \quad = \mathrm { P } \left( \frac { B - 3 } { 2 } , \alpha + 2 \right) ^ { - 1 } 2 ^ { - \alpha - 2 } \mu ^ { * - 1 } \mathbb { V } _ { \psi } ^ { ( 4 ) } ( G ) \{ \Lambda \} . } \end{array} +$$ + +Theorem G.9. $B ^ { \alpha } G ^ { \otimes 2 } \circ \mathsf { B } \ \uparrow \ \mathcal { H } _ { B } ^ { G }$ has the following eigendecomposition (note that here $\mu ^ { * }$ is still with respect to $\phi$ , not $\psi = \alpha \phi$ ) + +1. Eigenspace $\mathbb { R } G$ with eigenvalue + +$$ +\lambda _ { G } ^ { \mathsf { B } } : = \frac { \alpha ^ { 2 } } { B - 3 } +$$ + +2. Eigenspace L with eigenvalue + +$$ +\begin{array} { r l } & { \lambda _ { \mathbb { L } } ^ { \mathbb { B } } : = B ^ { \alpha } \mathrm { P } \left( \frac { B - 3 } { 2 } , \alpha + 2 \right) ^ { - 1 } 2 ^ { - \alpha - 2 } \mu ^ { * - 1 } 2 \lambda _ { \mathbb { L } } ^ { G , \psi } ( B , \alpha ) } \\ & { \quad = \frac { 2 \alpha ^ { 3 } ( B - 2 ) } { ( B - 3 ) ( B - 1 ) ( B - 1 + 2 \alpha ) } + \frac { 2 \alpha ^ { 2 } B J _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) } { ( B - 3 ) ( B - 1 ) ^ { 2 } ( B - 1 + 2 \alpha ) ( J _ { \phi } ( 1 ) - \mathrm { J } _ { \phi } \left( \frac { - 1 } { B - 1 } \right) ) } } \end{array} +$$ + +3. Eigenspace M with eigenvalue + +$$ +\begin{array} { c } { { \lambda _ { \mathbb { M } } ^ { \mathtt { B } } : = B ^ { \alpha } \mathrm { P } \left( \frac { B - 3 } { 2 } , \alpha + 2 \right) ^ { - 1 } 2 ^ { - \alpha - 2 } \mu ^ { * - 1 } 2 \lambda _ { \mathbb { M } } ^ { G , \psi } ( B , \alpha ) } } \\ { { = \frac { 2 \alpha ^ { 2 } B \mathrm { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) } { ( B - 3 ) ( B - 1 ) ( B - 1 + 2 \alpha ) ( \mathrm { J } _ { \phi } ( 1 ) - \mathrm { J } _ { \phi } \left( \frac { - 1 } { B - 1 } \right) ) } } } \end{array} +$$ + +Proof. The only thing to justify is the value of $\lambda _ { G } ^ { \mathsf { B } }$ . We have + +$$ +\begin{array} { r l } { \lambda _ { C } ^ { \mathbb { B } } = B ^ { \alpha } \mathrm { P } \Bigg ( \frac { B - 3 } { 2 } , \alpha + 2 \Bigg ) ^ { - 1 } 2 ^ { - \alpha - 2 } \mu ^ { * - 1 } } & { } \\ & { \qquad \times \left( ( B - 1 ) \mathrm { c } _ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha } \left( \mathrm { J } _ { \mathcal { S } } ( 1 ) - \mathrm { J } _ { \Psi } \left( \frac { - 1 } { B - 1 } \right) \right) + 2 \lambda _ { \Re } ^ { \mathcal { A } , \Re } ( B , \alpha ) \right) } \\ & { = B ^ { \alpha } \mathrm { P } \Bigg ( \frac { B - 3 } { 2 } , \alpha + 2 \Bigg ) ^ { - 1 } 2 ^ { - \alpha - 2 } \mu ^ { * - 1 } ( B - 1 + 2 \alpha ) \mathrm { c } _ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha } \left( \mathrm { J } _ { \Phi } ( 1 ) - \mathrm { J } _ { \Psi } \left( \frac { - 1 } { B - 1 } \right) \right. } \\ & { = \mathrm { c } _ { \alpha } ( B - 1 ) ^ { \alpha } ( B - 1 + 2 \alpha ) \mathrm { P } \left( \frac { B - 3 } { 2 } , \alpha + 2 \right) ^ { - 1 } 2 ^ { - \alpha - 2 } \mu ^ { * - 1 } \left( \mathrm { J } _ { \Psi } ( 1 ) - \mathrm { J } _ { \Psi } \left( \frac { - 1 } { B - 1 } \right) \right) } \\ & { = \mathrm { c } _ { \alpha } ( B - 1 ) ^ { \alpha } \mathrm { P } \left( \frac { B - 3 } { 2 } , \alpha + 1 \right) ^ { - 1 } 2 ^ { - \alpha - 1 } \mu ^ { * - 1 } \left( \mathrm { J } _ { \Phi } ( 1 ) - \mathrm { J } _ { \Psi } \left( \frac { - 1 } { B - 1 } \right) \right) } \\ & { = \mathrm { c } _ { \alpha } ( B - 1 ) ^ { \alpha } \mathrm { P } \left( \frac { B - 3 } { 2 } , \alpha + 1 \right) ^ { - 1 } 2 ^ { - \alpha - 1 } \mathrm { K } _ { \alpha , B } ^ { - 1 } \alpha ^ { 2 } } \end{array} +$$ + +# Computing C. By Lemma E.2, + +$$ +\begin{array} { l } { { \displaystyle { \sum _ { i } } \{ \Lambda \} = \mathbb { E } \left[ r ^ { - 2 ( \alpha + 1 ) } \left( D \Lambda g ^ { \tau } D + D y g ^ { \tau } \Lambda D \right) : y \sim \mathcal { N } ( 0 , \mu ^ { * } G ) \right] } } \\ { { \displaystyle \quad = \Gamma ( \alpha + 1 ) ^ { - 1 } \int \mathrm { d } s ~ s ^ { \alpha } \mathrm { d e t } ( I + 2 s \mu ^ { * } G ) ^ { - 1 / 2 } \mathbb { E } [ D \Lambda y g ^ { \tau } D + D y g ^ { \tau } \Lambda D : y \sim \mathcal { N } ( 0 , \frac { \mu ^ { * } } { 1 + 2 s \mu ^ { * } } \zeta ^ { * } } } \\ { { \displaystyle \quad \quad = \Gamma ( \alpha + 1 ) ^ { - 1 } \int \mathrm { d } s ~ s ^ { \alpha } ( 1 + 2 s \mu ^ { * } ) ^ { - ( B - 1 ) / 2 } \left( \frac { \mu ^ { * } } { 1 + 2 s \mu ^ { * } } \right) ^ { \alpha } \mathbb { E } [ D \Lambda g y ^ { \tau } D + D y g ^ { \tau } \Lambda D : y > \mathcal { N } } } \\ { { \displaystyle \quad = \Gamma ( \alpha + 1 ) ^ { - 1 } \int \mathrm { d } s ~ ( \mu ^ { * } s ) ^ { \alpha } ( 1 + 2 s \mu ^ { * } ) ^ { - ( B - 1 ) / 2 - \alpha } \mathbb { E } [ D \Lambda y g ^ { \tau } D + D y g ^ { \tau } \Lambda D : y \sim \mathcal { N } ( 0 , G ) ] } } \\ { { \displaystyle \quad = \Gamma ( \alpha + 1 ) ^ { - 1 } \mathrm { B e t a } \left( \frac { B - 3 } { 2 } , \alpha + 1 \right) 2 ^ { - \alpha - 1 } \mu ^ { * - 1 } \mathbb { E } [ D \Lambda y g ^ { \tau } D + D y g ^ { \tau } \Lambda D : y \sim \mathcal { N } ( 0 , G ) ] } } \\ { { \displaystyle \quad = \mathbb { P } \left( \frac { B - 3 } { 2 } , \alpha + 1 \right) ^ { - 1 } 2 ^ { - \alpha - 1 } \mu ^ { * - 1 } \mathbb { E } [ D \Lambda g y ^ { T } D + D y g ^ { \tau } \Lambda D : y \sim \mathcal { N } ( 0 , G ) ] } } \\ { { \displaystyle \quad = \mathbb { P } \left( \frac { B - 3 } { 2 } , \alpha + 1 \right) ^ { - 1 } 2 ^ { - \alpha - 1 } \mu ^ { * - 1 } \mathbb { E } [ D \Lambda g ^ { \tau } D + D y g ^ { \tau } \Lambda D : y \sim \mathcal { N } ( 0 , G ) ] } } \end \end{array} +$$ + +Lemma G.10. Suppose $\phi$ is degree $\alpha$ positive-homogeneous. Then for $\Lambda \in \mathcal { H } _ { B }$ , + +$$ +\begin{array} { l } { { \displaystyle \mathbb { E } [ D \otimes ( D y y ^ { T } ) + D y y ^ { T } \otimes D : y \sim \mathcal { N } ( 0 , \Sigma ) ] = \alpha \left. \frac { \mathrm { d } { \mathrm { V } _ { \phi } } ( \Pi \Sigma \Pi ) } { \mathrm { d } \Pi } \right. _ { \Pi = I } } } \\ { { \displaystyle \mathbb { E } [ D \Lambda y y ^ { T } D + D y y ^ { T } \Lambda D : y \sim \mathcal { N } ( 0 , \Sigma ) ] = \alpha \frac { \mathrm { d } { \mathrm { V } _ { \phi } } ( \Sigma ) } { \mathrm { d } \Sigma } \{ \Sigma \Lambda + \Lambda \Sigma \} } } \end{array} +$$ + +where $D = \operatorname { D i a g } ( \phi ^ { \prime } ( y ) )$ . + +Proof. Let $\Pi _ { t } , t \in ( - \epsilon , \epsilon )$ be a smooth path in $\mathcal { H } _ { B }$ with $\Pi _ { 0 } = I$ . Write $D _ { t } = \mathrm { D i a g } ( \phi ^ { \prime } ( \Pi _ { t } y ) )$ , so that $D _ { 0 } = D$ . Then, using $\dot { \Pi } _ { t }$ to denote $t$ derivative, + +$$ +\begin{array} { l } { \displaystyle \frac { d } { d t } \mathrm { V } _ { \phi } ( \Pi _ { t } \Sigma \Pi _ { t } ) = \frac { d } { d t } \mathbb { E } [ \phi ( \Pi _ { t } y ) \phi ( \Pi _ { t } y ) ^ { T } : y \sim \mathcal { N } ( 0 , \Sigma ) ] } \\ { \displaystyle \qquad = \mathbb { E } [ D _ { t } \dot { \Pi } _ { t } y \phi ( \Pi _ { t } y ) ^ { T } + \phi ( \Pi _ { t } y ) y ^ { T } \dot { \Pi } _ { t } D _ { t } : y \sim \mathcal { N } ( 0 , \Sigma ) ] } \\ { \displaystyle \frac { d } { d t } \mathrm { V } _ { \phi } ( \Pi _ { t } \Sigma \Pi _ { t } ) \Bigg | _ { t = 0 } = \mathbb { E } [ D \dot { \Pi } _ { 0 } y \phi ( y ) ^ { T } + \phi ( y ) y ^ { T } \dot { \Pi } _ { 0 } D : y \sim \mathcal { N } ( 0 , \Sigma ) ] } \end{array} +$$ + +Because for $x \in \mathbb { R }$ , $\alpha \phi ( x ) = x \phi ^ { \prime } ( x )$ , for $\boldsymbol { y } \in \mathbb { R } ^ { B }$ we can write $\phi ( y ) = \alpha ^ { - 1 } \mathrm { D i a g } ( \phi ^ { \prime } ( y ) ) y ,$ . Then + +$$ +\begin{array} { l } { \displaystyle \alpha \frac { \mathrm { d } \mathrm { V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } \{ \Sigma \dot { \Pi } _ { 0 } + \dot { \Pi } _ { 0 } \Sigma \} = \alpha \frac { \mathrm { d } \mathrm { V } _ { \phi } ( \Sigma ) } { \mathrm { d } \Sigma } \left\{ \frac { d \Pi _ { t } \Sigma \Pi _ { t } } { d t } \right\} } \\ { \displaystyle \qquad = \alpha \left. \frac { d } { d t } \mathrm { V } _ { \phi } ( \Pi _ { t } \Sigma \Pi _ { t } ) \right. _ { t = 0 } } \\ { \displaystyle \qquad = \mathbb { E } [ D \dot { \Pi } _ { 0 } y y ^ { T } D + D y y ^ { T } \dot { \Pi } _ { 0 } D : y \sim \mathcal { N } ( 0 , \Sigma ) ] } \end{array} +$$ + +Therefore, for $\Lambda \in \mathcal { H } _ { B } ^ { G }$ , + +$$ +\begin{array} { l } { { \displaystyle { \mathsf { C } } \{ \Lambda \} = { \mathsf { P } } \left( \frac { B - 3 } { 2 } , \alpha + 1 \right) ^ { - 1 } 2 ^ { - \alpha - 1 } \mu ^ { * - 1 } \alpha \left. \frac { \mathrm { d } { \mathsf { V } } _ { \phi } ( { \Sigma } ) } { \mathrm { d } \Sigma } \right. _ { { \Sigma } = G } \left\{ G \Lambda + \Lambda G \right\} } } \\ { { \displaystyle ~ = 2 \alpha { \mathsf { P } } \left( \frac { B - 3 } { 2 } , \alpha + 1 \right) ^ { - 1 } 2 ^ { - \alpha - 1 } \mu ^ { * - 1 } \left. \frac { \mathrm { d } { \mathsf { V } } _ { \phi } ( { \Sigma } ) } { \mathrm { d } \Sigma } \right. _ { { \Sigma } = G } \left\{ \Lambda \right\} } } \end{array} +$$ + +# So Thm F.31 gives + +Theorem G.11. $B ^ { \alpha } G ^ { \otimes 2 } \circ \mathsf C$ has the following eigendecomposition + +1. eigenspace $\mathbb { R } G$ with eigenvalue + +$$ +\begin{array} { l } { { \lambda _ { G } ^ { \mathrm { \tiny { C } } } : = 2 B ^ { \alpha } \alpha \mathrm { P } \left( \displaystyle \frac { B - 3 } { 2 } , \alpha + 1 \right) ^ { - 1 } 2 ^ { - \alpha - 1 } \mu ^ { * - 1 } \lambda _ { G } ^ { G , \phi } ( B , \alpha ) } } \\ { { \mathrm { \tiny { ~ \alpha ~ } } = \displaystyle \frac { 2 \alpha ^ { 2 } } { B - 3 } \qquad } } \end{array} +$$ + +2. eigenspace L with eigenvalue + +$$ +\begin{array} { r l } & { \lambda _ { \perp } ^ { \mathsf { C } } : = 2 B ^ { \alpha } \alpha \mathrm { P } \left( \frac { B - 3 } { 2 } , \alpha + 1 \right) ^ { - 1 } 2 ^ { - \alpha - 1 } \mu ^ { * - 1 } \lambda _ { \perp } ^ { G , \phi } ( B , \alpha ) } \\ & { \quad = \frac { 2 \alpha ^ { 2 } ( B - 2 ) } { ( B - 3 ) ( B - 1 ) } + \frac { 2 \alpha B \mathrm { J } _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) } { ( B - 3 ) ( B - 1 ) ^ { 2 } \left( \mathrm { J } _ { \phi } ( 1 ) - \mathrm { J } _ { \phi } \left( \frac { - 1 } { B - 1 } \right) \right) } } \end{array} +$$ + +3. eigenspace M with eigenvalue + +$$ +\begin{array} { l } { { \lambda _ { \mathbb { M } } ^ { \mathbb { C } } : = 2 B ^ { \alpha } \alpha \mathrm { P } \left( \frac { B - 3 } { 2 } , \alpha + 1 \right) ^ { - 1 } 2 ^ { - \alpha - 1 } \mu ^ { * - 1 } \lambda _ { \mathbb { M } } ^ { G , \phi } ( B , \alpha ) } } \\ { { \mathrm { ~ } = \frac { 2 \alpha B J _ { \phi } ^ { \prime } \left( \frac { - 1 } { B - 1 } \right) } { ( B - 3 ) ( B - 1 ) ( \mathrm { J } _ { \phi } ( 1 ) - \mathrm { J } _ { \phi } \left( \frac { - 1 } { B - 1 } \right) } } } \end{array} +$$ + +Altogether, by Eq. (53) and Thms G.6, G.9 and G.11, this implies + +Theorem G.12. $G ^ { \otimes 2 } \circ \mathcal { F } ( \Sigma ^ { * } ) : \mathcal { H } _ { B } ^ { G } \to \mathcal { H } _ { B } ^ { G }$ has eigenspaces $\mathbb { R } G , \mathbb { M } , \mathbb { L }$ respectively with the following eigenvalues + +$$ +\begin{array} { r l } { \lambda _ { 1 } ^ { \mu } = } & { \frac { \lambda ^ { \mu } } { 2 } ( \mu - 3 ) + 2 \lambda _ { 1 } ^ { \mu } ( \frac { \lambda _ { 2 } } { \mu } ( \frac { \lambda _ { 1 } ^ { \mu } } { \mu } ) + \frac { \lambda _ { 2 } ^ { \mu } } { \mu } ( \frac { \lambda _ { 2 } ^ { \mu } } { \mu } ) + \frac { \lambda _ { 3 } ^ { \mu } } { 2 } ( \mu - 3 ) \lambda _ { 2 } ^ { \mu } ) , \quad \psi _ { 2 } ^ { \mu } = 2 } \\ & { - \frac { \lambda _ { 3 } ^ { \mu } } { 2 } ( \mu - 3 ) ( \mu - 5 ) \beta _ { 3 } ^ { \mu } - 1 5 \zeta _ { 2 } ^ { \mu } ( \mu - 1 ) ( \frac { \lambda _ { 2 } ^ { \mu } } { \mu } ) ^ { \mu } } \\ & { - c ^ { \mu } ( \mu - 3 ) c ^ { \mu } ( \mu - 3 ) c ^ { \mu } ( \mu - 1 ) c ^ { \mu } ( \frac { \lambda _ { 3 } ^ { \mu } } { \mu } ) ^ { \mu } } \\ & { - \frac { \lambda _ { 3 } ^ { \mu } } { 2 } ( \mu - 3 ) ( \lambda _ { 3 } ^ { \mu } + 3 ) - \frac { \lambda _ { 3 } ^ { \mu } } { 2 } ( \mu - 1 ) c ^ { \mu } ( \frac { \lambda _ { 2 } ^ { \mu } } { \mu } ) ^ { \mu } } \\ & { - c ^ { \mu } ( \frac { \lambda _ { 3 } ^ { \mu } } { \mu } - 3 ) ( \lambda _ { 3 } ^ { \mu } + \frac { \lambda _ { 3 } ^ { \mu } } { 2 } ) ^ { \mu } , \quad \psi _ { 2 } ^ { \mu } = 2 } \\ & - c ^ { \mu } ( \frac { \lambda _ { 3 } ^ { \mu } } { 2 } c ^ { \mu } - 3 ) ( \frac { \lambda _ { 3 } ^ { \mu } } { 2 } \lambda _ { 3 } ^ { \mu } ) ^ { \mu } ( \frac { \lambda _ { 3 } ^ { \mu } } { \mu } ) ^ { \mu } ( \frac { \lambda _ { 2 } ^ { \mu } } { \mu } ) ^ { \mu } ( \frac { \lambda _ { 3 } ^ { \mu } } { \mu } ) ^ { \mu } \psi _ { 2 } ^ { \mu } ( \frac \end{array} +$$ + +# H CROSS BATCH: FORWARD DYNAMICS + +In this section, we study the generalization of Eq. (43) to multiple batches. + +Definition H.1. For linear operators $\mathcal T _ { i } : \mathcal X _ { i } \mathcal y _ { i } , i = 1 , . . . , k$ , we write $\oplus _ { i } \mathcal { T } _ { i } : \oplus _ { i } \mathcal { X } _ { i } \to \oplus _ { i } \mathcal { V } _ { i }$ for the operator $( x _ { 1 } , \ldots , x _ { k } ) \mapsto ( { \mathcal { T } } _ { 1 } ( x _ { 1 } ) , \ldots , { \mathcal { T } } _ { k } ( x _ { k } ) )$ . We also write $\mathcal { T } _ { 1 } ^ { \oplus n } : = \oplus _ { j = 1 } ^ { n } \mathcal { T } _ { 1 }$ for the direct sum of $n$ copies of $\mathcal { T } _ { 1 }$ . + +For $k \geq 2$ , now consider the extended ( $^ { 6 6 } k$ -batch”) dynamics on $\widetilde { \Sigma } \in { S } _ { k B }$ defined by + +$$ +\widetilde { \Sigma } ^ { l } = \mathrm { V } _ { B _ { \phi } ^ { \oplus k } } ( \widetilde { \Sigma } ^ { l - 1 } ) +$$ + +If we restrict the dynamics to just the upper left $B \times B$ submatrix (as well as any of the diagonal $B \times B$ blocks) of $\widetilde { \Sigma } ^ { l }$ , then we recover Eq. (43). + +# H.1 LIMIT POINTS + +In general, like in the case of Eq. (43), it is difficult to prove global convergence behavior. Thus we manually look for fixed points and the local convergence behaviors around them. A very natural extension of the notion of BSB1 matrices is the following + +Definition H.2. We say a matrix $\widetilde { \Sigma } \in { \cal S } _ { k B }$ is CBSB1 (short for “1-Step Cross-Batch Symmetry Breaking”) if $\widetilde { \Sigma }$ in block form $k \times k$ blocks, each of size $B \times B )$ has one common BSB1 block on the diagonal and one common constant block on the off-diagonal, i.e. + +$$ +\begin{array} { r } { \left( \begin{array} { c c c c } { \mathrm { B S B 1 } ( a , b ) } & { c { \bf 1 1 } ^ { T } } & { c { \bf 1 1 } ^ { T } } & { \cdots } \\ { c { \bf 1 1 } ^ { T } } & { \mathrm { B S B 1 } ( a , b ) } & { c { \bf 1 1 } ^ { T } } & { \cdots } \\ { c { \bf 1 1 } ^ { T } } & { c { \bf 1 1 } ^ { T } } & { \mathrm { B S B 1 } ( a , b ) } & { \cdots } \\ { \vdots } & { \vdots } & { \vdots } & { \ddots } \end{array} \right) } \end{array} +$$ + +We will study the fixed point $\widetilde { \Sigma } ^ { * }$ to Eq. (54) of CBSB1 form. + +H.1.1 SPHERICAL INTEGRATION AND GEGENBAUER EXPANSION + +Theorem H.3. Let $\widetilde { \Sigma } \ \in \ S _ { k B }$ . If $\widetilde { \Sigma }$ is CBSB1 then $\nabla _ { \boldsymbol { B } _ { \phi } ^ { \oplus k } } ( \widetilde { \Sigma } )$ is CBSB1 and equals $\widetilde { \Sigma } ^ { * } : =$ $\left( \begin{array} { c c } { { \Sigma ^ { * } } } & { { c ^ { * } } } \\ { { c ^ { * } } } & { { \Sigma ^ { * } } } \end{array} \right)$ where $\Sigma ^ { * }$ is the BSB1 fixed point of Thm F.5 and $c ^ { * } \geq 0$ with $\begin{array} { r l } { \sqrt { c ^ { * } } } & { { } = } \end{array}$ $\begin{array} { r } { \frac { \dot { \Gamma ( ( B - 1 ) / 2 ) } } { \Gamma ( ( B - 2 ) / 2 ) } \pi ^ { - 1 / 2 } \int _ { 0 } ^ { \pi } \mathrm { d } \theta \ \phi ( - \sqrt { B - 1 } \cos \theta ) \sin ^ { B - 3 } \theta } \end{array}$ . If $\phi ( { \sqrt { B - 1 } } x )$ has Gegenbauer expansion $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ , then $c ^ { * } = a _ { 0 } ^ { 2 }$ . + +Proof. We will prove for $k = 2$ . The general $k$ cases follow from the same reasoning. + +Let $\widetilde { \Sigma } = \left( \begin{array} { c c } { { \Sigma } } & { { c \mathbf { 1 } \mathbf { 1 } ^ { T } } } \\ { { c \mathbf { 1 } \mathbf { 1 } ^ { T } } } & { { \Sigma } } \end{array} \right)$ where $\Sigma = \mathrm { B S B } 1 ( a , b )$ . As remarked below Eq. (54), restricting to any diagonal blocks just recovers the dynamics of Eq. (43), which gives the claim about the diagonal blocks being $\Sigma ^ { * }$ through Thm F.5. + +We now look at the off-diagonal blocks. + +where $\begin{array} { r l } { \nabla _ { B _ { \phi } ^ { \oplus 2 } } ( \widetilde { \Sigma } ) _ { 1 : B , B + 1 : 2 B } = \mathbb { E } [ \mathcal { B } _ { \phi } ( z _ { 1 : B } ) \otimes \mathcal { B } _ { \phi } ( z _ { B + 1 : 2 B } ) : z \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ) ] } & { } \\ & { = \mathbb { E } [ \phi \circ \boldsymbol { \mathrm { n } } ( y _ { 1 : B } ) \otimes \phi \circ \boldsymbol { \mathrm { n } } ( y _ { B + 1 : 2 B } ) : y \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { G } ) ] } \\ { \widetilde { \Sigma } ^ { G } } & : = \begin{array} { r l } { G ^ { \oplus 2 } \widetilde { \Sigma } G ^ { \oplus 2 } } & { = \begin{array} { r l } { \binom { G } { 0 } \widetilde { \Sigma } \left( \begin{array} { c c } { G } & { 0 } \\ { 0 } & { G } \end{array} \right) \widetilde { \Sigma } \left( \begin{array} { c c } { G } & { 0 } \\ { 0 } & { G } \end{array} \right) } & { = \begin{array} { r l } { \binom { G \Sigma G } { c 4 1 ^ { T } G } } & { G \Sigma G \pmb { \mathrm { \Delta } } } \end{array} } \end{array} } \end{array} \end{array}$ $\left( \begin{array} { c c } { { ( a - b ) G } } & { { 0 } } \\ { { 0 } } & { { ( a - b ) G } } \end{array} \right)$ (the last step follows from Lemma E.52). Thus $y _ { 1 : B }$ is independent from $y _ { B + 1 : 2 B }$ , and + +$$ +\begin{array} { r } { \mathsf { V } _ { \mathcal { B } _ { \phi } ^ { \oplus 2 } } \bigl ( \widetilde { \Sigma } \bigr ) _ { 1 : B , B + 1 : 2 B } = \mathbb { E } [ \phi \circ \mathsf { n } ( x ) : x \sim \mathcal { N } ( 0 , ( a - b ) G ) ] ^ { \otimes 2 } } \end{array} +$$ + +By symmetry, + +$$ +\begin{array} { r } { \mathbb { E } [ \phi \circ \mathsf { n } ( x ) : x \sim \mathcal { N } ( 0 , ( a - b ) G ) ] = \sqrt { c ^ { * } } \mathbf { 1 } } \\ { \mathbb { E } [ \phi \circ \mathsf { n } ( x ) : x \sim \mathcal { N } ( 0 , ( a - b ) G ) ] ^ { \otimes 2 } = c ^ { * } \mathbf { 1 } \mathbf { 1 } ^ { T } } \end{array} +$$ + +where $\sqrt { c ^ { * } } : = \mathbb { E } [ \phi ( \mathfrak { n } ( x ) _ { 1 } ) : x \sim \mathcal { N } ( 0 , ( a - b ) G ) ]$ . We can compute + +$$ +\begin{array} { r l r } & { } & { = \mathbb { E } [ \phi ( { \mathsf { n } } ( \mathfrak { e x } ) _ { 1 } ) : x \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] \qquad \mathrm { w h e r e ~ e i s ~ a s ~ i n ~ E q . ~ } ( 3 3 ) } \\ & { } & { = \mathbb { E } [ \phi ( ( \mathsf { e n } ( x ) ) _ { 1 } ) : x \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] } \\ & { } & { = \mathbb { E } [ \phi ( \sqrt { B } ( \mathfrak { e v } ) _ { 1 } ) : x \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] \qquad \mathrm { b e c a u s e ~ \bar { e } ~ i s ~ a n ~ i s o m e t r y } } \\ & { } & \\ & { } & { = \mathbb { E } [ \phi ( \sqrt { B } ( \mathfrak { e v } ) _ { 1 } ) : x \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] \qquad \mathrm { w i t h ~ } v = x / \| x \| } \\ & { } & { = \frac { \Gamma ( ( B - 1 ) / 2 ) } { \Gamma ( ( B - 2 ) / 2 ) } \pi ^ { - 1 / 2 } \int _ { 0 } ^ { \pi } \mathrm { d } \theta \phi ( - \sqrt { B - 1 } \cos \theta ) \sin ^ { B - 3 } \theta } \end{array} +$$ + +by Lemma E.37. At the same time, by Proposition E.33, we can obtain the Gegenbauer expansion + +$$ +\begin{array} { r l } & { \sqrt { c ^ { * } } = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B - 1 } \hat { \mathbf { e } } _ { 1 , \tau } ) } \\ & { \quad = \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \displaystyle \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( \mathbf { e } _ { 1 , \tau } v ) } \\ & { \quad = \left. \displaystyle \sum _ { l = 0 } ^ { \infty } a _ { l } Z _ { \hat { \mathbf { e } } _ { 1 , z } } ^ { B - 2 , ( l ) } , 1 \right. } \\ & { \quad = a _ { 0 } Z _ { \hat { \mathbf { e } } _ { 1 , z } } ^ { B - 2 , ( 0 ) } ( \hat { \mathbf { e } } _ { 1 , z } ) } \\ & { \quad = a _ { 0 } . } \end{array} +$$ + +Corollary H.4. With the notation as in Thm $\cdot$ , if $\phi$ is positive-homogeneous of degree $\alpha$ and $\phi ( x ) = a \rho _ { \alpha } ( x ) - b \rho _ { \alpha } ( - x )$ , then $\begin{array} { r } { \sqrt { c ^ { * } } = ( a - b ) \frac { 1 } { 2 \sqrt { \pi } } ( B - 1 ) ^ { \alpha / 2 } \frac { \Gamma ( ( B - 1 ) / 2 ) \Gamma ( ( \alpha + 1 ) / 2 ) } { \Gamma ( ( \alpha + B - 1 ) / 2 ) } } \end{array}$ . + +Proof. We compute + +$$ +\int _ { 0 } ^ { \pi / 2 } \mathrm { d } \theta ( \cos \theta ) ^ { \alpha } \sin ^ { B - 3 } \theta = \frac { 1 } { 2 } \mathrm { B e t a } \left( 0 , 1 ; \frac { \alpha + 1 } { 2 } , \frac { B - 2 } { 2 } \right) +$$ + +$$ +\begin{array} { r l r } & { } & { = \displaystyle \frac { 1 } { 2 } \mathrm { B e t a } \left( \frac { \alpha + 1 } { 2 } , \frac { B - 2 } { 2 } \right) } \\ & { } & { \displaystyle \int _ { \pi / 2 } ^ { \pi } \mathrm { d } \theta \left( - \cos \theta \right) ^ { \alpha } \sin ^ { B - 3 } \theta = \int _ { 0 } ^ { \pi / 2 } \mathrm { d } \theta \left( \cos \theta \right) ^ { \alpha } \sin ^ { B - 3 } \theta } \\ & { } & { = \displaystyle \frac { 1 } { 2 } \mathrm { B e t a } \left( \frac { \alpha + 1 } { 2 } , \frac { B - 2 } { 2 } \right) } \end{array} +$$ + +So for a positive homogeneous function $\phi ( x ) = a \rho _ { \alpha } ( x ) - b \rho _ { \alpha } ( - x )$ , + +$$ +{ \begin{array} { r l } & { { \sqrt { c ^ { * } } } = { \frac { \Gamma ( ( B - 1 ) / 2 ) } { \Gamma \left( ( B - 2 ) / 2 \right) } } \pi ^ { - 1 / 2 } \left( a \int _ { \pi / 2 } ^ { \pi } \mathrm { d } \theta \left( - { \sqrt { B - 1 } } \cos \theta \right) ^ { \alpha } \sin ^ { B - 3 } \theta \right. } \\ & { \qquad \left. - b \int _ { 0 } ^ { \pi / 2 } \mathrm { d } \theta \left( { \sqrt { B - 1 } } \cos \theta \right) ^ { \alpha } \sin ^ { B - 3 } \theta \right) } \\ & { = ( a - b ) { \frac { \Gamma ( ( B - 1 ) / 2 ) } { \Gamma ( ( B - 2 ) / 2 ) } } \pi ^ { - 1 / 2 } ( B - 1 ) ^ { \alpha / 2 } { \frac { 1 } { 2 } } \operatorname { B e t a } \left( { \frac { \alpha + 1 } { 2 } } , { \frac { B - 2 } { 2 } } \right) } \\ & { = ( a - b ) { \frac { 1 } { 2 { \sqrt { \pi } } } } ( B - 1 ) ^ { \alpha / 2 } { \frac { \Gamma ( ( B - 1 ) / 2 ) \Gamma ( ( \alpha + 1 ) / 2 ) } { \Gamma ( ( \alpha + B - 1 ) / 2 ) } } } \end{array} } +$$ + +Expanding the beta function and combining with Thm H.3 gives the desired result. + +# H.1.2 LAPLACE METHOD + +While we don’t need to use the Laplace method to compute $c ^ { * }$ for positive homogeneous functions (since it is already given by Corollary H.4), going through the computation is instructive for the machinery for computing the eigenvalues in a later section. + +Lemma H.5 (The Laplace Method Cross Batch Master Equation). For $A , B , C \ \in \ \mathbb { N } ,$ , let $f :$ $\mathbb { R } ^ { A + B } \mathbb { R } ^ { C }$ and let $a , b \geq 0$ . Suppose for any $y \in \mathbb { R } ^ { A } , z \in \mathbb { R } ^ { B }$ , $\| f ( y , z ) \| \leq h ( \sqrt { \| y \| ^ { 2 } + \| z \| ^ { 2 } } )$ for some nondecreasing function $h : \mathbb { R } ^ { \geq 0 } \mathbb { R } ^ { \geq 0 }$ such that $\mathbb { E } [ h ( r \| z \| ) : z \in \mathcal { N } ( 0 , I _ { A + B } ) ]$ exists for every $r \geq 0$ . Define $\wp ( \Sigma ) : = \mathbb { E } [ \| y \| ^ { - 2 a } \| z \| ^ { - 2 b } f ( y , z ) : ( y , z ) \sim \mathcal { N } ( 0 , \Sigma ) ]$ ]. Then on $\{ \Sigma \in { \mathcal { S } } _ { A + B } : \operatorname { r a n k } \Sigma > 2 ( a + b ) \}$ , $\wp ( \Sigma )$ is well-defined and continuous, and furthermore satisfies + +$$ +\wp ( \Sigma ) = \Gamma ( a ) ^ { - 1 } \Gamma ( b ) ^ { - 1 } \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } \mathrm { d } t s ^ { a - 1 } t ^ { b - 1 } \operatorname * { d e t } ( I _ { A + B } + 2 \Omega ) ^ { - 1 / 2 } \underset { ( y , z ) \sim \mathcal { N } ( 0 , \Pi ) } { \mathbb { E } } f ( y , z ) +$$ + +Proof. If $\Sigma$ is full rank, we can show Fubini-Tonelli theorem is valid in the following computation by the same arguments of the proof of Lemma E.2. + +$$ +\begin{array} { r l } & { \quad \mathbb { E } [ \| y \| ^ { - 2 a } \| z \| ^ { - 2 b } f ( y , z ) : ( y , z ) \sim \mathcal { N } ( 0 , \Sigma ) ] } \\ & { = \mathbb { E } [ f ( y , z ) \int _ { 0 } ^ { \infty } \mathrm { d } s \Gamma ( a ) ^ { - 1 } s ^ { a - 1 } e ^ { - \| y \| ^ { 2 } s } \int _ { 0 } ^ { \infty } \mathrm { d } t \Gamma ( b ) ^ { - 1 } t ^ { b - 1 } e ^ { - \| y \| ^ { 2 } t } : ( y , z ) \sim \mathcal { N } ( 0 , \Sigma ) ] } \\ & { = ( 2 \pi ) ^ { - \frac { A + B } { 2 } } \Gamma ( a ) ^ { - 1 } \Gamma ( b ) ^ { - 1 } \operatorname* { d e t } \Sigma ^ { - 1 / 2 } } \\ & { \qquad \times \displaystyle \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } \mathrm { d } t s ^ { a - 1 } t ^ { b - 1 } \int _ { \mathbb { R } ^ { A + B } } \mathrm { d } y \mathrm { ~ d } z f ( y , z ) e ^ { - \frac { 1 } { 2 } ( y , z ) ( \Sigma ^ { - 1 } + 2 D ^ { 2 } ) ( y , z ) ^ { T } } } \end{array} +$$ + +$$ += \Gamma ( a ) ^ { - 1 } \Gamma ( b ) ^ { - 1 } \int _ { 0 } ^ { \infty } \mathrm { d } s \ \int _ { 0 } ^ { \infty } \mathrm { d } t \ s ^ { a - 1 } t ^ { b - 1 } \operatorname * { d e t } ( \Sigma ( \Sigma ^ { - 1 } + 2 D ^ { 2 } ) ) ^ { - 1 / 2 } \mathbb { E } f ( y , z ) +$$ + +where in the last line, $( y , z ) \sim \mathcal { N } ( 0 , ( \Sigma ^ { - 1 } + 2 D ^ { 2 } ) ^ { - 1 } )$ . We recover the equation in question with the following simplifications. + +$$ +\begin{array} { r l } & { ( \Sigma ^ { - 1 } + 2 D ^ { 2 } ) ^ { - 1 } = \Sigma ( I _ { A + B } + 2 D ^ { 2 } \Sigma ) ^ { - 1 } } \\ & { \qquad = \Sigma ( D ^ { - 1 } + 2 D \Sigma ) ^ { - 1 } D ^ { - 1 } } \\ & { \qquad = \Sigma D ( I _ { A + B } + 2 D \Sigma D ) ^ { - 1 } D ^ { - 1 } } \\ & { \qquad = D ^ { - 1 } \Omega ( I _ { A + B } + 2 \Omega ) ^ { - 1 } D ^ { - 1 } } \\ & { \qquad \mathrm { d e t } ( \Sigma ( \Sigma ^ { - 1 } + 2 D ^ { 2 } ) ) = \mathrm { d e t } ( I _ { A + B } + 2 \Sigma D ^ { 2 } ) } \\ & { \qquad = \mathrm { d e t } ( I _ { A + B } + 2 D \Sigma D ) } \\ & { \qquad = \mathrm { d e t } ( I _ { A + B } + 2 \Omega ) } \end{array} +$$ + +The case of general $\Sigma$ with $\operatorname { r a n k } \Sigma > 2 ( a + b )$ is given by the same continuity arguments as in Lemma E.2. + +Let $\widetilde { \Sigma } \in { \cal S } _ { 2 B } , \widetilde { \Sigma } = \left( \begin{array} { c c } { { \Sigma } } & { { \Xi } } \\ { { \Xi ^ { T } } } & { { \Sigma ^ { \prime } } } \end{array} \right)$ where $\Sigma , \Sigma ^ { \prime } \in { \cal S } _ { B }$ and $\Xi \in \mathbb { R } ^ { B \times B }$ . Consider the off-diagonal block of $\nabla _ { \boldsymbol { B } _ { \phi } ^ { \oplus 2 } } ( \widetilde { \Sigma } )$ . + +$$ +\begin{array} { r l r } { { \mathbb { E } [ \mathcal { B } _ { \phi } ( z ) \otimes \mathcal { B } _ { \phi } ( z ^ { \prime } ) : ( z , z ^ { \prime } ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ) ] } } \\ & { = \mathbb { E } [ \phi ( \mathbf { n } ( y ) ) \otimes \phi ( \mathbf { n } ( y ^ { \prime } ) ) : ( y , y ^ { \prime } ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { G } ) ] } \\ & { = B ^ { \alpha } \mathbb { E } [ \| y \| ^ { - \alpha } \| y ^ { \prime } \| ^ { - \alpha } \phi ( y ) \otimes \phi ( y ^ { \prime } ) : ( y , y ^ { \prime } ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { G } ) ] } \\ & { = B ^ { \alpha } \Gamma ( \alpha / 2 ) ^ { - 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } \mathrm { d } t ( s t ) ^ { \alpha / 2 - 1 } \mathrm { d e t } ( I _ { 2 B } + 2 \Omega ) ^ { - 1 / 2 } \underset { ( y , y ^ { \prime } ) \sim \mathcal { N } ( 0 , \Pi ) } { \mathbb { E } } \phi ( y ) \otimes \phi ( y ^ { \prime } ) } \end{array} +$$ + +$$ += B ^ { \alpha } \Gamma ( \alpha / 2 ) ^ { - 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \ \int _ { 0 } ^ { \infty } \mathrm { d } t \ ( s t ) ^ { \alpha / 2 - 1 } \operatorname * { d e t } ( I _ { 2 B } + 2 \Omega ) ^ { - 1 / 2 } \mathrm { V } _ { \phi } ( \Pi ) _ { 1 : B , B + 1 : 2 B } +$$ + +$\Omega = \left( \begin{array} { c c } { { s \Sigma ^ { G } } } & { { \sqrt { s t } \Xi ^ { G } } } \\ { { \sqrt { s t } ( \Xi ^ { G } ) ^ { T } } } & { { t \Sigma ^ { \prime G } } } \end{array} \right)$ and $\Pi = D ^ { - 1 } \Omega ( I + 2 \Omega ) ^ { - 1 } D ^ { - 1 }$ with $D = \sqrt { s } I _ { B } \oplus \sqrt { t } I _ { B } =$ $\left( \begin{array} { c c } { { \sqrt { s } I _ { B } } } & { { \mathrm { ~ ~ 0 ~ } } } \\ { { 0 } } & { { \sqrt { t } I _ { B } } } \end{array} \right) .$ + +Theorem H.6 (Rephrasing and adding to Corollary H.4). Let $\widetilde { \Sigma } \in { S } _ { k B }$ and $\phi$ be positive homogeneous of degree $\alpha$ . If $\widetilde { \Sigma }$ is CBSB1 then $\nabla _ { \boldsymbol { B } _ { \phi } ^ { \oplus k } } ( \widetilde { \Sigma } )$ is CBSB1 and equals $\widetilde { \Sigma } ^ { * } = \left( \begin{array} { c c } { { \Sigma ^ { * } } } & { { c ^ { * } { \bf 1 1 } ^ { T } } } \\ { { c ^ { * } { \bf 1 1 } ^ { T } } } & { { \Sigma ^ { * } } } \end{array} \right)$ where $\Sigma ^ { * }$ is the BSB1 fixed point of Thm F.5 and + +$$ +\begin{array} { l } { { c ^ { * } = { \tt c } _ { \alpha } \left( \frac { B - 1 } { 2 } \right) ^ { \alpha } \mathrm { P } \left( \frac { B - 1 } { 2 } , \frac { \alpha } { 2 } \right) ^ { - 2 } { \tt J } _ { \phi } ( 0 ) } } \\ { { \mathrm { = } ( a - b ) ^ { 2 } \displaystyle \frac { 1 } { 4 \pi } ( B - 1 ) ^ { \alpha } \left( \displaystyle \frac { \Gamma ( ( B - 1 ) / 2 ) \Gamma ( ( \alpha + 1 ) / 2 ) } { \Gamma ( ( \alpha + B - 1 ) / 2 ) } \right) ^ { 2 } } } \end{array} +$$ + +Proof. Like in the proof of Corollary H.4, we only need to compute the cross-batch block, which is given above by Eq. (56). Recall that Σ = $\widetilde { \Sigma } = \left( \begin{array} { c c } { { \Sigma } } & { { c \mathbf { 1 } \mathbf { 1 } ^ { T } } } \\ { { c \mathbf { 1 } \mathbf { 1 } ^ { T } } } & { { \Sigma } } \end{array} \right)$ where $\Sigma = \mathrm { B S B } 1 ( a , b )$ . Set $\mu : = a - b$ . Note that because the off-diagonal block $\boldsymbol { \Xi } = c \mathbf { 1 1 } ^ { T }$ by assumption, $\Xi ^ { G } = 0$ , so that + +$$ +\begin{array} { c } { { \Omega = s \Sigma ^ { G } \oplus t \Sigma ^ { G } } } \\ { { = s \mu G \oplus t \mu G } } \\ { { \Omega ( I + 2 \Omega ) ^ { - 1 } = \displaystyle \frac { s \mu } { 1 + 2 s \mu } G \oplus \displaystyle \frac { t \mu } { 1 + 2 t \mu } G } } \\ { { \Pi = \displaystyle \frac { \mu } { 1 + 2 s \mu } G \oplus \displaystyle \frac { \mu } { 1 + 2 t \mu } G } } \end{array} +$$ + +Therefore, + +$$ +\begin{array} { c c } { { \operatorname * { d e t } ( I + 2 \Omega ) = ( 1 + 2 s \mu ) ^ { B - 1 } ( 1 + 2 t \mu ) ^ { B - 1 } } } & { { \mathrm { ~ } } } \\ { { \mathrm { V } _ { \phi } ( \Pi ) = \mu ^ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha } \Delta \mathrm { V } _ { \phi } \left( \begin{array} { c c } { { \mathrm { B S B 1 } ( 1 , \frac { - 1 } { B - 1 } ) } } & { { 0 } } \\ { { 0 } } & { { \mathrm { B S B 1 } ( 1 , \frac { - 1 } { B - 1 } ) } } \end{array} \right) \Delta \mathrm { V } _ { \phi } ( \Pi ) V _ { \phi } ( \Pi ) } } & { { \mathrm { ~ a ~ n ~ d ~ } } } \\ { { \mathrm { ~ } } } & { { \mathrm { ~ a ~ n ~ d ~ } \left( \frac { B - 1 } { B - 1 } \right) ^ { \alpha } \Delta \mathrm { V } _ { \phi } \left( \begin{array} { c c } { { \mathrm { B S B 1 } ( 1 , \frac { - 1 } { B - 1 } ) } } & { { 0 } } \\ { { 0 } } & { { \mathrm { B S B 1 } ( 1 , \frac { - 1 } { B - 1 } ) } } \end{array} \right) \Delta \mathrm { V } _ { \phi } ( \Pi ) V _ { \phi } ( \Pi ) } } \end{array} +$$ + +$\Delta = \left( \begin{array} { c c } { { ( 1 + 2 s \mu ) ^ { - \alpha / 2 } I _ { B } } } & { { 0 } } \\ { { 0 } } & { { ( 1 + 2 t \mu ) ^ { - \alpha / 2 } I _ { B } } } \end{array} \right)$ . In particular, the off-diagonal block is constant with entry $\begin{array} { r } { \mathsf { c } _ { \alpha } \mu ^ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha } ( 1 + 2 s \mu ) ^ { - \alpha / 2 } ( 1 + 2 t \mu ) ^ { - \alpha / 2 } \mathsf { J } _ { \phi } ( 0 ) . } \end{array}$ + +Finally, by Eq. (56), + +$$ +\begin{array} { l } { { \displaystyle \mathbb { E } [ B _ { \phi } ( z ) \otimes B _ { \phi } ( z ^ { \prime } ) : ( z , z ^ { \prime } ) \sim \mathcal { N } ( 0 , \tilde { \Sigma } ) ] } } \\ { { \displaystyle = B ^ { \alpha } \Gamma ( \alpha / 2 ) ^ { - 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } \mathrm { d } t ( s t ) ^ { \alpha / 2 - 1 } \mathrm { d e t } ( I _ { 2 B } + 2 \Omega ) ^ { - 1 / 2 } \mathrm { V } _ { \phi } ( \Pi ) _ { 1 : B , B + 1 : 2 B } } } \\ { { \displaystyle = B ^ { \alpha } \Gamma ( \alpha / 2 ) ^ { - 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } \mathrm { d } t ( s t ) ^ { \alpha / 2 - 1 } [ ( 1 + 2 s \mu ) ( 1 + 2 t \mu ) ] ^ { - \frac { B - 1 } { 2 } } } } \\ { { \displaystyle ~ \times \mathrm { c } _ { \alpha } \mu ^ { \alpha } \left( \frac { B - 1 } { B } \right) ^ { \alpha } [ ( 1 + 2 s \mu ) ( 1 + 2 t \mu ) ] ^ { - \alpha / 2 } \mathrm { J } _ { \phi } ( 0 ) \mathbf { 1 } ^ { T } } } \\ { { \displaystyle = \mathrm { c } _ { \alpha } ( B - 1 ) ^ { \alpha } \Gamma ( \alpha / 2 ) ^ { - 2 } \mathrm { J } _ { \phi } ( 0 ) \mathbf { 1 } ^ { 1 T } \int _ { 0 } ^ { \infty } \mathrm { d } \sigma \int _ { 0 } ^ { \infty } \mathrm { d } \sigma \left( \sigma \tau \right) ^ { \alpha / 2 - 1 } ( 1 + 2 \sigma ) ^ { - \frac { B - 1 + \alpha } { 2 } } ( 1 + 2 \tau ) ^ { - \frac { B - 1 + \alpha } { 2 } } } } \end{array} +$$ + +Simplification with Defn E.7 and Proposition E.21 gives the result. + +# H.2 LOCAL CONVERGENCE + +We now look at the linearized dynamics around CBSB1 fixed points of Eq. (54), and investigate the eigen-properties of $\left. \frac { \mathrm { d } \mathrm { V } _ { B _ { \phi } ^ { \oplus 2 } } } { \mathrm { d } \widetilde { \Sigma } } \right| _ { \widetilde { \Sigma } = \widetilde { \Sigma } ^ { * } }$ . By Lemma F.17, its nonzero eigenvalues are exactly those of $\begin{array} { r } { \tilde { \mathcal { F } } : = ( G ^ { \oplus 2 } ) ^ { \otimes 2 } \circ \frac { \mathrm { d } \mathrm { V } _ { ( \phi \circ \mathrm { m } ) \oplus 2 } ( \tilde { \Sigma } ^ { \sharp G } ) } { \mathrm { d } \tilde { \Sigma } ^ { G } } \bigg | _ { \tilde { \Sigma } ^ { G } = \mu ^ { * } G ^ { \oplus 2 } } , \tilde { \mathcal { F } } \{ \Lambda \} = G ^ { \oplus 2 } \left( \left. \frac { \mathrm { d } \mathrm { V } _ { ( \phi \circ \mathrm { m } ) \oplus 2 } } { \mathrm { d } \tilde { \Sigma } ^ { G } } \right| _ { \mu ^ { * } G ^ { \oplus 2 } } \{ \Lambda \} \right) G ^ { \oplus 2 } . } \end{array}$ . Here $\widetilde { \Sigma } ^ { G } = G ^ { \oplus 2 } \widetilde { \Sigma } G ^ { \oplus 2 }$ e. Thus it suffices to obtain the eigendecomposition of $\tilde { \mathcal { F } }$ . + +rst, notice that $\tilde { \mathcal { F } }$ acts on the diagonal b s $G ^ { \otimes 2 } \circ \frac { \mathrm { d V } _ { \phi \circ \mathsf { n } } } { \mathrm { d } \Sigma ^ { G } }$ , and the off-diagonal components of $\Lambda$ ${ \tilde { \mathcal { F } } } \{ \Lambda \}$ + +Definition H.7. Let $\mathcal { T } : \mathcal { H } _ { k B } \mathcal { H } _ { k B }$ be a linear operator such that for any block matrix $\widetilde { \Sigma } \in \mathcal { H } _ { k B }$ with $k \times k$ blocks $\widetilde { \Sigma } ^ { i j } , i , j \in [ k ]$ , of size $B$ , we have + +$$ +\begin{array} { l } { T \{ \widetilde { \Sigma } \} ^ { i i } = { \mathcal { U } } \{ \widetilde { \Sigma } ^ { i i } \} } \\ { T \{ \widetilde { \Sigma } \} ^ { i j } = \mathcal { V } \{ \widetilde { \Sigma } ^ { i i } \} + \mathcal { V } \{ \widetilde { \Sigma } ^ { j j } \} ^ { T } + \mathcal { W } \{ \widetilde { \Sigma } ^ { i j } \} , \forall i \not = j , } \end{array} +$$ + +where $\mathcal { U } : \mathcal { H } _ { B } \mathcal { H } _ { B } , \mathcal { V } : \mathcal { H } _ { B } \mathbb { R } ^ { B \times B }$ , and $\mathcal { W } : \mathbb { R } ^ { B \times B } \mathbb { R } ^ { B \times B }$ are linear operators, and ${ \mathcal { T } } \{ { \widetilde { \Sigma } } \} ^ { i j }$ denotes the $( i , j )$ th block of $\mathcal { T } \{ \widetilde { \Sigma } \}$ . We say that $\tau$ is blockwise diagonal-off-diagonal semidirect, or BDOS for short. We write more specifically $\mathcal { T } = \mathrm { B D O S } ( \mathcal { U } , \mathcal { V } , \mathcal { W } )$ . + +We have $\tilde { \mathcal { F } } = \mathrm { B D O S } ( \mathcal { U } , \mathcal { V } , \mathcal { W } )$ where + +$$ +\begin{array} { l } { { \displaystyle \mathcal { U } = G ^ { \otimes 2 } \circ \left. \frac { \mathrm { d } \nabla _ { \phi \circ \mathbf { n } } } { \mathrm { d } \Sigma ^ { G } } \right. _ { \Sigma ^ { G } = \mu ^ { * } G } } } \\ { { \displaystyle \mathcal { V } = G ^ { \otimes 2 } \circ \left. \frac { \mathrm { d } } { \mathrm { d } ( \widetilde { \Sigma } ^ { G } ) ^ { 1 1 } } \underset { ( h , h ^ { \prime } ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { G } ) } { \mathbb { E } } \phi ( \mathbf { n } ( h ) ) \otimes \phi ( \mathbf { n } ( h ^ { \prime } ) ) \right. _ { \widetilde { \Sigma } ^ { G } = \mu ^ { * } G ^ { \oplus 2 } } } } \\ { { \displaystyle \mathcal { W } = 2 G ^ { \otimes 2 } \circ \left. \frac { \mathrm { d } } { \mathrm { d } ( \widetilde { \Sigma } ^ { G } ) ^ { 1 2 } } \underset { ( h , h ^ { \prime } ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { G } ) } { \mathbb { E } } \phi ( \mathbf { n } ( h ) ) \otimes \phi ( \mathbf { n } ( h ^ { \prime } ) ) \right. _ { \widetilde { \Sigma } ^ { G } = \mu ^ { * } G ^ { \oplus 2 } } . } } \end{array} +$$ + +Note that the factor 2 in $\mathcal { W }$ is due to the contribution of $( \widetilde { \Sigma } ^ { G } ) ^ { 2 1 }$ by symmetry. We will first show that $\mathcal { W }$ is multiplication by a constant. + +H.2.1 SPHERICAL INTEGRATION AND GEGENBAUER EXPANSION + +Theorem H.8. Suppose $\phi ( { \sqrt { B - 1 } } x )$ has Gegenbauer expansion $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ . Then satisfying $G \Lambda G = \Lambda ,$ + +$$ +\begin{array} { l } { { \displaystyle \mathcal { W } \{ \Lambda \} = 2 a _ { 1 } ^ { 2 } { \mu ^ { * } } ^ { \mathrm { - 1 } } \mathrm { P } \left( \frac { B - 1 } { 2 } , \frac 1 2 \right) ^ { 2 } \frac { B } { B - 1 } \Lambda } } \\ { { \displaystyle \quad = \frac { a _ { 1 } ^ { 2 } \frac { 1 } { 2 } \mathrm { P } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { - 2 } B ( B - 1 ) } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } \Lambda } } \end{array} +$$ + +Proof. Let $\begin{array} { r } { J = \left. \frac { \mathsf { d } } { \mathsf { d } ( \widetilde { \Sigma } ^ { G } ) ^ { 1 2 } } \mathbb { E } _ { ( h , h ^ { \prime } ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { G } ) } \phi ( \mathsf { n } ( h ) ) \otimes \phi ( \mathsf { n } ( h ^ { \prime } ) ) \right. _ { \widetilde { \Sigma } ^ { G } = \mu ^ { * } G ^ { \oplus 2 } } . \mathrm { C o m b i n i n g } . } \end{array}$ Lemma F.18 with projection by $\mathbb { P }$ , we get + +$$ +\begin{array} { r l } & { J \{ \Lambda \} = \displaystyle \frac { 1 } { 2 } _ { y , y ^ { \prime } \sim N \mathbb { 0 } , \mu ^ { * } I B = y ) } \phi \left( \mathfrak { n } ( \mathtt { e } y ) \right) \otimes \phi \left( \mathfrak { n } ( \mathtt { e } y ^ { \prime } ) \right) \langle \mu ^ { * - 2 } y y ^ { \prime T } - \mu ^ { * - 1 } I _ { B - 1 } , \mathtt { e } ^ { T } \Lambda \mathtt { e } \rangle } \\ & { \qquad = \displaystyle \frac { 1 } { 2 } \mu ^ { * - 1 } _ { y , y ^ { \prime } \sim N \mathbb { 0 } , I _ { B - 1 } ) } \overline { { \Phi } } \left( \mathfrak { n } ( \mathtt { e } y ) \right) \otimes \phi \left( \mathfrak { n } ( \mathtt { e } y ^ { \prime } ) \right) \left( y ^ { \prime T } \mathfrak { e } ^ { T } \Lambda \mathtt { e } y - \mathrm { t r } \Lambda \right) } \\ & { \qquad = \displaystyle \frac { 1 } { 2 } \mu ^ { * - 1 } \bigg ( \left( \sqrt { 2 } \mathrm { P } \left( \frac { B - 1 } { 2 } , \frac { 1 } { 2 } \right) \right) ^ { 2 } _ { v , v ^ { \prime } \sim S ^ { B - 2 } } \phi \left( \sqrt { B } \mathfrak { e } v \right) \otimes \phi \left( \sqrt { B } \mathfrak { e } v ^ { \prime } \right) \left( v ^ { \prime T } \mathfrak { e } ^ { T } \Lambda \mathtt { e } v \right) } \\ & { \qquad - \left( \mathrm { t r } \Lambda \right) _ { v , v ^ { \prime } \sim S ^ { B - 2 } } \phi \left( \sqrt { B } \mathfrak { e } v \right) \otimes \phi \left( \sqrt { B } \mathfrak { e } v ^ { \prime } \right) \bigg ) } \end{array} +$$ + +# by Proposition E.33. + +Note that Ev,v0∼SB−2 φ √Bev ⊗ φ √Bev0 = Ev∼SB−2 φ √Bev⊗2 is a constant matrix, because of the independence of $v$ from and the spherical symmetry of the integrand. Thus this term drops out in $G ^ { \mathrm { \hat { \otimes } 2 } } \circ J \{ \Lambda \}$ . + +Next, we analyze $H \{ \Lambda \} = \mathbb { E } _ { v , v ^ { \prime } \sim S ^ { B - 2 } } \phi \left( { \sqrt { B } } \Phi v \right) \otimes \phi \left( { \sqrt { B } } \Phi v ^ { \prime } \right) \left( v ^ { \prime T } \Phi ^ { T } \Lambda \mathbb { e } v \right)$ . This is a linear function in $\Lambda$ with coefficients + +$$ +H _ { a b | c d } = \frac { B - 1 } { B } \left( \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B - 1 } \hat { \mathbf { e } } _ { b , : } v ) ( \hat { \mathbf { e } } _ { c , : } v ) \right) \left( \underset { v \sim S ^ { B - 2 } } { \mathbb { E } } \phi ( \sqrt { B - 1 } \hat { \mathbf { e } } _ { a , : } v ) ( \hat { \mathbf { e } } _ { d , : } v ) \right) . +$$ + +By the usual zonal spherical harmonics argument, + +$$ +\begin{array} { r l } & { \quad \underset { s \sim \delta = 2 } { \overset { \mathbb { R } } { \sim } } \phi ( \sqrt { B } - 1 \hat { \Phi } _ { b , \sigma } ) \big ( \hat { e } _ { e , \sigma } , y \big ) } \\ & { = \bigg \langle \overset { \sum } { i = 1 } a _ { 1 } \frac { 1 } { c B - 1 , i } C _ { 1 } ^ { ( \frac { 1 - \alpha - 3 } { 2 } ) } \big ( \langle \hat { e } _ { b , \sigma } , \cdot \rangle \big ) , \langle \hat { e } _ { e , \sigma } , \cdot \rangle \bigg \rangle _ { S ^ { B - 1 } } } \\ & { = \bigg \langle \overset { \sum } { i = 1 } a _ { 1 } Z _ { k , \sigma } ^ { B - 2 , ( b ) } , \frac { 1 } { B - 1 } Z _ { e , \sigma } ^ { B - 2 , ( b ) } \bigg \rangle _ { S ^ { B - 2 } } } \\ & { = \frac { a _ { 1 } } { B - 1 } Z _ { e , \sigma } ^ { B - 2 , ( 1 ) } ( \hat { e } _ { b , \sigma } ) } \\ & { = \frac { a _ { 1 } } { B - 1 } \frac { 1 } { c B - 1 , 1 } C _ { 1 } ^ { ( \frac { 1 - \alpha - 3 } { 2 } ) } \big ( \langle \hat { e } _ { b , \sigma } , \hat { e } _ { e , \cdot } \rangle \big ) } \\ & { = \bigg \{ \frac { a _ { 1 } } { B - 1 } \frac { 1 } { c B - 1 , 1 } \big ( \mathrm { f i } b = c } \\ & { = \bigg \{ \frac { a _ { 1 } } { B - 1 } , \quad \mathrm { f t b } = c } \\ & { \geq \frac { a _ { 1 } } { \delta - 1 } , \quad \mathrm { o t h e r w i s e } . } \end{array} +$$ + +Thus, + +$$ +\begin{array} { l } { { \displaystyle H \{ \Lambda \} _ { a b } = \frac { B - 1 } { B } \Bigg ( \left( \frac { - a _ { 1 } } { B - 1 } \right) ^ { 2 } \sum _ { c , d \in [ B ] } ^ { \mathrm { ~ \sum ~ \sum ~ } _ { h < d } } } } \\ { { \displaystyle ~ + \left( a _ { 1 } - \frac { - a _ { 1 } } { B - 1 } \right) \left( \frac { - a _ { 1 } } { B - 1 } \right) \sum _ { c - b \in \mathbb { S } ^ { d - a } } \Lambda _ { c d } } } \\ { { \displaystyle ~ + \left( a _ { 1 } ^ { 2 } - \frac { a _ { 1 } ^ { 2 } } { ( B - 1 ) ^ { 2 } } \right) \Lambda _ { a b } \Bigg ) } } \\ { { \displaystyle = \frac { B - 1 } { B } \left( - 2 \left( a _ { 1 } - \frac { - a _ { 1 } } { B - 1 } \right) \left( \frac { - a _ { 1 } } { B - 1 } \right) \Lambda _ { a b } + \left( a _ { 1 } ^ { 2 } - \frac { a _ { 1 } ^ { 2 } } { ( B - 1 ) ^ { 2 } } \right) \Lambda _ { a b } \right) } } \\ { { \displaystyle ~ \mathrm { b e c a u s c r o w s ~ a n d ~ c o l u m s ~ o f ~ A s ~ s u m ~ t o ~ } } } \\ { { \displaystyle = a _ { 1 } ^ { 2 } \frac { B } { B - 1 } \Lambda _ { a b } . } } \end{array} +$$ + +Therefore, + +$$ +\begin{array} { r l } & { \mathcal { W } \{ \Lambda \} = 2 G ^ { \otimes 2 } \circ J \{ \Lambda \} } \\ & { \qquad = 2 a _ { 1 } ^ { 2 } \mu ^ { * - 1 } \mathrm { P } \left( \frac { B - 1 } { 2 } , \frac { 1 } { 2 } \right) ^ { 2 } \frac { B } { B - 1 } \Lambda } \\ & { \qquad = \frac { a _ { 1 } ^ { 2 } \frac { 1 } { 2 } \mathrm { P } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { - 2 } B ( B - 1 ) } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } \Lambda } \end{array} +$$ + +where we used the identity $\begin{array} { r } { \mathrm { P } \left( \frac { B - 1 } { 2 } , \frac { 1 } { 2 } \right) \left( \frac { 2 } { B - 1 } \right) = \mathrm { P } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { - 1 } } \end{array}$ . + +Next, we step back a bit to give a full treatment of the eigen-properties of BDOS operators. Then we specialize back to our case and give the result for $\tilde { \mathcal { F } }$ . + +Definition H.9. Let $\mathcal { H } _ { k B } ^ { G } : = \{ \widetilde { \Sigma } \in \mathcal { H } _ { k B } : G ^ { \oplus k } \widetilde { \Sigma } G ^ { \oplus k } = \widetilde { \Sigma } \}$ be the subspace of $\mathcal { H } _ { k B }$ stable under + +Definition H.10. Let $\widetilde { \mathbb { M } } _ { k B } \in \mathcal { H } _ { k B } ^ { G }$ be the space of block matrices $\widetilde { \Sigma } \in \mathcal { H } _ { k B } ^ { G }$ where the diagonal blocks $\widetilde { \Sigma } ^ { i i }$ are all zero. Likewise, let $\overline { { \mathbb { M } } } _ { k B } \in \mathcal { H } _ { k B }$ be the space of block matrices $\widetilde { \Sigma } \in \mathcal { H } _ { k B }$ where the diagonal blocks $\widetilde { \Sigma } ^ { i i }$ are all zero. We suppress the subscript $k B$ when it is clear from the context. + +Definition H.11. Define the block $L$ -shaped matrix + +$$ +\begin{array} { r } { \overline { { \mathrm { L } } } _ { k B } ( \Sigma , \Lambda ) : = \left( \begin{array} { c c c } { \Sigma } & { - \Lambda } & { \cdots } \\ { - \Lambda ^ { T } } & { 0 } & { \cdots } \\ { \vdots } & { \vdots } & { \ddots } \end{array} \right) } \end{array} +$$ + +BDOS operators of the form $\mathrm { B D O S } ( \mathcal { U } , \mathcal { V } , w )$ , where $w$ stands for multiplication by a constant $w$ , like $\tilde { \mathcal { F } }$ , have simple eigendecomposition, just like DOS operators. + +Theorem H.12. If $\gamma - w I _ { B }$ is not singular, then $\operatorname { B D O S } ( \mathcal { V } , \mathcal { U } , w ) : \mathcal { H } _ { k B } \to \mathcal { H } _ { k B }$ has the following eigenspaces and eigenvectors + +1. Me with associated eigenvalue $w$ . + +2. For each eigenvalue $\lambda$ and eigenvector $\Sigma \in \mathcal { H } _ { B }$ of $\mathcal { U }$ , the block matrix + +$$ +\begin{array} { r } { \overline { { \mathbf { L } } } _ { k B } ( \Sigma , - ( \lambda - w ) ^ { - 1 } \mathcal { V } \{ \Sigma \} ) = \left( ( \lambda - w ) ^ { - 1 } \mathcal { V } \{ \Sigma \} ^ { T } \begin{array} { c c c } { ( \lambda - w ) ^ { - 1 } \mathcal { V } \{ \Sigma \} } & { \cdots } \\ { ( \lambda - w ) ^ { - 1 } \mathcal { V } \{ \Sigma \} ^ { T } } & { 0 } & { \cdots } \\ { \vdots } & { \vdots } & { \ddots } \end{array} \right) } \end{array} +$$ + +along with any matrix obtained from permutating its block columns and rows simultaneously. The associated eigenvalue is $\lambda$ . + +$I f \mathrm { B D O S } ( \mathcal { V } , \mathcal { U } , w )$ also restricts to $\mathcal { H } _ { k B } ^ { G } \mathcal { H } _ { k B } ^ { G }$ , then with respect to this signature, the eigendecomposition is obtained by replacing $\mathcal { H } _ { k B }$ with $\mathcal { H } _ { k B } ^ { G }$ in the above: + +1. Block diagonal matrices $\widetilde { \Sigma } \in \mathcal { H } _ { k B } ^ { G }$ where the diagonal blocks $\widetilde { \Sigma } ^ { i i }$ are all zero. The associated eigenvalue is $w$ . + +2. For each eigenvalue $\lambda$ and eigenvector $\begin{array} { r l r l } { \Sigma } & { { } \in } & { { \mathcal { H } } _ { B } ^ { G } } \end{array}$ of $\mathcal { U }$ , the matrix $\overline { { \mathrm { L } } } _ { k B } \big ( \Sigma , - ( \lambda - w ) ^ { - 1 } \mathcal { V } \{ \Sigma \} \big )$ along with any matrix obtained from permutating its block columns and rows simultaneously. The associated eigenvalue is $\lambda$ . + +Proof. The proofs are similar in both cases: One can easily verify that these are eigenvectors and eigenvalues. Then by dimensionality considerations, these must be all of them. □ + +By Thms H.8 and H.12, we easily obtain + +Theorem H.13. Suppose $\phi ( { \sqrt { B - 1 } } x )$ has Gegenbauer expansion $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ . The eigendecomposition of $\tilde { \mathcal { F } } : \mathcal { H } _ { 2 B } ^ { G } \to \mathcal { H } _ { 2 B } ^ { G }$ is given below, as long as $\lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } \neq \lambda _ { \mathbb { L } } ^ { \uparrow } , \lambda _ { \mathbb { M } } ^ { \uparrow }$ . + +1. $\smash { \widetilde { \mathbb { M } } }$ with associated eigenvalue + +$$ +\begin{array} { r l } & { \lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } : = 2 a _ { 1 } ^ { 2 } \mu ^ { * } ^ { - 1 } \mathrm { P } \left( \frac { B - 1 } { 2 } , \frac { 1 } { 2 } \right) ^ { 2 } \frac { B } { B - 1 } } \\ & { \quad \quad = \frac { a _ { 1 } ^ { 2 } \frac { 1 } { 2 } \mathrm { P } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { - 2 } B ( B - 1 ) } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } . } \end{array} +$$ + +2. The space generated by $\overline { { \mathrm { L } } } _ { 2 B } \big ( G , - \big ( \lambda _ { \mathbb { L } } ^ { \uparrow } - \lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } \big ) ^ { - 1 } \mathcal { V } \{ G \} \big )$ along with any matrix obtained efrom permutating its block columns and rows simultaneously. The associated eigenvalue is 0. + +3. The space generated by $\overline { { \mathrm { L } } } _ { 2 B } ( L , - ( \lambda _ { \mathbb { L } } ^ { \uparrow } - \lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } ) ^ { - 1 } \mathcal { V } \{ L \} )$ , where $L = \mathrm { L } _ { B } ( B - 2 , 1 )$ , along ewith any matrix obtained from permutating its block columns and rows simultaneously. The associated eigenvalue is $\lambda _ { \mathbb { L } } ^ { \uparrow }$ . + +4. The space generated by $\overline { { \mathrm { L } } } _ { 2 B } \big ( M , - \big ( \lambda _ { \mathbb { M } } ^ { \uparrow } - \lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } \big ) ^ { - 1 } \mathcal { V } \{ M \} \big )$ , for any $M \in \mathbb { M } ,$ , along with eany matrix obtained from permutating its block columns and rows simultaneously. The associated eigenvalue is $\lambda _ { \mathbb { M } } ^ { \uparrow }$ . + +Here, $\nu$ is as in Eq. (57). The local convergence dynamics of Eq. (54) to the CBSB1 fixed point $\widetilde { \Sigma } ^ { * }$ has the same eigenvalues $\lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } , \lambda _ { \mathbb { L } } ^ { \uparrow } , \lambda _ { \mathbb { M } } ^ { \uparrow }$ , and $\boldsymbol { \theta }$ . + +Note that, to reason about $\tilde { \mathcal { F } }$ extended over $k$ batches, one can replace $2 B$ with $k B$ everywhere in $\cdot$ and the theorem will still hold. As we will see below, $\lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow }$ is the same as the eigenvalue egoverning the dynamics of the cross batch correlations in the gradient propagation. + +# H.2.2 LAPLACE METHOD + +As with the single batch case, with positive homogeneous $\phi$ , we can obtain a more direct answer of the new eigenvalue $\lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow }$ in terms of the J function of $\phi$ . + +Theorem H.14. Let $\phi$ be degree α positive homogeneous. Then the eigenvalue of $\tilde { \mathcal { F } }$ on M is + +$$ +\begin{array} { l } { { \displaystyle \lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } = \frac { B } { 2 } \left( \frac { B - 1 } { 2 } \right) ^ { \alpha - 1 } \mathrm { P } \left( \frac { B } { 2 } , \frac { \alpha } { 2 } \right) ^ { - 2 } \mu ^ { * - 1 } \mathrm { c } _ { \alpha } \mathrm { J } _ { \phi } ^ { \prime } ( 0 ) } } \\ { { \displaystyle \quad = \frac { B } { B - 1 } \frac { \mathrm { P } \left( \frac { B - 1 } { 2 } , \alpha \right) } { \mathrm { P } \left( \frac { B } { 2 } , \frac { \alpha } { 2 } \right) ^ { 2 } } \frac { \mathrm { J } _ { \phi } ^ { \prime } ( 0 ) } { \mathrm { J } _ { \phi } ( 1 ) - \mathrm { J } _ { \phi } \left( \frac { - 1 } { B - 1 } \right) } } } \end{array} +$$ + +Proof. Let $\widetilde { \Sigma } ^ { * } = \left( \begin{array} { c c } { { \Sigma ^ { * } } } & { { c ^ { * } { \bf 1 1 } ^ { T } } } \\ { { c ^ { * } { \bf 1 1 } ^ { T } } } & { { \Sigma ^ { * } } } \end{array} \right)$ be the CBSB1 fixed point in $\mathrm { T h m } \mathrm { H } . 6$ . Take a smooth path $\widetilde { \Sigma } _ { \tau } = \left( \begin{array} { l l } { \Sigma _ { \tau } } & { \Xi _ { \tau } } \\ { \Xi _ { \tau } ^ { T } } & { \Sigma _ { \tau } ^ { \prime } } \end{array} \right) \in \mathcal { S } _ { 2 B } , \tau \in ( - \epsilon , \epsilon )$ such that $\begin{array} { r } { \widetilde { \Sigma } _ { 0 } = \widetilde { \Sigma } ^ { * } , \dot { \widetilde { \Sigma } } _ { 0 } : = \frac { d } { d \tau } \widetilde { \Sigma } = \left( \begin{array} { r l } { 0 } & { \Upsilon } \\ { \Upsilon ^ { T } } & { 0 } \end{array} \right) } \end{array}$ for some Υ ∈ RB×B. Then with Ωτ =  sΣGτ√ st(ΞGτ )T and $\Pi _ { \tau } = D ^ { - 1 } \Omega _ { \tau } ( I + 2 \Omega _ { \tau } ) ^ { - 1 } D ^ { - 1 }$ with $D = \sqrt { s } I _ { B } \oplus \sqrt { t } I _ { B } = \left( \begin{array} { c c } { { \sqrt { s } I _ { B } } } & { { 0 } } \\ { { 0 } } & { { \sqrt { t } I _ { B } } } \end{array} \right)$ + +$$ +\begin{array} { r l } & { \quad \displaystyle \frac { d } { d \tau } \mathbb { E } [ B _ { \phi } ( z ) \otimes B _ { \phi } ( z ^ { \prime } ) : ( z , z ^ { \prime } ) \sim \mathcal { N } ( 0 , \bar { \Sigma } ) ] _ { \tau = 0 } } \\ & { = \displaystyle \frac { d } { d \tau } b ^ { \alpha } \Gamma ( \alpha / 2 ) ^ { - 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } \mathrm { d } t \left( s \right) ^ { \alpha / 2 - 1 } \mathrm { d } \mathrm { e t } ( ( 2 g _ { R } + 2 \Omega ) ^ { - 1 / 2 } \mathbb { V } _ { \phi } ( \mathrm { H } ) _ { 1 : B , R + 1 : 2 R } \bigg | _ { \tau = 0 } } \\ & { = B ^ { \alpha } \Gamma ( \alpha / 2 ) ^ { - 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } \mathrm { d } t \left( s \right) ^ { \alpha / 2 - 1 } \bigg ( \displaystyle \frac { d } { d \tau } \mathrm { d } \mathrm { e t } ( I _ { 2 B } + 2 \Omega ) ^ { - 1 / 2 } \bigg | _ { \tau = 0 } V _ { \phi } ( \mathrm { H } ) _ { 1 : B , R + 1 : 2 B } } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { = \displaystyle \mathrm { d } \mathrm { e t } ( I _ { 2 B } + 2 \Omega ) ^ { - 1 / 2 } \frac { d } { d \tau } \displaystyle \frac { V _ { \phi } ( \mathrm { H } ) _ { 1 : B , B + 1 : 2 B } } { d \tau } \bigg | _ { \tau = 0 } \bigg ) } \\ & { = B ^ { \alpha } \Gamma ( \alpha / 2 ) ^ { - 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } \mathrm { d } t \left( s \right) ^ { \alpha / 2 - 1 } \bigg ( - \mathrm { d } \mathrm { e t } ( I + 2 \Omega _ { 0 } ) ^ { - 1 / 2 } \mathrm { t r } \left( ( I + 2 \Omega _ { 0 } ) ^ { - 1 } \hat { \Omega } _ { 0 } \right) \mathbb { V } _ { \phi } ( \mathrm { H } ) } \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ \end{array} +$$ + +by chain rule and Lemma F.26, where $\begin{array} { r } { \mathcal { I } = \frac { \mathrm { d } \mathrm { V } _ { \phi } } { \mathrm { d } \Sigma } \bigg | _ { \Sigma = \Omega _ { 0 } ( I + 2 \Omega _ { 0 } ) ^ { - 1 } } } \end{array}$ . We compute + +$$ +\begin{array} { c } { { \Omega _ { 0 } = \mu ^ { * } ( s G \circledast t G ) } } \\ { { \dot { \Omega } _ { 0 } = \sqrt { s t } \left( \begin{array} { c c } { { 0 } } & { { \Upsilon ^ { G } } } \\ { { ( \Upsilon ^ { G } ) ^ { T } } } & { { 0 } } \end{array} \right) } } \\ { { \dot { ( I + 2 \Omega _ { 0 } ) ^ { - 1 } = ( ( 1 + 2 s \mu ^ { * } ) ^ { - 1 } G + \frac { 1 } { B } { \bf 1 1 } ^ { T } ) \circledast ( ( 1 + 2 t \mu ^ { * } ) ^ { - 1 } G + \frac { 1 } { B } { \bf 1 1 } ^ { T } ) } } } \\ { { \operatorname* { d e t } ( I + 2 \Omega _ { 0 } ) = ( 1 + 2 s \mu ^ { * } ) ^ { B - 1 } ( 1 + 2 t \mu ^ { * } ) ^ { B - 1 } } } \\ { { \Omega _ { 0 } ( I + 2 \Omega _ { 0 } ) ^ { - 1 } = \frac { s \mu ^ { * } } { 1 + 2 s \mu ^ { * } } G \circledast \frac { t \mu ^ { * } } { 1 + 2 t \mu ^ { * } } G . } } \end{array} +$$ + +So then + +$$ +7 = \left( \left( \frac { s \mu ^ { * } } { 1 + 2 s \mu ^ { * } } \right) ^ { ( \alpha - 1 ) / 2 } I _ { B } \oplus \left( \frac { t \mu ^ { * } } { 1 + 2 t \mu ^ { * } } \right) ^ { ( \alpha - 1 ) / 2 } I _ { B } \right) ^ { \otimes 2 } \circ \left( \frac { B - 1 } { B } \right) ^ { \alpha - 1 } \frac { \mathrm { d } \mathrm { V } _ { \phi } } { \mathrm { d } \Sigma } \bigg | _ { \Sigma = \mathrm { B S B } 1 ( 1 , \frac { - 1 } { B - 1 } ) } , +$$ + +With + +$$ +\begin{array} { c c } { { \displaystyle \left. \frac { d } { d \tau } \Omega ( I + 2 \Omega ) ^ { - 1 } \right| _ { \tau = 0 } = ( I + 2 \Omega _ { 0 } ) ^ { - 1 } \dot { \Omega } _ { 0 } ( I + 2 \Omega _ { 0 } ) ^ { - 1 } } } & { { } } \\ { { } } & { { ( \mathrm { b y } } } \\ { { } } & { { = \displaystyle \frac { \sqrt { s t } } { ( 1 + 2 s \mu ^ { * } ) ( 1 + 2 t \mu ^ { * } ) } \left( { \begin{array} { c c } { { 0 } } & { { \Upsilon ^ { G } } } \\ { { ( \Upsilon ^ { G } ) ^ { T } } } & { { 0 } } \end{array} } \right) } } \end{array} +$$ + +we have + +$$ +\begin{array} { r l } & { \quad \mathcal { I } \left\{ \displaystyle \frac { d } { d \tau } \Omega ( I + 2 \Omega ) ^ { - 1 } \bigg | _ { \tau = 0 } \right\} } \\ & { = \left( \frac { B - 1 } { B } \right) ^ { \alpha - 1 } \displaystyle \frac { \sqrt { s t } } { ( 1 + 2 s \mu ^ { * } ) ( 1 + 2 t \mu ^ { * } ) } \mathrm { c } _ { \alpha } J _ { \phi } ^ { \prime } ( 0 ) } \\ & { \qquad \times \left( \frac { s \mu ^ { * } } { 1 + 2 s \mu ^ { * } ) } ^ { ( \alpha - 1 ) / 2 } \left( \frac { t \mu ^ { * } } { 1 + 2 t \mu ^ { * } } \right) ^ { ( \alpha - 1 ) / 2 } \left( { ( \Upsilon ^ { G } ) } ^ { T } \right) ^ { G } \right) } \\ & { = \left( \frac { B - 1 } { B } \right) ^ { \alpha - 1 } \displaystyle \frac { \left( s t \right) ^ { \alpha / 2 } \mu ^ { * \alpha - 1 } \mathrm { c } _ { \alpha } J _ { \phi } ^ { \prime } ( 0 ) } { ( 1 + 2 s \mu ^ { * } ) ( \alpha + 1 ) ^ { \prime } 2 ( 1 + 2 t \mu ^ { * } ) ^ { ( \alpha + 1 ) / 2 } \left( ( \Upsilon ^ { G } ) ^ { T } \right. \left. \begin{array} { l l } { \mathrm { T } ^ { G } } \\ { \mathrm { 0 } } \end{array} \right) } . } \end{array} +$$ + +Thus the product $( I + 2 \Omega _ { 0 } ) ^ { - 1 } \dot { \Omega } _ { 0 }$ has zero diagonal blocks so that its trace is 0. Therefore, only the second term in the sum in Eq. (58) above survives, and we have + +$$ +\begin{array} { r l } & { \frac { d } { d \tau } \mathbb { E } [ B _ { + } | \hat { z } | \mathcal { E } \theta _ { + } | \hat { z } \rangle \otimes B _ { + } ( z ) | \hat { z } \rangle \langle z , z | ^ { 2 } \rangle \sim \mathcal { N } ( 0 , \hat { z } ) ] _ { | \omega | } } \\ & = B ^ { \ast } \Gamma ( \alpha / 2 ) ^ { - 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } d \tau \mathrm { d } s ^ { \mathrm { i n } / 2 - 1 } \{ 1 + 2 g \mu ^ { s } \} ^ { - \frac { D - 1 } { 2 } ( 1 + 2 g \mu ^ { s } ) ^ { - \frac { D - 1 } { 2 } } ( 1 + 2 g \mu ^ { s } ) ^ { - \frac { D - 1 } { 2 } } \times } \\ & { \qquad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & = B ( B - 1 ) ^ { \ast - 1 } \Gamma ( \kappa / 2 ) ^ { - 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } d \tau ( s ) ^ { \omega / 2 - 1 } \frac { ( \omega ^ { \omega - 1 } ) ^ { \omega - 1 } ( \omega ^ { \omega - 1 } ) ^ { \omega - 1 } ( \omega ^ { \omega - 1 } ) ^ { \omega } ( 1 + 2 g \mu ^ { s } ) ^ { - \frac { D - 1 } { 2 } } ( 1 + 2 g \mu ^ { s } ) ^ { - \frac { D - 1 } { 2 } } } \\ & = B ( B - 1 ) ^ { \ast - 1 } \Gamma ( \kappa / 2 ) ^ { - 2 } \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } \mathrm { d } s \int _ { 0 } ^ { \infty } d \tau ( s ) ^ { \omega / 2 - 1 } \frac { ( \omega ^ { \omega - 1 } - \omega \hat { \omega } ^ { \omega } ) ( 0 ) } ( 1 + 2 g \mu ^ { s } ) ^ { 1 / 2 } ( 1 - 2 g \mu ^ { s } ) ^ { \omega - 1 } ( 1 + 2 g \mu ^ { s } ) ^ { \omega - 1 } ( \frac \ \end{array} +$$ + +# I CROSS BATCH: BACKWARD DYNAMICS + +For $k \geq 2$ , we wish to study the following generalization of Eq. (51), + +$$ +\tilde { \Pi } ^ { l } = \mathrm { V } _ { { B _ { \phi } ^ { \oplus k ^ { \prime } } } } ( \widetilde { \Sigma } ^ { l } ) ^ { \dagger } \big \{ \tilde { \Pi } ^ { l + 1 } \big \} . +$$ + +As in the single batch case, we approximate it by taking $\widetilde { \Sigma } ^ { l }$ to its CBSB1 limit $\widetilde { \Sigma } ^ { * }$ , so that we analyze + +$$ +\tilde { \Pi } ^ { l } = \mathrm { V } _ { B _ { \phi } ^ { \oplus k ^ { \prime } } } ( \widetilde { \Sigma } ^ { * } ) ^ { \dagger } \{ \tilde { \Pi } ^ { l + 1 } \} +$$ + +Its dynamics is then given by the eigendecomposition of $\nabla _ { B _ { \phi } ^ { \oplus _ { k ^ { \prime } } } } ( \widetilde { \Sigma } ^ { * } )$ , in parallel to the single batch scenario. + +WLOG, we only need to consider the case $k = 2$ . + +$$ +\begin{array} { r l } { \mathrm { V } _ { B _ { \phi } ^ { \Phi ^ { 2 } } } / \widetilde { \Sigma } ^ { * } \big ) = \underset { ( x , y ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { * } ) } { \mathbb { E } } [ ( B _ { \phi } ^ { \prime } ( x ) \oplus B _ { \phi } ^ { \prime } ( y ) ) ^ { \otimes 2 } ] } & { } \\ { \mathrm { V } _ { B _ { \phi } ^ { \Phi ^ { 2 } \prime } } ( \widetilde { \Sigma } ^ { * } ) \left\{ \left( \begin{array} { l l } { \Sigma } & { \Xi } \\ { \Xi ^ { T } } & { \Lambda } \end{array} \right) \right\} = \underset { ( x , y ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { * } ) } { \mathbb { E } } \left[ \left( B _ { \phi } ^ { \prime } ( x ) \Sigma B _ { \phi } ^ { \prime } ( x ) ^ { T } \quad B _ { \phi } ^ { \prime } ( x ) \Xi B _ { \phi } ^ { \prime } ( y ) ^ { T } \right) \right] } & { } \\ { \mathrm { V } _ { B _ { \phi } ^ { \Phi ^ { 2 } \prime } } ( \widetilde { \Sigma } ^ { * } ) \left\{ \left( \begin{array} { l l } { \Sigma } & { \Lambda } \\ { \Xi ^ { T } } & { \Lambda } \end{array} \right) \right\} = \underset { ( x , y ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { * } ) } { \mathbb { E } } \left[ \left( B _ { \phi } ^ { \prime } ( y ) \Xi ^ { T } B _ { \phi } ^ { \prime } ( x ) ^ { T } \quad B _ { \phi } ^ { \prime } ( y ) \Lambda B _ { \phi } ^ { \prime } ( y ) ^ { T } \right) \right] } & { } \end{array} +$$ + +From this one sees that $\mathrm { V } _ { \left[ ( B _ { \phi } ^ { \oplus 2 } ) ^ { \prime } \right] }$ acts independently on each block, and consequently so does its adjoint. The diagonal blocks of Eq. (59) evolves according to Eq. (51) which we studied in Appendix G. In this section we will study the evolution of the off-diagonal blocks, which is given by + +$$ +\begin{array} { r l r } { \Xi ^ { l } = \underset { ( x , y ) \sim \mathcal { N } ( 0 , \tilde { \Sigma } ^ { * } ) } { \mathbb { E } } [ \mathcal { B } _ { \phi } ^ { \prime } ( x ) \otimes \mathcal { B } _ { \phi } ^ { \prime } ( y ) ] ^ { \dagger } \{ \Xi ^ { l + 1 } \} } & { = \underset { ( x , y ) \sim \mathcal { N } ( 0 , \tilde { \Sigma } ^ { * } ) } { \mathbb { E } } [ \mathcal { B } _ { \phi } ^ { \prime } ( y ) ^ { T } \otimes \mathcal { B } _ { \phi } ^ { \prime } ( x ) ^ { T } ] \{ \Xi ^ { l + 1 } \} } \\ & { = \underset { ( x , y ) \sim \mathcal { N } ( 0 , \tilde { \Sigma } ^ { * } ) } { \mathbb { E } } [ \mathcal { B } _ { \phi } ^ { \prime } ( y ) ^ { T } \Xi ^ { l + 1 } \mathcal { B } _ { \phi } ^ { \prime } ( x ) ^ { T } ] } \end{array} +$$ + +Definition I.1. Let $\mathcal { M } _ { B } : = \mathbb { R } ^ { B \times B }$ be the set of all $B \times B$ real matrices, and let $\mathcal { M } _ { B } ^ { G } : = \{ \Xi \in$ $\mathcal { M } _ { B } : G \Xi G = \Xi \}$ . + +Define $\mathcal { T } : = \mathbb { E } _ { ( x , y ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { * } ) } [ \mathcal { B } _ { \phi } ^ { \prime } ( x ) \otimes \mathcal { B } _ { \phi } ^ { \prime } ( y ) ]$ . As in the single batch case, after one step of backprop, $\Xi ^ { L - 1 } = \mathcal { V } ^ { \dagger } \{ \Xi ^ { L } \}$ is in $\mathcal { M } _ { B } ^ { G }$ . Thus it suffices to study the eigendecomposition of $G ^ { \otimes 2 } \circ T ^ { \dagger } \mid { \mathcal { M } } _ { B } ^ { G } :$ $\mathcal { M } _ { B } ^ { G } \mathcal { M } _ { B } ^ { G }$ . Below, we will show that its adjoint $\mathcal { T } \circ G ^ { \otimes 2 }$ acting on $\mathcal { M } _ { B } ^ { G }$ is in fact multiplication by a constant, and thus so is it. + +We first compute + +$$ +\begin{array} { r } { \mathcal { T } = \underset { ( \xi , \eta ) \sim \mathcal { N } ( 0 , G ^ { \oplus 2 } \widetilde { \Sigma } ^ { * } G ^ { \oplus 2 } ) } { \mathbb { E } } \left[ \left. \frac { \mathrm { d } \phi \circ \mathsf { n } ( z ) } { \mathrm { d } z } \right| _ { z = \xi } \otimes \left. \frac { \mathrm { d } \phi \circ \mathsf { n } ( z ) } { \mathrm { d } z } \right| _ { z = \eta } \right] \circ G ^ { \otimes 2 } } \\ { = \underset { ( \xi , \eta ) \sim \mathcal { N } ( 0 , \mu ^ { * } \left( \begin{array} { l l } { G } & { G } \\ { 0 } & { G } \end{array} \right) ) } { \mathbb { E } } \left[ \left. \frac { \mathrm { d } \phi \circ \mathsf { n } ( z ) } { \mathrm { d } z } \right| _ { z = \xi } \otimes \left. \frac { \mathrm { d } \phi \circ \mathsf { n } ( z ) } { \mathrm { d } z } \right| _ { z = \eta } \right] \circ G ^ { \otimes 2 } } \end{array} +$$ + +since $\widetilde { \Sigma } ^ { * }$ is CBSB1 with diagonal blocks $\Sigma ^ { * }$ (Corollary H.4). Then $\xi$ is independent from $\eta$ , so this is just + +$$ +\begin{array} { c } { { { \displaystyle \mathbb { E } } } } \\ { { { \displaystyle ( x , y ) \sim \mathcal { N } ( 0 , \widetilde { \Sigma } ^ { * } ) } } } \\ { { \displaystyle = \sum _ { \xi \sim \mathcal { N } ( 0 , \mu ^ { * } G ) } \mathbb { E } _ { \xi } ^ { \prime } ( y ) ] = \sum _ { \xi \sim \mathcal { N } ( 0 , \mu ^ { * } G ) \atop \xi \sim \mathcal { N } ( 0 , \mu ^ { * } G ) } [ \frac { \mathrm { d } \phi \circ \mathsf { n } ( z ) } { \mathrm { d } z } | _ { z = \xi } ] ^ { \otimes 2 } \circ G ^ { \otimes 2 } \ } } \\ { { { \displaystyle = \sum _ { \xi \sim \mathcal { N } ( 0 , \mu ^ { * } G ) } \mathbb { E } [ \sqrt { B } D r ^ { - 1 } ( I - v v v ^ { T } ) ] ^ { \otimes 2 } \circ G ^ { \otimes 2 } \ } } } \end{array} +$$ + +# (by Proposition G.4) + +where $\begin{array} { r c l r c l } { D } & { = } & { \mathrm { D i a g } ( \phi ^ { \prime } ( \sqrt { B } v ) ) , r } & { = } & { \| \xi \| , v } & { = } & { \xi / \| \xi \| } \end{array}$ . But notice that $T : =$ $\mathbb { E } _ { \xi \sim \mathcal { N } ( 0 , \mu ^ { * } G ) } \left[ D r ^ { - 1 } ( I - v v ^ { T } ) \right]$ is actually BSB1, with diagonal + +$$ +_ { r v \sim \mathcal { N } ( 0 , \mu ^ { * } G ) } r ^ { - 1 } ( 1 - v _ { i } ^ { 2 } ) \phi ^ { \prime } ( \sqrt { B } v _ { i } ) = \underset { r v \sim \mathcal { N } ( 0 , \mu ^ { * } G ) } { \mathbb { E } } r ^ { - 1 } ( 1 - v _ { 1 } ^ { 2 } ) \phi ^ { \prime } ( \sqrt { B } v _ { 1 } ) , \forall i \in [ B ] +$$ + +and off-diagonal + +$$ +\begin{array} { r } { \underset { r v \sim \mathcal { N } ( 0 , \mu ^ { * } G ) } { \mathbb { E } } r ^ { - 1 } ( - v _ { i } v _ { j } ) \phi ^ { \prime } ( \sqrt { B } v _ { i } ) = \underset { r v \sim \mathcal { N } ( 0 , \mu ^ { * } G ) } { \mathbb { E } } r ^ { - 1 } ( - v _ { 1 } v _ { 2 } ) \phi ^ { \prime } ( \sqrt { B } v _ { 1 } ) , \forall i \neq j \in [ B ] . } \end{array} +$$ + +Thus by Lemma E.51, $T$ has two eigenspaces, $\{ x : G x = x \}$ and $\mathbb { R } \mathbf { 1 }$ , with respective eigenvalues $T _ { 1 1 } - T _ { 1 2 }$ and $T _ { 1 1 } + ( B - 1 ) T _ { 1 2 }$ . However, because of the composition with $G ^ { \overleftarrow { \otimes } 2 }$ , only the former eigenspace survives with nonzero eigenvalue in $G ^ { \otimes 2 } \circ \tau$ . + +$$ +\begin{array} { r l } & { \mathrm { T h e o r e m } \qquad \mathbf { I . 2 . } \qquad \tau \qquad \ o n \qquad \mathcal { M } _ { B } ^ { G } \qquad i s \qquad j u s t \qquad m u l t i p l i c a t i o n \qquad b y \qquad \lambda _ { \widetilde { \mathbb { M } } } ^ { \pm } } \\ & { \frac { 1 } { 2 \pi } B \mu ^ { * - 1 } \left( \int _ { 0 } ^ { \pi } \mathrm { d } \theta \ \sin ^ { B - 1 } \theta \phi ^ { \prime } \bigl ( - \sqrt { B - 1 } \cos \theta \bigr ) \right) ^ { 2 } . } \end{array} +$$ + +Proof. By the above reasoning, the sole eigenspace with nonzero eigenvalue is $\{ x : G x = x \} ^ { \otimes 2 } =$ $\mathcal { M } _ { B } ^ { G }$ . It has eigenvalue + +$$ +\begin{array} { r l } & { \quad B \left( \mathbb { E } [ r ^ { - 1 } ( 1 - v _ { 1 } ^ { 2 } + v _ { 1 } v _ { 2 } ) \phi ^ { \prime } ( \sqrt { B } v _ { 1 } ) : r v \sim \mathcal { N } ( 0 , \mu ^ { * } G ) ] \right) ^ { 2 } } \\ & { = B \mu ^ { * - 1 } \left( \mathbb { E } [ r ^ { - 1 } ( 1 - v _ { 1 } ^ { 2 } + v _ { 1 } v _ { 2 } ) \phi ^ { \prime } ( \sqrt { B } v _ { 1 } ) : r v \sim \mathcal { N } ( 0 , G ) ] \right) ^ { 2 } } \\ & { = B \mu ^ { * - 1 } \left( \mathbb { E } [ r ^ { - 1 } ( 1 - ( \exp ) _ { 1 } ^ { 2 } + ( \exp ) _ { 1 } ( \exp ) _ { 2 } ) \phi ^ { \prime } ( \sqrt { B } ( \exp ) _ { 1 } ) : r w \sim \mathcal { N } ( 0 , I _ { B - 1 } ) ] \right) ^ { 2 } } \\ & { = B \mu ^ { * - 1 } \biggl ( \frac { \Gamma ( ( B - 2 ) / 2 ) } { \Gamma ( ( B - 3 ) / 2 ) } 2 ^ { - 1 / 2 } \pi ^ { - 1 } } \\ & { \qquad \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 2 } \left( 1 - \zeta ( \theta _ { 1 } ) ^ { 2 } + \zeta ( \theta _ { 1 } ) \omega ( \theta _ { 1 } , \theta _ { 2 } ) \right) \phi ^ { \prime } ( \sqrt { B } \zeta ( \theta _ { 1 } ) ) \sin ^ { B - 3 } \theta _ { 1 } \sin ^ { B - 4 } \theta _ { 2 } \biggr ) ^ { 2 } } \end{array} +$$ + +where we applied Lemma E.36 with $\begin{array} { r } { \zeta ( \theta ) = - \sqrt { \frac { B - 1 } { B } } \cos \theta } \end{array}$ and $\begin{array} { r } { \omega ( \theta _ { 1 } , \theta _ { 2 } ) = \frac { 1 } { \sqrt { B ( B - 1 ) } } \cos \theta _ { 1 } - } \end{array}$ $\begin{array} { r } { \sqrt { \frac { B - 2 } { B - 1 } } \sin \theta _ { 1 } \cos \theta _ { 2 } . } \end{array}$ + +We can further simplify + +$$ +\begin{array} { r l r } { { \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 2 } ( 1 - \zeta ( \theta _ { 1 } ) ^ { 2 } + \zeta ( \theta _ { 1 } ) \omega ( \theta _ { 1 } , \theta _ { 2 } ) ) \phi ^ { \prime } ( \sqrt { B } \zeta ( \theta _ { 1 } ) ) \sin ^ { B - 3 } \theta _ { 1 } \sin ^ { B - 4 } \theta _ { 2 } } } \\ & { } & { = \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } \sin ^ { B - 3 } \theta _ { 1 } \phi ^ { \prime } ( \sqrt { B } \zeta ( \theta _ { 1 } ) ) \Bigg ( \Bigg ( 1 - \zeta ( \theta _ { 1 } ) ^ { 2 } + \zeta ( \theta _ { 1 } ) \frac { 1 } { \sqrt { B ( B - 1 ) } } \cos \theta _ { 1 } \Bigg ) \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 2 } \sin ^ { B - 4 } \theta _ { 2 } } \\ & { } & { \quad - \sqrt { \frac { B - 2 } { B - 1 } } \sin \theta _ { 1 } \zeta ( \theta _ { 1 } ) \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 2 } \sin ^ { B - 4 } \theta _ { 2 } \cos \theta _ { 2 } \Bigg ) } \\ & { } & { = \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } \sin ^ { B - 3 } \theta _ { 1 } \phi ^ { \prime } ( \sqrt { B } \zeta ( \theta _ { 1 } ) ) \Bigg ( \Bigg ( 1 - \zeta ( \theta _ { 1 } ) ^ { 2 } + \zeta ( \theta _ { 1 } ) \frac { 1 } { \sqrt { B ( B - 1 ) } } \cos \theta _ { 1 } \Bigg ) \mathrm { B e t a } \Bigg ( \frac { B - 3 } { 2 } , \frac { 1 } { 2 } \Bigg ) \Bigg ) . } \end{array} +$$ + +(by Lemma E.29) + +so the cross-batch backward off-diagonal eigenvalue is + +$$ +\begin{array} { c } { { B \mu ^ { * - 1 } \left( \displaystyle \frac { 1 } { \sqrt { 2 \pi } } \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } ~ \mathrm { s i n } ^ { B - 1 } \theta _ { 1 } \phi ^ { \prime } ( - \sqrt { B - 1 } \cos \theta _ { 1 } ) \right) ^ { 2 } } } \\ { { = \displaystyle \frac { 1 } { 2 \pi } B \mu ^ { * - 1 } \left( \int _ { 0 } ^ { \pi } \mathrm { d } \theta _ { 1 } ~ \mathrm { s i n } ^ { B - 1 } \theta _ { 1 } \phi ^ { \prime } ( - \sqrt { B - 1 } \cos \theta _ { 1 } ) \right) ^ { 2 } } } \end{array} +$$ + +For positive homogeneous functions we can evaluate the eigenvalues explicitly. + +Theorem I.3. If $\phi$ is positive-homogeneous of degree $\alpha$ with $\phi ( c ) = a \rho _ { \alpha } ( c ) - b \rho _ { \alpha } ( - c )$ , then $\mathcal { T } : \mathcal { M } _ { B } ^ { G } \to \mathcal { M } _ { B } ^ { \ddot { G } }$ is just multiplication by + +$$ +\lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow } = \frac { 1 } { 8 \pi } B ( B - 1 ) ^ { \alpha - 1 } \mu ^ { * - 1 } \alpha ^ { 2 } ( a + b ) ^ { 2 } \mathrm { B e t a } \left( \frac { \alpha } { 2 } , \frac { B } { 2 } \right) ^ { 2 } . +$$ + +Proof. As in the proof of Corollary H.4, + +$$ +\int _ { 0 } ^ { \pi / 2 } \mathrm { d } \theta ( \cos \theta ) ^ { \alpha - 1 } \sin ^ { B - 1 } \theta = \int _ { \pi / 2 } ^ { \pi } \mathrm { d } \theta ( - \cos \theta ) ^ { \alpha - 1 } \sin ^ { B - 1 } \theta = { \frac { 1 } { 2 } } \operatorname { B e t a } \left( { \frac { \alpha } { 2 } } , { \frac { B } { 2 } } \right) +$$ + +So + +$$ +\begin{array} { l } { { { \frac { 1 } { 2 \pi } } B \mu ^ { * - 1 } \left( \displaystyle \int _ { 0 } ^ { \pi } { \mathrm { d } \theta _ { 1 } \sin ^ { B - 1 } \theta _ { 1 } \phi ^ { \prime } ( - \sqrt { B - 1 } \cos \theta _ { 1 } ) } \right) ^ { 2 } } } \\ { { = { \frac { 1 } { 2 \pi } } B \mu ^ { * - 1 } ( B - 1 ) ^ { \alpha - 1 } \alpha ^ { 2 } \left( a \displaystyle \int _ { \pi / 2 } ^ { \pi } { \mathrm { d } \theta \left( - \cos \theta \right) ^ { \alpha - 1 } \sin ^ { B - 1 } \theta } + b \displaystyle \int _ { 0 } ^ { \pi / 2 } { \mathrm { d } \theta \left( \cos \theta \right) ^ { \alpha - 1 } \sin ^ { B - 1 } \theta } \right) ^ { 2 } } } \\ { { = { \frac { 1 } { 2 \pi } } B \mu ^ { * - 1 } ( B - 1 ) ^ { \alpha - 1 } \alpha ^ { 2 } ( a + b ) ^ { 2 } 2 ^ { - 2 } \mathrm { B e t a } \left( { \frac { \alpha } { 2 } } , { \frac { B } { 2 } } \right) ^ { 2 } } } \\ { { = { \frac { 1 } { 8 \pi } } B ( B - 1 ) ^ { \alpha - 1 } \mu ^ { * - 1 } \alpha ^ { 2 } ( a + b ) ^ { 2 } \mathrm { B e t a } \left( { \frac { \alpha } { 2 } } , { \frac { B } { 2 } } \right) ^ { 2 } } } \end{array} +$$ + +It is also straightforward to obtain an expression of the eigenvalue in terms of Gegenbauer coefficients. + +Theorem I.4. If φ(√B − 1x) has Gegenbauer expansion P∞l=0 al 1cB−1,l C $\begin{array} { r } { \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( x ) } \end{array}$ , then $\tau$ $\mathcal { M } _ { B } ^ { G } \mathcal { M } _ { B } ^ { G }$ is just multiplication by + +$$ +\begin{array} { l } { { \displaystyle \lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow } = \frac { 1 } { 2 \pi } B ( B - 1 ) \mu ^ { \ast ^ { - 1 } } a _ { 1 } ^ { 2 } \mathrm { B e t a } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { 2 } } } \\ { { \displaystyle \quad = \frac { \frac { 1 } { 2 \pi } B ( B - 1 ) a _ { 1 } ^ { 2 } \mathrm { B e t a } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { 2 } } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } } } \\ { { \displaystyle \quad = \lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } . } } \end{array} +$$ + +Proof. It suffices to express the integral in $\mathrm { T h m } \ : \mathrm { I } . 2$ in Gegenbauer coefficients. Changing coordinates $x = - \cos \theta$ , so that $\mathrm { d } x = \sin \theta \mathrm { d } \theta$ , we get + +$$ +\int _ { 0 } ^ { \pi } \mathrm { d } \theta \sin ^ { B - 1 } \theta \phi ^ { \prime } ( - \sqrt { B - 1 } \cos \theta ) = \int _ { - 1 } ^ { 1 } \mathrm { d } x ( 1 - x ^ { 2 } ) ^ { \frac { B - 2 } { 2 } } \phi ^ { \prime } ( \sqrt { B - 1 } x ) . +$$ + +By Eq. (35), + +$$ +\begin{array} { l } { { \displaystyle { \phi ^ { \prime } ( \sqrt { B - 1 } x ) = ( B - 1 ) ^ { - 1 / 2 } \frac { d } { d x } \phi ( \sqrt { B - 1 } x ) } } } \\ { { \displaystyle ~ = ( B - 1 ) ^ { - 1 / 2 } \sum _ { l = 0 } ^ { \infty } a _ { l } \frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \prime ( x ) } } \\ { { \displaystyle ~ = ( B - 1 ) ^ { - 1 / 2 } \sum _ { l = 1 } ^ { \infty } a _ { l } \frac { B - 3 } { c _ { B - 1 , l } } C _ { l - 1 } ^ { ( \frac { B - 1 } { 2 } ) } ( x ) } . } \end{array} +$$ + +Then by the orthogonality relations among $\{ C _ { l - 1 } ^ { ( \frac { B - 1 } { 2 } ) } \} _ { l \ge 1 }$ , + +$$ +\begin{array} { r l } & { \displaystyle \int _ { 0 } ^ { \pi } \mathrm { d } \theta \sin ^ { B - 1 } \theta \phi ^ { \prime } ( - \sqrt { B - 1 } \cos \theta ) = \int _ { - 1 } ^ { 1 } \mathrm { d } x ( 1 - x ^ { 2 } ) ^ { \frac { B - 2 } { 2 } } \phi ^ { \prime } ( \sqrt { B - 1 } x ) } \\ & { \quad \quad \quad \quad = ( B - 1 ) ^ { - 1 / 2 } a _ { 1 } \frac { B - 3 } { c _ { B - 1 , 1 } } \int _ { - 1 } ^ { 1 } \left[ C _ { 0 } ^ { ( \frac { B - 1 } { 2 } ) } ( x ) \right] ^ { 2 } ( 1 - x ^ { 2 } ) ^ { \frac { B - 2 } { 2 } } d x } \\ & { \quad \quad \quad = ( B - 1 ) ^ { - 1 / 2 } a _ { 1 } \frac { B - 3 } { c _ { B - 1 , 1 } } \frac { \sqrt { \pi } \Gamma \left( \frac { B } { 2 } \right) } { \Gamma \left( \frac { B + 1 } { 2 } \right) } } \\ & { \quad \quad \quad = a _ { 1 } \sqrt { B - 1 } \mathrm { B e t a } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) . } \end{array} +$$ + +To summarize, + +Theorem I.5. The backward cross batch dynamics Eq. (59) over $\mathcal { H } _ { k B } ^ { G }$ is separable over each individual $B \times B$ block. The diagonal blocks evolve as in Eq. (51), and this linear dynamics has eigenvalues $\lambda _ { G } ^ { \downarrow } , \lambda _ { \mathbb { L } } ^ { \downarrow } , \lambda _ { \mathbb { M } } ^ { \downarrow }$ as described by Appendix G. The off-diagonal blocks dynamics is multiplication by $\lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow } = \lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow }$ . + +Note that + +$$ +\begin{array} { r l } & { \lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow } = \lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } = \frac { \frac { 1 } { 2 \pi } B ( B - 1 ) a _ { 1 } ^ { 2 } \mathrm { B e t a } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { 2 } } { \sum _ { l = 0 } ^ { \infty } a _ { l } ^ { 2 } \frac { 1 } { c _ { B - 1 , l } } \left( C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } } \\ & { \qquad \leq \frac { \frac { 1 } { 2 \pi } B ( B - 1 ) \mathrm { B e t a } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { 2 } } { \frac { 1 } { c _ { B - 1 , 1 } } \left( C _ { 1 } ^ { ( \frac { B - 3 } { 2 } ) } ( 1 ) - C _ { 1 } ^ { ( \frac { B - 3 } { 2 } ) } \left( \frac { - 1 } { B - 1 } \right) \right) } } \\ & { \qquad = \frac { B - 1 } { 2 } \mathrm { P } \left( \frac { B } { 2 } , \frac { 1 } { 2 } \right) ^ { - 2 } } \end{array} +$$ + +which increases to 1 from below as $B \infty$ . Thus, + +Corollary I.6. For any $\phi$ inducing Eq. (54) to converge to a CBSB1 fixed point, $\lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow } = \lambda _ { \widetilde { \mathbb { M } } } ^ { \uparrow } <$ 1. For any fixed batch size $B$ , $\lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow }$ is maximized by $\phi \ : = \ : \mathrm { i d } .$ . Furthermore, for $\phi = \mathrm { i d } , \ : \lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow } =$ $\textstyle { \frac { B - 1 } { 2 } } \operatorname { P } \left( { \frac { B } { 2 } } , { \frac { 1 } { 2 } } \right) ^ { - 2 }$ and $\begin{array} { r } { \operatorname* { l i m } _ { B \to \infty } \lambda _ { \widetilde { \mathbb { M } } } ^ { \downarrow } = 1 } \end{array}$ , increasing to the limit from below. + +This corollary indicates that stacking batchnorm in a deep neural network will always cause chaotic behavior, in the sense that cross batch correlation, both forward and backward, decreases to 0 exponentially with depth. The $\phi$ that can maximally ameliorate the exponential loss of information is linear. + +# J UNCOMMON REGIMES + +In the above exposition of the mean field theory of batchnorm, we have assumed $B \geq 4$ and that $\phi$ induces BSB1 fixed points in Eq. (8). + +Small Batch Size What happens when $B \ < \ 4 2$ It is clear that for $B = 1$ , batchnorm is not well-defined. For $B = 2$ , ${ \cal B } _ { \phi } ( h ) = ( \pm 1 , \mp 1 )$ depending on the signs of $h$ . Thus, the gradient of a batchnorm network with $B = 2$ is 0. Therefore, we see an abrupt phase transition from the immediate gradient vanishing of $B = 2$ to the gradient explosion of $B \geq 4$ . We empirically see that $B = 3$ suffers similar gradient explosion as $B = 4$ and conjecture that a generalization of $\cdot$ holds for $B = 3$ . + +Batch Symmetry Breaking What about other nonlinearities? Empirically, we observe that if the fixed point is not BSB1, then it is BSB2, like in Fig. 1, where a submatrix of $\Sigma$ (the dominant block) is much larger in magnitude than everything else (see Defn K.1). If the initial $\Sigma ^ { 0 }$ is permutationinvariant, then convergence to this fixed point requires spontaneous symmetry breaking, as the dominant block can appear in any part of $\Sigma$ along the diagonal. This symmetry breaking is lost when we take the mean field limit, but in real networks, the symmetry is broken by the network weight randomness. Because small fluctuation in the input can also direct the dynamics toward one BSB1 fixed point against others, causing large change in the output, the gradient is intuitively large as well. Additionally, at the BSB2 fixed point, we expect the dominant block goes through a similar dynamics as if it were a BSB1 fixed point for a smaller $B$ , thus suffering from similar gradient explosion. Appendix K discusses several results on our current understanding of BSB2 fixed points. A specific form of BSB2 fixed point with a $1 \times 1$ dominant block can be analyzed much further, and this is done in Appendix K.1. + +Finite width effect For certain nonlinearities, the favored fixed point can be different between the large it and small width. For example, for $\phi = C _ { 5 } ^ { ( \frac { B - 3 } { 2 } ) } ( x / \sqrt { B - 1 } ) - C _ { 7 } ^ { ( \frac { B - 3 } { 2 } ) } ( x / \sqrt { B - 1 } )$ $B = 1 0$ , one can observe that for width 100, Eq. (43) favors BSB2 fixed points, but for width 1000 and more, Eq. (43) favors BSB1 fixed points. + +# K TOWARD UNDERSTANDING BSB2 FIXED POINTS + +Definition K.1. For $B _ { 1 } , B _ { 2 } \geq 2$ , a BSB2 matrix is one of the form + +$$ +\mathrm { B S B 2 } _ { B _ { 1 } , B _ { 2 } } ( d _ { 1 } , f _ { 1 } , d _ { 2 } , f _ { 2 } , c ) : = \left( \begin{array} { c c } { { \mathrm { B S B 1 } _ { B _ { 1 } } ( d _ { 1 } , f _ { 1 } ) } } & { { c } } \\ { { c } } & \mathrm { B S B 1 } _ { B _ { 2 } } ( d _ { 2 } , f _ { 2 } ) \right) \end{array} +$$ + +up to simultaneous permutation of rows and columns. A $\widehat { \mathrm { B S B 2 } }$ matrix is a specific kind of BSB2 matrix $\widehat { \mathrm { B S B 2 } } _ { B } ^ { B ^ { \prime } } ( d , f , c , b ) = \mathrm { B S B 2 } _ { B ^ { \prime } , B - B ^ { \prime } } ( d , f , c , b , b )$ , where $B ^ { \prime } \in [ 2 , B - 1 ]$ (where for $B ^ { \prime } =$ $B { - } 1$ the lower right hand block is a scalar $b$ ). We call the upper left hand block of $\widehat { \mathrm { B S B 2 } } _ { B } ^ { B ^ { \prime } } ( d , f , c , b )$ its main block. + +When $\phi$ grows quickly, Eq. (43) can converge to BSB2 fixed points of the form $\widehat { \mathrm { B S B 2 } } _ { B } ^ { B ^ { \prime } } ( d , f , c , b )$ , up to simultaneous permutation of rows and columns. In this section we present several results on BSB2 fixed points that sheds light on their structure. We leave the study of the forward Eq. (43) and backward dynamics Eq. (51) to future work. + +First, we give the eigendecomposition of BSB2 matrices. + +Theorem K.2. Let $\Sigma = \mathrm { B S B 2 } _ { B _ { 1 } , B _ { 2 } } ( d _ { 1 } , f _ { 1 } , d _ { 2 } , f _ { 2 } , c )$ , where $B _ { 1 } , B _ { 2 } \ \geq \ 2$ . Then ${ \cal G } ^ { \otimes 2 } \{ \Sigma \} =$ $G \Sigma G = \Sigma ^ { G }$ has the following eigenspaces and eigenvalues. + +1. $( B _ { 1 } - 1 )$ -dimensional eigenspace $\mathbb { Z } _ { 1 } : = \{ ( x _ { 1 } , \ldots , x _ { B _ { 1 } } , 0 , \ldots , 0 ) : \textstyle \sum _ { i = 1 } ^ { B _ { 1 } } x _ { i } = 0 \}$ , with eigenvalue $d _ { 1 } - f _ { 1 }$ . +2. $( B _ { 2 } - 1 )$ -dimensional eigenspace $\mathbb { Z } _ { 2 } : = \{ ( 0 , \ldots , 0 , y _ { 1 } , \ldots , y _ { B _ { 2 } } ) : \textstyle \sum _ { j = 1 } ^ { B _ { 2 } } y _ { j } = 0 \} _ { }$ , with eigenvalue $d _ { 2 } - f _ { 2 }$ . + +3. $^ { l }$ -dimensional $\mathbb { R } q$ with eigenvalue $\frac { ( d _ { 2 } - f _ { 2 } ) B _ { 1 } + ( d _ { 1 } - f _ { 1 } ) B _ { 2 } + ( f _ { 1 } + f _ { 2 } - 2 c ) B _ { 1 } B _ { 2 } } { B _ { 1 } + B _ { 2 } }$ , where $\begin{array} { r l } { \pmb { q } } & { { } = } \end{array}$ $( \overbrace { B _ { 1 } ^ { - 1 } , \ldots , B _ { 1 } ^ { - 1 } } ^ { B _ { 1 } } , \overbrace { - B _ { 2 } ^ { - 1 } , \ldots , - B _ { 2 } ^ { - 1 } } ^ { B _ { 2 } } ) .$ + +4. $^ { l }$ -dimensional $\mathbb { R } \mathbf { 1 }$ with eigenvalue $O$ . + +Proof. We can verify all eigenspaces and their eigenvalues in a straightforward manner. These then must be all of them by a dimensionality argument. □ + +Specializing to $\widehat { \mathrm { B S B 2 } }$ matrices, we get +Corollary K.3. Let $\Sigma = \widehat { \mathrm { B S B 2 } } _ { B } ^ { B ^ { \prime } } ( d , f , c , b )$ with $B ^ { \prime } \geq 2$ . Then $G ^ { \otimes 2 } \{ \Sigma \} = G \Sigma G = \Sigma ^ { G }$ has the +following eigenspaces and eigenvalues + +$( B ^ { \prime } - 1 )$ -de ional eigenspace. $\mathbb { Z } _ { 1 } : = \{ ( x _ { 1 } , \ldots , x _ { B ^ { \prime } } , 0 , \ldots , 0 ) : \textstyle \sum _ { i = 1 } ^ { B ^ { \prime } } x _ { i } = 0 \} _ { }$ , with $d - f$ + +• $^ { l }$ -dimensional $\mathbb { R } q$ with eigenvalue $\frac { ( d - f ) ( B - B ^ { \prime } ) + ( f - c ) B ( B - B ^ { \prime } ) } { B }$ where q = , $B ^ { \prime }$ B $( \overbrace { { B ^ { \prime } } ^ { - 1 } , \ldots , { B ^ { \prime } } ^ { - 1 } } ^ { B } , \overbrace { - { B _ { 2 } ^ { - 1 } } , \ldots , - { B _ { 2 } ^ { - 1 } } } ^ { B _ { 2 } } )$ where $B _ { 2 } = B - B ^ { \prime }$ . + +B 0 • (B − B0)-dimensional eigenspace {(z }| {−µ, . . . , −µ, y1, . . . , yB−B0 : µ = 1B−B0 PB−B j=1 with eigenvalue $O$ . + +Note also that when $B ^ { \prime } = B$ , then $\mathbb { Z } _ { 1 }$ and $\mathbb { R } q$ become the usual eigenspaces of $\operatorname { B S B 1 } ( d , f )$ . + +For fixed $B ^ { \prime } < B$ , specializing the forward dynamics Eq. (43) to $\widehat { \mathrm { B S B 2 } } _ { B } ^ { B ^ { \prime } }$ fixed points yields a 2-dimensional dynamics on the eigenvalues for the eigenspaces $\mathbb { Z } _ { 1 }$ and $\mathbb { R } q$ . This dynamics in general is not degenerate, so that the fixed point seems difficult to obtain analytically. Moreoever, the Gegenbauer method, essential for proving that gradient explosion is unavoidable when Eq. (43) has a BSB1 fixed point, does not immediately generalize to the BSB2 case, since $\mathbb { Z } _ { 1 }$ and $\mathbb { R } q$ in general do not have the same eigenvalues so that we cannot reduce the integrals to that on a sphere. For this reason, at present we do not have any rigorous result on the BSB2 case. + +However, we expect that the main block of the $\widehat { \mathrm { B S B 2 } }$ fixed point should undergo a similar dynamics to that of a BSB1 fixed point, leading to similar gradient explosion. + +# K.1 $\widehat { \mathrm { B S B 2 } } _ { B } ^ { 1 }$ FIXED POINTS + +When $\phi$ grows extremely rapidly, for example $\phi ( x ) = \mathrm { r e l u } ( x ) ^ { 3 0 }$ , and the width of the network is not too large, then we sometimes observe $\widehat { \mathrm { B S B 2 } } _ { B } ^ { 1 }$ fixed points + +$$ +\widehat { \mathbf { B S B 2 } } _ { B } ^ { 1 } ( d , c , b ) = \left( \begin{array} { c c } { d } & { c \mathbf { 1 } ^ { T } } \\ { c \mathbf { 1 } } & { b \mathbf { 1 } \mathbf { 1 } ^ { T } } \end{array} \right) +$$ + +where $\mathbf { 1 } \in \mathbb { R } ^ { B - 1 }$ is the all 1s column vector. In these situations we can in fact see pathological gradient vanishing, in a way reminiscent of the gradient vanishing for $B = 2$ batchnorm, as we shall see shortly. + +We can extend Corollary K.3 to the $B ^ { \prime } = 1$ case. + +Theorem K.4. $\widehat { \mathrm { B S B 2 } } _ { B } ^ { 1 } ( d , c , b ) ^ { G }$ is rank $^ { l }$ and equal to $\lambda \hat { \pmb q } ^ { \otimes 2 }$ , where $\begin{array} { r } { \lambda = \frac { B - 1 } { B } ( d - 2 c + b ) } \end{array}$ and $\begin{array} { r } { \hat { \pmb q } = ( 1 , \frac { - 1 } { B - 1 } , \ldots , \frac { - 1 } { B - 1 } ) \sqrt { \frac { B - 1 } { B } } } \end{array}$ . + +Proof. Straightforward verification. + +Because $\widehat { \mathrm { B S B 2 } } _ { B } ^ { 1 }$ under $G$ -projection is rank 1, such a fixed point of Eq. (43) can be determined $\widehat { \mathrm { B S B 2 } } _ { B } ^ { B ^ { \prime } }$ +(unlike the general case). + +Theorem K.5. There is a unique $\widehat { \mathrm { B S B 2 } } _ { B } ^ { 1 }$ fixed point of Eq. (43), $\widehat { \mathrm { B S B 2 } } _ { B } ^ { 1 } ( d ^ { * } , c ^ { * } , b ^ { * } )$ where + +$$ +\begin{array} { l } { \displaystyle d ^ { * } = \frac { 1 } { 2 } \left( \phi ( \sqrt { B - 1 } ) ^ { 2 } + \phi ( - \sqrt { B - 1 } ) ^ { 2 } \right) } \\ { \displaystyle c ^ { * } = \frac { 1 } { 2 } \left( \phi ( \sqrt { B - 1 } ) \phi ( \frac { - 1 } { \sqrt { B - 1 } } ) + \phi ( - \sqrt { B - 1 } ) \phi ( \frac { 1 } { \sqrt { B - 1 } } ) \right) } \\ { \displaystyle b ^ { * } = \frac { 1 } { 2 } \left( \phi ( \frac { - 1 } { \sqrt { B - 1 } } ) ^ { 2 } + \phi ( \frac { 1 } { \sqrt { B - 1 } } ) ^ { 2 } \right) . } \end{array} +$$ + +Additionally, $\widehat { \mathrm { B S B 2 } } _ { B } ^ { 1 } ( d ^ { * } , c ^ { * } , b ^ { * } ) ^ { G } = \lambda ^ { * } \hat { \pmb q } ^ { \otimes 2 }$ , where + +$$ +\lambda ^ { * } = \frac { B - 1 } { 2 B } \left[ \left( \phi ( \sqrt { B - 1 } ) - \phi ( \frac { - 1 } { \sqrt { B - 1 } } ) \right) ^ { 2 } + \left( \phi ( - \sqrt { B - 1 } ) - \phi ( \frac { 1 } { \sqrt { B - 1 } } ) \right) ^ { 2 } \right] +$$ + +Proof. If $\Sigma = { \widehat { \mathrm { B S B 2 } } } _ { B } ^ { 1 } ( d , c , b )$ , then + +$$ +\begin{array} { l } { { \displaystyle \mathrm { V } _ { \mathcal { B } _ { \phi } } ( { \Sigma } ) = \underset { y \sim \mathcal { N } ( 0 , G \Sigma G ) } { \mathbb { E } } \phi ( { \mathsf { n } } ( y ) ) ^ { \otimes 2 } } } \\ { { \displaystyle = \frac { 1 } { 2 } \left[ \phi ( \sqrt { B } \hat { \pmb q } ) ^ { \otimes 2 } + \phi ( - \sqrt { B } \hat { \pmb q } ) ^ { \otimes 2 } \right] } } \end{array} +$$ + +since $G \Sigma G = \lambda \hat { q } ^ { \otimes 2 }$ for some $\lambda$ by Thm K.4. The rest then follows from straightforward computation and another application of Thm K.4. □ + +The rank-1 nature of the $\widehat { \mathrm { B S B 2 } } _ { B } ^ { 1 }$ fixed point under $G$ -projection is reminiscent of the case of $B = 2$ batchnorm. As we will see, gradients of a certain form will vanish similarly. + +Let $\delta$ be a gradient vector. Then by Proposition G.4, for $\Sigma ^ { * }$ being the $\widehat { \mathrm { B S B 2 } } _ { B } ^ { 1 }$ fixed point, one step of backpropagation yields + +$$ +\frac { \mathrm { d } B _ { \phi } } { \mathrm { d } \Sigma } \bigg | _ { \Sigma = \Sigma ^ { * } } ^ { \dagger } \delta = \sqrt { B } r ^ { - 1 } G ( I - v v ^ { T } ) D \delta +$$ + +where $D = \mathrm { D i a g } ( \phi ^ { \prime } ( { \mathsf { n } } ( y ) ) ) , r = \| y \| , v = y / \| y \|$ , and $y \sim { \mathcal { N } } ( 0 , G \Sigma ^ { * } G )$ . Because $G \Sigma ^ { * } G =$ $\lambda ^ { * } \hat { \pmb q } ^ { \otimes 2 }$ by $\cdot$ , $\boldsymbol { \mathsf { n } } ( y ) = \sqrt { B } \boldsymbol { v } = \pm \sqrt { B } \boldsymbol { \hat { q } }$ with equal probability and $r \sim | \mathcal { N } ( 0 , \lambda ^ { * } ) |$ (the absolute value of a Gaussian). Furthermore, $G ( \bar { I } - \hat { q } ^ { \otimes 2 } ) ^ { \bullet } = \acute { G } - \hat { q } ^ { \otimes 2 }$ because $G \hat { \pmb q } = \hat { \pmb q }$ . Thus + +$$ +\frac { \mathrm { d } \mathcal { B } _ { \phi } } { \mathrm { d } \Sigma } \bigg | _ { \Sigma = \Sigma ^ { * } } ^ { \dagger } \delta = \sqrt { B } | r | ^ { - 1 } ( G - \hat { q } ^ { \otimes 2 } ) \big ( \phi ^ { \prime } ( \sqrt { B } \mathrm { s g n } ( r ) \hat { q } ) \odot \delta ) , r \sim \mathcal { N } ( 0 , \lambda ^ { * } ) +$$ + +Now notice the following + +$$ +G _ { B } - \hat { q } ^ { \otimes 2 } = \left( \begin{array} { c c } { { 0 } } & { { 0 } } \\ { { 0 } } & { { G _ { B - 1 } } } \end{array} \right) = \left( \begin{array} { c c c c c } { { 0 } } & { { 0 } } & { { 0 } } & { { \cdots } } & { { 0 } } \\ { { 0 } } & { { 1 - \frac { 1 } { B - 1 } } } & { { \frac { - 1 } { B - 1 } } } & { { \cdots } } & { { \frac { - 1 } { B - 1 } } } \\ { { 0 } } & { { \frac { - 1 } { B - 1 } } } & { { 1 - \frac { 1 } { B - 1 } } } & { { \cdots } } & { { \frac { - 1 } { B - 1 } } } \\ { { 0 } } & { { \frac { - 1 } { B - 1 } } } & { { \frac { - 1 } { B - 1 } } } & { { \cdots } } & { { 1 - \frac { 1 } { B - 1 } } } \end{array} \right) . +$$ + +Proof. Straightforward computation. + +Thus if $\delta = ( \delta _ { 1 } , \delta _ { 2 } ) ^ { T } \in \mathbb { R } ^ { 1 } \times \mathbb { R } ^ { B - 1 }$ , then + +$$ +\frac { \mathrm { d } \mathcal { B } _ { \phi } } { \mathrm { d } \Sigma } \bigg | _ { \Sigma = \Sigma ^ { * } } ^ { \dagger } \delta = \sqrt { B } \phi ^ { \prime } \left( \frac { - \mathrm { s g n } ( r ) } { \sqrt { B - 1 } } \right) | r | ^ { - 1 } ( 0 , G _ { B - 1 } \delta _ { 2 } ) ^ { T } , r \sim \mathcal { N } ( 0 , \lambda ^ { * } ) +$$ + +If $\delta _ { 2 } = 0$ , then this is deterministically 0. Otherwise, this random variable does not have finite mean because of the $| r | ^ { - 1 }$ term, and this is solely due to $\Sigma ^ { * }$ being rank 1. Thus as long as the gradient $\delta$ is not constant in $\delta _ { 2 }$ , then we expect severe gradient explosion; otherwise we expect severe gradient vanishing. + +Remark K.7. We would like to note that, in pytorch, with either Float Tensors or Double Tensors, the gradient empirically vanishes rapidly with generic gradient $\delta$ , when $\phi$ is rapidly increasing and leads to a $\widehat { \mathrm { B S B 2 } } _ { B } ^ { 1 }$ fixed point, e.g. $\phi ( x ) = \mathrm { r e l u } ( x ) ^ { 3 0 }$ . This is due to numerical issues rather than mathematical issues, where the computation of $( \dot { G } - \dot { \pmb q } ^ { \otimes 2 } ) \delta$ in backprop does not kill $\delta _ { 1 }$ completely. + +# L WEIGHT GRADIENTS + +Recall weight gradient and the backpropagation equation Eq. (16) of the main text. + +$$ +\frac { \partial \mathcal { L } } { \partial W _ { \alpha \beta } ^ { l } } = \sum _ { i } \delta _ { \alpha i } ^ { l } x _ { \beta i } ^ { l - 1 } , \qquad \delta _ { \alpha i } ^ { l } = \sum _ { \beta j } \frac { \partial x _ { \alpha j } ^ { l } } { \partial h _ { \alpha i } ^ { l } } W _ { \beta \alpha } ^ { l + 1 } \delta _ { \beta j } ^ { l + 1 } +$$ + +where we have used $\boldsymbol { x } _ { \beta i } ^ { l }$ to denote $B _ { \phi } ( { \pmb h } _ { \beta } ^ { l } ) _ { i }$ , $\delta _ { i } ^ { l } = \nabla _ { h _ { i } ^ { l } } \mathcal { L }$ , and subscript $i$ denotes slice over sample $i$ in the batch, and subscript $\alpha$ denotes slice over neuron $\alpha$ in the layer. Then + +$$ +\begin{array} { r l r } { { \mathbb { E } [ ( \frac { \partial \mathcal { L } } { \partial W _ { \alpha \beta } ^ { l } } ) ^ { 2 } ] = \mathbb { E } [ ( \sum _ { i } \delta _ { \alpha i } ^ { l } x _ { \beta i } ^ { l - 1 } ) ^ { 2 } ] } } \\ & { } & { = \mathbb { E } [ \sum _ { i , j } \delta _ { \alpha i } ^ { l } x _ { \beta i } ^ { l - 1 } \delta _ { \alpha j } ^ { l } x _ { \beta j } ^ { l - 1 } ] . } \end{array} +$$ + +By gradient independence assumption (Appendix B), + +$$ +\begin{array} { r l } & { \mathbb { E } \delta _ { \alpha i } ^ { l } x _ { \beta i } ^ { l - 1 } \delta _ { \alpha j } ^ { l } x _ { \beta j } ^ { l - 1 } = \mathbb { E } [ \delta _ { \alpha i } ^ { l } \delta _ { \alpha j } ^ { l } ] \mathbb { E } [ x _ { \beta i } ^ { l - 1 } x _ { \beta j } ^ { l - 1 } ] } \\ & { \qquad = \Pi _ { i j } ^ { l } \Sigma _ { i j } ^ { l - 1 } . } \end{array} +$$ + +Thus + +$$ +\mathbb { E } \left[ \left( \frac { \partial \mathcal { L } } { \partial W _ { \alpha \beta } ^ { l } } \right) ^ { 2 } \right] = \sum _ { i , j } \Pi _ { i j } ^ { l } \Sigma _ { i j } ^ { l - 1 } = \langle \Pi ^ { l } , \Sigma ^ { l - 1 } \rangle . +$$ + +If $l$ is large and $\phi$ is well-behaved, then we expect $\Sigma ^ { l - 1 } \approx \Sigma ^ { * }$ , the unique BSB1 fixed point of Eq. (43). Assuming equality, we then have + +$$ +\begin{array} { r l r } { { \mathbb { E } [ ( \frac { \partial \mathcal { L } } { \partial W _ { \alpha \beta } ^ { l } } ) ^ { 2 } ] = \langle \Pi ^ { l } , \Sigma ^ { * } \rangle = \langle G ^ { \otimes 2 } \{ \Pi ^ { l } \} , \Sigma ^ { * } \rangle } } \\ & { } & { = \langle \Pi ^ { l } , G ^ { \otimes 2 } \{ \Sigma ^ { * } \} \rangle = \langle \Pi ^ { l } , \mu ^ { * } G \rangle } \\ & { } & { = \mu ^ { * } \operatorname { t r } \Pi ^ { l } . } \end{array} +$$ + +Thus in general, we expect the magnitude of weight gradients to follow the magnitude of hidden unit gradient $\delta _ { i } ^ { l }$ . \ No newline at end of file diff --git a/parse/train/SyMDXnCcF7/SyMDXnCcF7_content_list.json b/parse/train/SyMDXnCcF7/SyMDXnCcF7_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..5d0454b5af48c42cb18deba6d46c5cada782a3e8 --- /dev/null +++ b/parse/train/SyMDXnCcF7/SyMDXnCcF7_content_list.json @@ -0,0 +1,14907 @@ +[ + { + "type": "text", + "text": "A MEAN FIELD THEORY OF BATCH NORMALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 98, + 815, + 122 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Greg Yang†∗, Jeffrey Pennington◦, Vinay Rao◦, Jascha Sohl-Dickstein◦, & Samuel S. Schoenholz◦ ", + "bbox": [ + 183, + 143, + 852, + 160 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Microsoft Research AI, $\\circ$ Google Brain gregyang@microsoft.com, {jpennin,vinaysrao,jaschasd,schsam}@google.com ", + "bbox": [ + 186, + 172, + 856, + 202 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 238, + 544, + 252 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We develop a mean field theory for batch normalization in fully-connected feedforward neural networks. In so doing, we provide a precise characterization of signal propagation and gradient backpropagation in wide batch-normalized networks at initialization. Our theory shows that gradient signals grow exponentially in depth and that these exploding gradients cannot be eliminated by tuning the initial weight variances or by adjusting the nonlinear activation function. Indeed, batch normalization itself is the cause of gradient explosion. As a result, vanilla batch-normalized networks without skip connections are not trainable at large depths for common initialization schemes, a prediction that we verify with a variety of empirical simulations. While gradient explosion cannot be eliminated, it can be reduced by tuning the network close to the linear regime, which improves the trainability of deep batch-normalized networks without residual connections. Finally, we investigate the learning dynamics of batch-normalized networks and observe that after a single step of optimization the networks achieve a relatively stable equilibrium in which gradients have dramatically smaller dynamic range. Our theory leverages Laplace, Fourier, and Gegenbauer transforms and we derive new identities that may be of independent interest. ", + "bbox": [ + 233, + 268, + 764, + 503 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 531, + 336, + 546 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural networks have been enormously successful across a broad range of disciplines. These successes are often driven by architectural innovations. For example, the combination of convolutions (LeCun et al., 1990), residual connections (He et al., 2015), and batch normalization (Ioffe & Szegedy, 2015) has allowed for the training of very deep networks and these components have become essential parts of models in vision (Zoph et al.), language (Chen & Wu, 2017), and reinforcement learning (Silver et al., 2017). However, a fundamental problem that has accompanied this rapid progress is a lack of theoretical clarity. An important consequence of this gap between theory and experiment is that two important issues become conflated. In particular, it is generally unclear whether novel neural network components improve generalization or whether they merely increase the fraction of hyperparameter configurations where good generalization can be achieved. Resolving this confusion has the promise of allowing researchers to more effectively and deliberately design neural networks. ", + "bbox": [ + 174, + 563, + 825, + 728 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recently, progress has been made (Poole et al., 2016; Schoenholz et al., 2016; Daniely et al., 2016; Pennington et al., 2017; Hanin & Rolnick, 2018; Yang, 2019) in this direction by considering neural networks at initialization, before any training has occurred. In this case, the parameters of the network are random variables which induces a distribution of the activations of the network as well as the gradients. Studying these distributions is equivalent to understanding the prior over functions that these random neural networks compute. Picking hyperparameters that correspond to well-conditioned priors ensures that the neural network will be trainable and this fact has been extensively verified experimentally. However, to fulfill its promise of making neural network design less of a black box, these techniques must be applied to neural network architectures that are used in practice. Over the past year, this gap has closed significantly and theory for networks with skip connections (Yang & Schoenholz, 2017; 2018), convolutional networks (Xiao et al., 2018), and gated recurrent networks (Chen et al., 2018; Gilboa et al., 2019) have been developed. More recently, Yang (2019) devised a formalism that extends this approach to include an even wider range of architectures. ", + "bbox": [ + 173, + 736, + 825, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 821, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Before state-of-the-art models can be analyzed in this framework, a slowly-decreasing number of architectural innovations must be studied. One particularly important component that has thus-far remained elusive is batch normalization. ", + "bbox": [ + 176, + 152, + 823, + 194 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our Contributions. In this paper, we develop a theory of fully-connected networks with batch normalization whose weights and biases are randomly distributed. A significant complication in the case of batch normalization (compared to e.g. layer normalization or weight normalization) is that the statistics of the network depend non-locally on the entire batch. Thus, our first main result is to recast the theory for random fully-connected networks so that it can be applied to batches of data. We then extend the theory to include batch normalization explicitly and validate this theory against Monte-Carlo simulations. We show that as in previous cases we can leverage our theory to predict valid hyperparameter configurations. ", + "bbox": [ + 174, + 213, + 825, + 325 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In the process of our investigation, we identify a number of previously unknown properties of batch normalization that make training unstable. In particular, for most nonlinearities used in practice, batchnorm in a deep, randomly initialized network induces high degree of symmetry in the embeddings of the samples in the batch $( \\mathrm { T h m } \\ 3 . 4 )$ . Whenever this symmetry takes hold, we show that for any choice of nonlinearity, gradients of fully-connected networks with batch normalization explode exponentially in the depth of the network $\\mathrm { T h m } \\ 3 . 9 $ ). This imposes strong limits on the maximum trainable depth of batch normalized networks. This limit can be lifted partially but not completely by pushing activation functions to be more linear at initialization. It might seem that such gradient explosion ought to lead to learning dynamics that are unfavorable. However, we show that networks with batch normalization causes the scale of the gradients to naturally equilibrate after a single step of gradient descent (provided the initial gradients are not so large as to cause numerical instabilities). For shallower networks, this equilibrating effect is sufficient to allow adequate training. ", + "bbox": [ + 174, + 332, + 825, + 498 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Finally, we note that there is a related vein of research that has emerged that leverages the prior over functions induced by random networks to perform exact Bayesian inference (Lee et al., 2017; de G. Matthews et al., 2018; Novak et al., 2019; Garriga-Alonso et al., 2019; Yang, 2019). One of the natural consequences of this work is that the prior for networks with batch normalization can be computed exactly in the wide network limit. As such, it is now possible to perform exact Bayesian inference in the case of wide neural networks with batch normalization. ", + "bbox": [ + 174, + 506, + 825, + 588 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 612, + 343, + 628 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Batch normalization has rapidly become an essential part of the deep learning toolkit. Since then, a number of similar modifications have been proposed including layer normalization (Ba et al., 2016) and weight normalization (Salimans & Kingma, 2016). Comparisons of performance between these different schemes have been challenging and inconclusive (Gitman & Ginsburg, 2017). The original introduction of batchnorm in Ioffe & Szegedy (2015) proposed that batchnorm prevents “internal covariate shift” as an explanation for its effectiveness. Since then, several papers have approached batchnorm from a theoretical angle, especially following Ali Rahimi’s catalyzing call to action at NIPS 2017. Balduzzi et al. (2017) found that batchnorm in resnets allow deep gradient signal propagation in contrast to the case without batchnorm. Santurkar et al. (2018) found that batchnorm does not help covariate shift but helps by smoothing loss landscape. Bjorck et al. (2018) reached the opposite conclusion as our paper for residual networks with batchnorm, that batchnorm works in this setting because it induces beneficial gradient dynamics and thus allows a much bigger learning rate. Luo et al. (2018) explores similar ideas that batchnorm allows large learning rates and likewise uses random matrix theory to support their claims. Kohler et al. (2018) identified situations in which batchnorm can provably induce acceleration in training. Of the above that mathematically analyze batchnorm, all but Santurkar et al. (2018) make simplifying assumptions on the form of batchnorm and typically do not have gradients flowing through the batch variance. Even Santurkar et al. (2018) only analyzes a vanilla network which gets added a single batchnorm at a single moment in training. Our analysis here on the other hand works for networks with arbitrarily many batchnorm layers with very general activation functions, and indeed, this deep stacking of batchnorm is precisely what leads to gradient explosion. It is an initialization time analysis for networks of infinite width, but we show experimentally that the resulting insights predict both training and test time behavior. ", + "bbox": [ + 173, + 645, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We remark that Philipp & Carbonell (2018) has also empirically noted gradient explosion happens in deep batchnorm networks with various nonlinearities. In this work, in contrast, we develop a precise theoretical characterization of gradient statistics and, as a result, we are able to make significantly stronger conclusions. ", + "bbox": [ + 174, + 138, + 825, + 194 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 THEORY ", + "text_level": 1, + "bbox": [ + 174, + 214, + 276, + 231 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We begin with a brief recapitulation of mean field theory in the fully-connected setting. In addition to recounting earlier results, we rephrase the formalism developed previously to compute statistics of neural networks over a batch of data. Later, we will extend the theory to include batch normalization. We consider a fully-connected network of depth $L$ whose layers have width $N _ { l }$ , activation function1 $\\phi$ , weights $W ^ { l } \\bar { \\mathbf { \\Xi } } \\in \\mathbb { R } ^ { N _ { l - 1 } \\times N _ { l } }$ , and biases $\\hat { b } ^ { l } \\in \\mathbb { R } ^ { N _ { l } }$ . Given a batch of $B$ inputs2 $\\{ x _ { i } : x _ { i } \\in \\mathbb { R } ^ { N _ { 0 } } \\} _ { i = 1 , \\cdots , B }$ , the pre-activations of the network are defined by the recurrence relation, ", + "bbox": [ + 174, + 246, + 825, + 330 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/de103cad3e807d0b2af9e0edc19a08374f12495b7cb44bb13491fa4531b7f652.jpg", + "text": "$$\n\\pmb { h } _ { i } ^ { 1 } = W ^ { 1 } \\pmb { x } _ { i } + b ^ { 1 } \\qquad \\mathrm { a n d } \\qquad \\pmb { h } _ { i } ^ { l } = W ^ { l } \\phi ( \\pmb { h } _ { i } ^ { l - 1 } ) + b ^ { l } \\quad \\forall l > 1 .\n$$", + "text_format": "latex", + "bbox": [ + 277, + 337, + 718, + 357 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "At initialization, we choose the weights and biases to be i.i.d. as $W _ { \\alpha \\beta } ^ { l } \\sim \\mathcal { N } ( 0 , \\sigma _ { w } ^ { 2 } / N _ { l - 1 } )$ and $b _ { \\alpha } ^ { l } \\sim \\mathcal { N } ( 0 , \\sigma _ { b } ^ { 2 } )$ . In the following, we use $\\alpha , \\beta , \\ldots$ for the neuron index, and $i , j , \\dots$ for the batch index. We will be concerned with understanding the statistics of the pre-activations and the gradients induced by the randomness in the weights and biases. For ease of exposition we will typically take the network to have constant width $N _ { l } = N$ . ", + "bbox": [ + 174, + 363, + 825, + 438 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the mean field approximation, we iteratively replace the pre-activations in Eq. (2) by Gaussian random variables with matching first and second moments. In the infinite width limit this approximation becomes exact (Lee et al., 2017; de G. Matthews et al., 2018; Yang, 2019). Since the weights are i.i.d. with zero mean it follows that the mean of each pre-activation is zero and the covariance between distinct neurons are zero. The pre-activation statistics are therefore given $( h _ { \\alpha _ { 1 } 1 } ^ { l } , \\cdot \\cdot \\cdot , h _ { \\alpha _ { B } B } ^ { l } ) \\xrightarrow { N _ { l - 1 } \\infty } \\mathcal { N } ( 0 , \\Sigma ^ { l } \\delta _ { \\alpha _ { 1 } \\cdots \\alpha _ { B } } )$ Nl−1→∞ −−−−−−→ N (0, Σlδα1···αB ) where Σl are B × B covariance matrices and $\\bar { \\partial } _ { \\alpha _ { 1 } , \\cdots \\alpha _ { B } }$ $\\delta$ $\\alpha _ { 1 } = \\alpha _ { 2 } = \\cdot \\cdot \\cdot = \\alpha _ { B }$ ", + "bbox": [ + 173, + 444, + 825, + 549 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Definition 3.1. Let $\\mathrm { V }$ be the operator on functions $\\phi$ such that $\\mathrm { V } _ { \\phi } ( \\Sigma ) = \\mathbb { E } [ \\phi ( h ) \\phi ( h ) ^ { T } : h \\sim$ $\\mathcal { N } ( 0 , \\Sigma ) ]$ computes the matrix of uncentered second moments of $\\phi ( h )$ for $h \\sim { \\mathcal { N } } ( 0 , \\Sigma )$ . ", + "bbox": [ + 171, + 553, + 821, + 583 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Using the above notation, we can express the covariance matrices by the recurrence relation, ", + "bbox": [ + 173, + 593, + 779, + 609 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/3b20566bcd661960138e14a79f65ca3f9e114a9a880bde84d8521ddfa8e85b80.jpg", + "text": "$$\n\\Sigma ^ { l } = \\sigma _ { w } ^ { 2 } \\mathrm { V } _ { \\phi } ( \\Sigma ^ { l - 1 } ) + \\sigma _ { b } ^ { 2 } \\mathbf { 1 } \\mathbf { 1 } ^ { T }\n$$", + "text_format": "latex", + "bbox": [ + 400, + 614, + 598, + 635 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "At first Eq. (2) may seem challenging since the expectation involves a Gaussian integral in $\\mathbb { R } ^ { B }$ . However, each term in the expectation of $\\mathrm { V } _ { \\phi }$ involves at most a pair of pre-activations and so the expectation may be reduced to the evaluation of $\\mathcal { O } ( B ^ { 2 } )$ two-dimensional integrals. These integrals can either be performed analytically (Cho & Saul, 2009; Williams, 1997) or efficiently approximated numerically (Lee et al., 2017), and so Eq. (2) defines a computationally efficient method for computing the statistics of neural networks after random initialization. This theme of dimensionality reduction will play a prominent role in the forthcoming discussion on batch normalization. ", + "bbox": [ + 173, + 641, + 825, + 741 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Eq. (2) defines a dynamical system over the space of covariance matrices. Studying the statistics of random feed-forward networks therefore amounts to investigating this dynamical system and is an enormous simplification compared with studying the pre-activations of the network directly. As is common in the dynamical systems literature, a significant amount of insight can be gained by investigating the behavior of Eq. (2) in the vicinity of its fixed points. For most common activation functions, Eq. (2) has a fixed point at some $\\Sigma ^ { * }$ . Moreover, when the inputs are non-degenerate, this fixed point generally has a simple structure with $\\Sigma ^ { * } = q ^ { * } [ ( 1 - c ^ { * } ) I + \\dot { c } ^ { * } \\mathbf { 1 } \\mathbf { 1 } ^ { T } ]$ owing to permutation symmetry among elements of the batch. We refer to fixed points with such symmetry as BSB1 (Batch Symmetry Breaking $^ { l }$ or $^ { l }$ Block Symmetry Breaking) fixed points. As we will discuss later, in the context of batch normalization other fixed points with fewer symmetries (“BSBk fixed points”) ", + "bbox": [ + 173, + 747, + 825, + 886 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "may become preferred. In the fully-connected setting fixed points may efficiently be computed by solving the fixed point equation induced by Eq. (2) in the special case $B = 2$ . The structure of this fixed point implies that in asymptotically deep feed-forward neural networks all inputs yield pre-activations of identical norm with identical angle between them. Neural networks that are deep enough so that their pre-activation statistics lie in this regime have been shown to be untrainable (Schoenholz et al., 2016). ", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Notation As we often talk about matrices and also linear operators over matrices, we write $\\mathcal { T } \\{ \\Sigma \\}$ for an operator $\\tau$ applied to a matrix $\\Sigma$ , and matrix multiplication is still written as juxtaposition. Composition of matrix operators are denoted with $\\mathcal { T } _ { 1 } \\circ \\mathcal { T } _ { 2 }$ . ", + "bbox": [ + 174, + 203, + 823, + 246 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Local Convergence to BSB1 Fixed Point. To understand the behavior of Eq. (2) near its BSB1 fixed point we can consider the Taylor series in the deviation from the fixed point, $\\Delta \\Sigma ^ { l } = \\Sigma ^ { l } - \\Sigma ^ { * }$ . To lowest order we generically find, ", + "bbox": [ + 173, + 261, + 825, + 304 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/3f2bc40e45da36e965b2037b707e055256c210be6d712981681b97d6835c700d.jpg", + "text": "$$\n\\Delta \\Sigma ^ { l } = J \\{ \\Delta \\Sigma ^ { l - 1 } \\}\n$$", + "text_format": "latex", + "bbox": [ + 434, + 311, + 563, + 330 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { J = \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } } { \\mathrm { d } \\Sigma } \\big | _ { \\Sigma = \\Sigma ^ { * } } } \\end{array}$ is the $B ^ { 2 } \\times B ^ { 2 }$ Jacobian of $\\mathrm { V } _ { \\phi }$ . In most prior work where $\\phi$ was a pointwise non-linearity, Eq. (3) reduces to the case $B = 2$ which naturally gave rise to linearized dynamics in $q ^ { l } = \\mathbb { E } [ ( h _ { i } ^ { l } ) ^ { \\overline { { 2 } } } ]$ and $c ^ { l } = \\mathbb { E } [ h _ { i } ^ { l } z _ { j } ^ { l } ] / q ^ { l }$ . However, in the case of batch normalization we will see that one must consider the evolution of Eq. (3) as a whole. This is qualitatively reminiscent of the case of convolutional networks studied in Xiao et al. (2018) where the evolution of the entire pixel $\\times$ pixel covariance matrix had to be evaluated. ", + "bbox": [ + 173, + 338, + 825, + 425 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The dynamics induced by Eq. (3) will be controlled by the eigenvalues of $J$ : Suppose $J$ has eigenvalues $\\lambda _ { i }$ — ordered such that $\\lambda _ { 1 } \\geq \\lambda _ { 2 } \\geq \\cdot \\cdot \\cdot \\geq \\lambda _ { B ^ { 2 } }$ — with associated eigen“vectors” $e _ { i }$ (note that the $e _ { i }$ will themselves be $B \\times B$ matrices). It follows that if $\\Delta \\Sigma ^ { 0 } = \\textstyle \\sum _ { i } c _ { i } e _ { i }$ for some choice of constants $c _ { i }$ then $\\begin{array} { r } { \\Delta \\Sigma ^ { l } = \\sum _ { i } c _ { i } \\lambda _ { i } ^ { l } e _ { i } } \\end{array}$ . Thus, if $\\lambda _ { i } < 1$ for all $i$ , $\\Delta \\Sigma ^ { l }$ will approach zero exponentially and the fixed-point will be stable. The number of layers over which $\\Sigma$ will approach $\\Sigma ^ { * }$ will be given by $- 1 / \\log ( \\lambda _ { 1 } )$ . By contrast if $\\lambda _ { i } > 1$ for any $i$ then the fixed point will be unstable. In this case, there is typically a different, stable, fixed point that must be identified. It follows that if the eigenvalues of $J$ can be computed then the dynamics will follow immediately. ", + "bbox": [ + 173, + 431, + 825, + 546 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "While $J$ may appear to be a complicated object at first, a moment’s thought shows that each diagonal element $J \\{ \\Delta \\Sigma \\} _ { i i }$ is only affected by the corresponding diagonal element $\\Delta \\Sigma _ { i i }$ , and each off-diagonal element $\\dot { J } \\{ \\Delta \\Sigma \\} _ { i j }$ is only affected by $\\Delta \\Sigma _ { i i } , \\Delta \\Sigma _ { j j }$ , and $\\Delta \\Sigma _ { i j }$ . Such a Diagonal-Offdiagonal Semidirect $( D O S )$ operator has a simple eigendecomposition with two eigenspaces corresponding to changes in the off-diagonal and changes in the diagonal $\\big ( \\begin{array} { l l l l l l } \\end{array} \\begin{array} { r l } \\end{array} \\begin{array} { r l } \\end{array} \\begin{array} { r l } \\end{array} \\big )$ . The associated eigenvalues are precisely those calculated by Schoenholz et al. (2016) in a simplified analysis. DOS operators are a particularly simple form of a more general operator possessing an abundance of symmetries called ultrasymmetric operators, which will play a prominent role in the analysis of batchnorm below. ", + "bbox": [ + 173, + 551, + 825, + 678 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Gradient Dynamics. Similar arguments allow us to develop a theory for the statistics of gradients. The backpropogation algorithm gives an efficient method of propagating gradients from the end of the network to the earlier layers as, ", + "bbox": [ + 174, + 693, + 825, + 736 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5df6832f69ccdafd959fbfe11c5d85310d465f55aafb936628bc301289e46da5.jpg", + "text": "$$\n\\frac { \\mathrm { d } \\mathcal { L } } { \\mathrm { d } W ^ { l } } = \\sum _ { i } \\delta _ { i } ^ { l } \\phi ( \\pmb { h } _ { i } ^ { l - 1 } ) ^ { T } \\qquad \\delta _ { \\alpha i } ^ { l } = \\phi ^ { \\prime } ( h _ { \\alpha i } ^ { l } ) \\sum _ { \\beta } W _ { \\beta \\alpha } ^ { l + 1 } \\delta _ { \\beta i } ^ { l + 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 297, + 742, + 699, + 782 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Here $\\mathcal { L }$ is the loss function and $\\delta _ { i } ^ { l } \\ = \\ \\nabla _ { h _ { i } ^ { l } } \\mathcal { L }$ are $N _ { l }$ -dimensional vectors that describe the error signal from neurons in the $\\mathbf { \\xi } _ { l } \\mathbf { \\cdot } \\mathbf { \\xi } _ { l } ,$ th layer due to the $\\because$ ’th element of the batch. The preceding discussion gave a precise characterization of the statistics of the $h _ { i } ^ { l }$ that we can leverage to understand the statistics of $\\delta _ { i } ^ { l }$ . Assuming gradient independence, that is, that an iid set of weights are used during backpropagation (see Appendix B for more discussions), it is easy to see that $\\mathbb { E } [ \\delta _ { \\alpha i } ^ { l } ] = 0$ and $\\mathbb { E } [ \\delta _ { \\alpha i } ^ { l } \\delta _ { \\beta j } ^ { l } ] = \\Pi _ { i j } ^ { l }$ if $\\alpha = \\beta$ and 0 otherwise, where $\\Pi ^ { l }$ is a covariance matrix and we may once again drop the neuron index. We can construct a recurrence relation to compute $\\Pi ^ { l }$ , ", + "bbox": [ + 173, + 790, + 825, + 900 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/43e605a5fb15cdc6bc05c011ce83e52e4c817ff77ceb2bc8f8c39f2f2efc9a32.jpg", + "text": "$$\n\\Pi ^ { l } = \\sigma _ { w } ^ { 2 } \\mathrm { V } _ { \\phi ^ { \\prime } } ( \\Sigma ^ { l } ) \\odot \\Pi ^ { l + 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 411, + 906, + 586, + 926 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Typically, we will be interested in understanding the dynamics of $\\Pi ^ { l }$ when $\\Sigma ^ { l }$ has converged exponentially towards its fixed point. Thus, we study the approximation, ", + "bbox": [ + 169, + 102, + 821, + 132 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c2e41a4f7a760b82189b8577c6f6e2f4375462d8c337ff97c25b5c78865a48f7.jpg", + "text": "$$\n\\Pi ^ { l } \\approx \\sigma _ { w } ^ { 2 } \\mathsf { V } _ { \\phi ^ { \\prime } } ( \\Sigma ^ { * } ) \\odot \\Pi ^ { l + 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 408, + 138, + 588, + 159 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Since these dynamics are linear (and in fact componentwise), explosion and vanishing of gradients will be controlled by $\\mathrm { V } _ { \\phi ^ { \\prime } } ( \\Sigma ^ { * } )$ . ", + "bbox": [ + 171, + 165, + 825, + 195 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.1 BATCH NORMALIZATION ", + "text_level": 1, + "bbox": [ + 174, + 212, + 388, + 227 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We now extend the mean field formalism to include batch normalization. Here, the definition for the neural network is modified to be the coupled equations, ", + "bbox": [ + 173, + 237, + 823, + 267 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/6c48df2a177bdd1c6641c6db80f080c3112c1e612e5ac5c3736bf817ec6bdf34.jpg", + "text": "$$\n\\pmb { h } _ { i } ^ { l } = W ^ { l } \\phi ( \\widetilde { \\pmb { h } } _ { i } ^ { l - 1 } ) + b ^ { l } \\qquad \\widetilde { \\pmb { h } } _ { \\alpha i } ^ { l } = \\gamma _ { \\alpha } \\frac { h _ { \\alpha i } ^ { l } - \\mu _ { \\alpha } } { \\sigma _ { \\alpha } } + \\beta _ { \\alpha }\n$$", + "text_format": "latex", + "bbox": [ + 320, + 273, + 678, + 308 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\gamma _ { \\alpha }$ and $\\beta _ { \\alpha }$ are parameters, and $\\begin{array} { r } { \\mu _ { \\alpha } = \\frac { 1 } { N _ { l } } \\sum _ { i } h _ { \\alpha i } } \\end{array}$ and $\\begin{array} { r } { \\sigma _ { \\alpha } ^ { 2 } = \\sqrt { \\frac { 1 } { N _ { l } } \\sum _ { i } ( h _ { \\alpha i } - \\mu _ { \\alpha } ) ^ { 2 } + \\epsilon } } \\end{array}$ are the per-neuron batch statistics. In practice $\\epsilon \\approx 1 0 ^ { - 5 }$ or so to prevent division by zero, but in this paper, unless stated otherwise (in the last few sections), $\\epsilon$ is assumed to be 0. Unlike in the case of vanilla fully-connected networks, here the pre-activations are invariant to $\\sigma _ { w } ^ { 2 }$ and $\\sigma _ { b } ^ { 2 }$ . Without a loss of generality, we therefore set $\\sigma _ { w } ^ { 2 } = 1$ and $\\sigma _ { b } ^ { 2 } = 0$ for the remainder of the text. In principal, batch normalization additionally yields a pair of hyperparameters $\\gamma$ and $\\beta$ which are set to be constants. However, these may be incorporated into the nonlinearity and so without a loss of generality we set $\\gamma = 1$ and $\\beta = 0$ . In order to avoid degenerate results, we assume $B \\geq 4$ unless stated otherwise; we shall discuss the small $B$ regime in Appendix J. ", + "bbox": [ + 173, + 316, + 826, + 452 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "If one treats batchnorm as a “batchwise nonlinearity”, then the arguments from the previous section can proceed identically and we conclude that as the width of the network grows, the pre-activations will be jointly Gaussian with identically distributed neurons. Thus, we arrive at an analogous expression to Eq. (2), ", + "bbox": [ + 174, + 458, + 825, + 515 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/fe3c15d61939a131d64b422a84a38ef25a554a20001262faf712b449b00744d5.jpg", + "text": "$$\n\\Sigma ^ { l } = \\mathsf { V } _ { \\mathcal { B } _ { \\phi } } ( \\Sigma ^ { l - 1 } ) \\qquad \\mathrm { w h e r e } \\qquad \\mathcal { B } _ { \\phi } : \\mathbb { R } ^ { B } \\to \\mathbb { R } ^ { B } , \\mathcal { B } _ { \\phi } ( h ) = \\phi \\left( \\frac { \\sqrt { B } G h } { \\vert \\vert G h \\vert \\vert } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 243, + 521, + 756, + 564 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Here we have introduced the projection operator $\\begin{array} { r } { G = I - \\frac { 1 } { B } \\mathbf { 1 } \\mathbf { 1 } ^ { T } } \\end{array}$ which is defined such that $G x =$ $x - \\mu \\mathbf { 1 }$ with $\\mu = \\textstyle \\sum _ { i } x _ { i } / B$ . Unlike $\\phi$ , $B _ { \\phi }$ does not act component-wise on $h$ . It is therefore not obvious whether $\\mathrm { V } _ { B _ { \\phi } }$ can be evaluated without performing a $B$ -dimensional Gaussian integral. ", + "bbox": [ + 173, + 570, + 823, + 614 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theoretical tools. In this paper, we present several ways to analyze high dimensional integrals like the above: 1. the Laplace method 2. the Fourier method 3. spherical integration 4. and the Gegenbauer method. The former two use the Laplace and Fourier transforms to simplify expressions like the above, where the Laplace method requires that $\\phi$ be positive homogeneous. Often the Laplace method will give clean, closed form answers for such $\\phi$ . Because batchnorm can be thought of as a linear projection $( G )$ followed by projection to the sphere of radius $\\sqrt { B }$ , spherical integration techniques are often very useful and in fact is typically the most straightforward way of numerically evaluating quantities. Lastly, the Gegenbauer method expresses objects in terms of the Gegenbauer coefficients of $\\phi$ . Briefly, Gegenbauer polynomials $\\{ C _ { l } ^ { ( \\bar { \\alpha } ) } ( x ) \\} _ { l = 0 } ^ { \\infty }$ are orthogonal polynomials with respect to the measure $( 1 - x ^ { 2 } ) ^ { \\alpha - \\frac { 1 } { 2 } }$ on $[ - 1 , 1 ]$ . They are intimately related to spherical harmonics (a natural basis for functions on a sphere), which explains their appearance in this context. The Gegenbauer method is the most illuminating amongst them all, and is what allows us to conclude that gradient explosion happens regardless of nonlinearity under general conditions. See Appendix D for a more in-depth discussion of these techniques. In what follows, we will mostly present results by the Laplace method for $\\phi =$ relu and by the Gegenbauer method for general $\\phi$ , but give pointers to the appendix for others. ", + "bbox": [ + 173, + 630, + 825, + 861 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Back to the topic of Eq. (8), we present a pair of results that expresses Eq. (8) in a more manageable form. From previous work (Poole et al., 2016), $\\mathrm { V } _ { \\phi }$ can be expressed in terms of a two-dimensional Gaussian integrals independent of $B$ . When $\\phi$ is degree- $\\alpha$ positive homogeneous (e.g. rectified linear activations) we can relate $\\mathrm { V } _ { \\phi }$ and $\\mathrm { V } _ { B _ { \\phi } }$ by the Laplace transform. (see Thm E.5). ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 3.2. Suppose $\\phi : \\mathbb { R } \\mathbb { R }$ is degree-α positive homogeneous. For any positive semi-definite matrix $\\Sigma$ define the projection $\\Sigma ^ { G } = G \\bar { \\Sigma } G$ . Then ", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/1cfb460584c864c2d17f6b2d5623d9ecb7b828ffb42a695500ca6e04077a0dfb.jpg", + "text": "$$\n\\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( \\Sigma ) = \\frac { B ^ { \\alpha } } { \\Gamma ( \\alpha ) } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s ~ s ^ { \\alpha - 1 } \\frac { \\mathrm { V } _ { \\phi } ( \\Sigma ^ { G } ( I + 2 s \\Sigma ^ { G } ) ^ { - 1 } ) } { \\sqrt { \\operatorname* { d e t } ( I + 2 s \\Sigma ^ { G } ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 313, + 137, + 683, + 175 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "whenever the integral exists. ", + "bbox": [ + 174, + 179, + 361, + 194 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Using this parameterization, when $\\mathrm { V } _ { \\phi }$ has a closed form solution (like ReLU), $\\mathrm { V } _ { B _ { \\phi } }$ involves only a single integral. A similar expression can be derived for general $\\phi$ by applying Fourier transform; see Appendix E.2. Next we express Eq. (8) as a spherical integral (see Proposition E.34) ", + "bbox": [ + 174, + 205, + 825, + 248 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3.3. Let $\\Phi \\in \\mathbb { R } ^ { B \\times B - 1 }$ have as columns a set of orthonormal basis vectors of $\\{ \\boldsymbol { x } \\in \\mathbb { R } ^ { B }$ $G x = x \\}$ . Then with $S ^ { B - 2 } \\subseteq \\mathbb { R } ^ { B - 1 }$ denoting the $( B - 2 )$ -dimensional sphere, ", + "bbox": [ + 171, + 251, + 821, + 280 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/14c7a898b93f578109bf94f0ad8b16fb13e3b63ba3851dfc33a90c9587854cf5.jpg", + "text": "$$\n\\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( \\Sigma ) = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\frac { \\phi ( \\sqrt { B } \\mathrm { e } v ) ^ { \\otimes 2 } } { \\sqrt { \\operatorname* { d e t } \\Phi ^ { T } \\Sigma \\mathrm { e } } ( v ^ { T } ( \\mathrm { e } ^ { T } \\Sigma \\mathrm { e } ) ^ { - 1 } v ) ^ { \\frac { B - 1 } { 2 } } } .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 286, + 678, + 324 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Together these theorems provide analytic recurrence relations for random neural networks with batch normalization over a wide range of activation functions. By analogy to the fully-connected case we would like to study the dynamical system over covariance matrices induced by these equations. ", + "bbox": [ + 174, + 337, + 825, + 378 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We begin by investigating the fixed point structure of Eq. (8). As in the case of feed-forward networks, permutation symmetry implies that there exist BSB1 fixed points $\\Sigma ^ { * } = q ^ { * } [ ( 1 - c ^ { * } ) I + c ^ { * } \\mathbf { 1 } \\mathbf { 1 } ^ { T } ]$ . We will see that this fixed point is in fact unique, and a clean expression of $q ^ { * }$ and $c ^ { * }$ can be obtained in terms of Gegenbauer basis (see Thm F.13). ", + "bbox": [ + 174, + 385, + 825, + 441 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3.4 (Gegenbauer expansions of BSB1 fixed point). If $\\phi ( \\sqrt { B - 1 } x )$ has Gegenbauer expansion P∞l=0 al 1cB−1,l $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ where $\\begin{array} { r } { c _ { B - 1 , l } = \\frac { B - 3 } { B - 3 + 2 l } } \\end{array}$ , then ", + "bbox": [ + 173, + 445, + 823, + 484 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/1d476b357a15cae179e75d28e1fef06a2387f372ad6736ec42af48cc16f756fe.jpg", + "text": "$$\nq ^ { * } = \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) , \\qquad q ^ { * } c ^ { * } = \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 491, + 766, + 532 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Thus the entries of the BSB1 fixed point are diagonal quadratic forms of the Gegenbauer coefficients of $\\phi ( \\sqrt { B - 1 } x )$ . Even more concise closed forms are available when the activation functions are degree $\\alpha$ positive homogeneous (see Thm F.8). In particular, for ReLU we arrive at the following ", + "bbox": [ + 173, + 544, + 825, + 588 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Theorem 3.5 (BSB1 fixed point for ReLU). When $\\phi =$ relu, then ", + "bbox": [ + 174, + 592, + 607, + 606 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/1b73bec2ad6b53e0dd6990fe185ebffc31823b0dec586a9ba66015ad2dc0613a.jpg", + "text": "$$\nq ^ { * } = { \\frac { 1 } { 2 } } , \\qquad c ^ { * } = \\operatorname { J } _ { 1 } \\left( { \\frac { - 1 } { B - 1 } } \\right) = { \\frac { 1 } { \\pi } } - { \\frac { 1 } { 2 ( B - 1 ) } } + O \\left( { \\frac { 1 } { ( B - 1 ) ^ { 2 } } } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 263, + 611, + 733, + 646 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\mathrm { J _ { 1 } } ( c ) = \\frac { 1 } { \\pi } ( \\sqrt { 1 - c ^ { 2 } } + ( \\pi - \\operatorname { a r c c o s } ( c ) ) c ) } \\end{array}$ is the arccosine kernel (Cho & Saul, 2009). ", + "bbox": [ + 168, + 652, + 777, + 670 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the appendix, $\\cdot$ also describes a trigonometric integral formula for general nonlinearities. ", + "bbox": [ + 173, + 679, + 820, + 695 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In the presence of batch normalization, when the activation function grows quickly, a winner-takeall phenomenon can occur where a subset of samples in the batch have much bigger activations than others. This causes the covariance matrix to form blocks of differing magnitude, breaking the BSB1 symmetry. One notices this, for example, as the degree $\\alpha$ of $\\alpha$ -ReLU (i.e. αth power of ReLU) increases past a point $\\alpha _ { \\mathrm { t r a n s i t i o n } } ( B )$ depending on the batch size $B$ (see Fig. 1). However, we observe, through simulations, that by far most of the nonlinearities used in practice, like ReLU, leaky ReLU, tanh, sigmoid, etc, all lead to BSB1 fixed points, and we can prove this rigorously for $\\phi = \\mathrm { i d }$ (see Corollary F.3). We discuss symmetry-broken fixed points (BSB2 fixed points) in Appendices J and K, but in the main text, from here on, ", + "bbox": [ + 173, + 700, + 825, + 827 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Assumption 1. Unless stated otherwise, we assume that any nonlinearity $\\phi$ mentioned induces $\\Sigma ^ { l }$ to converge to a BSB1 fixed point under the dynamics of Eq. (8). ", + "bbox": [ + 173, + 828, + 823, + 856 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1.1 LINEARIZED DYNAMICS ", + "text_level": 1, + "bbox": [ + 174, + 871, + 398, + 886 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "With the fixed point structure for batch normalized networks having been described, we now investigate the linearized dynamics of Eq. (8) in the vicinity of these fixed points. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To determine the eigenvalues of $\\frac { \\mathrm { d } \\mathrm { V } _ { B _ { \\phi } } } { \\mathrm { d } \\Sigma } \\Big | _ { \\Sigma = \\Sigma ^ { * } }$ it is helpful to consider the action of batch normalization in more detail. In particular, we notice that $B _ { \\phi }$ can be decomposed into the composition of three separate operations, $B _ { \\phi } = \\phi \\circ \\mathsf { n o } G$ . As discussed above, $G h$ subtracts the mean from $h$ and we introduce the new function $\\mathsf { n } ( h ) = \\sqrt { B } h / | | h | |$ which normalizes a centered $h$ by its standard deviation. Applying the chain rule, we can rewrite the Jacobian as, ", + "bbox": [ + 173, + 101, + 825, + 179 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/7f73c5f271c310cf61285786a7628e598d576a20622467f33f4a28273127f688.jpg", + "text": "$$\n\\frac { \\mathrm { d } \\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( \\Sigma ) } { \\mathrm { d } \\Sigma } = \\frac { \\mathrm { d } \\mathrm { V } _ { [ \\phi \\circ \\boldsymbol { \\mathsf { n } } ] } ( \\Sigma ^ { G } ) } { \\mathrm { d } \\Sigma ^ { G } } \\circ G ^ { \\otimes 2 }\n$$", + "text_format": "latex", + "bbox": [ + 385, + 186, + 612, + 220 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\circ$ denotes composition and $G ^ { \\otimes 2 }$ is the natural extension of $G$ to act on matrices as ${ \\cal G } ^ { \\otimes 2 } \\{ \\Sigma \\} =$ $G \\Sigma G = \\Sigma ^ { G }$ . It ends up being advantageous to study $\\begin{array} { r } { G ^ { \\otimes 2 } \\circ \\left. \\frac { \\mathrm { d } \\mathrm { V } _ { [ \\phi \\circ \\mathrm { n } ] } } { \\mathrm { d } \\Sigma } \\right| _ { \\Sigma = \\Sigma ^ { * } } \\circ G ^ { \\otimes 2 } = : G ^ { \\otimes 2 } \\circ \\hat { J } \\circ G ^ { \\otimes 2 } } \\end{array}$ and to note that the nonzero eigenvalues of this object are identical to the nonzero eigenvalues of the Jacobian (see Lemma F.17). ", + "bbox": [ + 173, + 227, + 825, + 295 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "At face value, this is a complicated object since it simultaneously has large dimension and possesses an intricate block structure. However, the permutation symmetry of the BSB1 $\\Sigma ^ { * }$ induces strong symmetries in $\\hat { J }$ that significantly simplify the analysis (see Appendix F.3). In particular while $\\hat { J } _ { i j k l }$ is a four-index object, we have $\\hat { \\hat { J } } _ { i j k l } = \\hat { J } _ { \\pi ( i ) \\pi ( j ) \\pi ( k ) \\pi ( l ) }$ for all permutations $\\pi$ on $B$ and $\\hat { J } _ { i j k l } = \\hat { J } _ { j i l k }$ . We call linear operators possessing such symmetries ultrasymmetric (Defn E.53) and show that all ultrasymmetric operators conjugated by $G ^ { \\otimes 2 }$ admit an eigendecomposition that contains three distinct eigenspaces with associated eigenvalues (see Thm E.62). ", + "bbox": [ + 173, + 300, + 825, + 409 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 3.6. Let $\\tau$ be an ultrasymmetric matrix operator. Then on the space of symmetric matrices, $G ^ { \\otimes 2 } \\circ \\mathcal { T } \\circ G ^ { \\otimes 2 }$ has the following orthogonal (under trace inner product) eigendecomposition, ", + "bbox": [ + 174, + 412, + 823, + 441 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "1. an eigenspace $\\{ \\Sigma : \\Sigma ^ { G } = 0 \\}$ with eigenvalue $O$ . ", + "bbox": [ + 212, + 452, + 547, + 469 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "2. a $^ { l }$ -dimensional eigenspace $\\mathbb { R } G$ with eigenvalue $\\lambda _ { G } ^ { G , T }$ . ", + "bbox": [ + 210, + 477, + 594, + 497 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3. a $( B - 1 )$ -dimensional eigenspace $\\mathbb { L } : = \\{ D ^ { G } : D$ diagonal, tr $D = 0 \\}$ , with eigenvalue $\\lambda _ { \\mathbb { L } } ^ { G , T }$ . ", + "bbox": [ + 207, + 506, + 823, + 540 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4. B(B−3)2 -dimensional eigenspace M := {Σ : ΣG = Σ, DiagΣ = 0}, with eigenvalue $\\lambda _ { \\mathbb { M } } ^ { G , T }$ . ", + "bbox": [ + 209, + 549, + 825, + 585 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The specific forms of the eigenvalues can be obtained as linear functions of the entries of $\\tau$ ; see $\\mathrm { T h m } \\mathrm { E } . 6 2 $ for details. Note that, as the eigenspaces are orthogonal, this implies that $G ^ { \\otimes 2 } \\circ T \\circ G ^ { \\otimes 2 }$ is self-adjoint (even when $\\tau$ is not). ", + "bbox": [ + 176, + 597, + 825, + 640 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "In our context with $\\mathcal { T } = \\hat { J }$ , the eigenspaces can be roughly interpreted as follows: The deviation $\\Delta \\Sigma = \\Sigma - \\Sigma ^ { * }$ from the fixed point decomposes as a linear combination of components in each of the eigenspaces. The $\\mathbb { R } G$ -component captures the average norm of elements of the batch (the trace of $\\Delta \\Sigma$ ), the $\\mathbb { L }$ -component captures the fluctuation of such norms, and the $\\mathbb { M }$ -component captures the covariances between elements of the batch. ", + "bbox": [ + 173, + 647, + 825, + 718 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Because of the explicit normalization of batchnorm, one sees immediately that the $\\mathbb { R } G$ -component goes to 0 after 1 step. For positive homogeneous $\\phi$ , we can use the Laplace method to obtain closed form expressions for the other eigenvalues (see Thm F.33). The below theorem shows that, as the batch size becomes larger, a deep ReLU-batchnorm network takes more layers to converge to a BSB1 fixed point. ", + "bbox": [ + 173, + 724, + 826, + 795 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Theorem 3.7. Let $\\phi =$ relu and $B > 3$ . The eigenvalues of $G ^ { \\otimes 2 } \\circ \\hat { J } \\circ G ^ { \\otimes 2 }$ for L and M are ", + "bbox": [ + 171, + 799, + 782, + 818 + ], + "page_idx": 6 + }, + { + "type": "equation", + "img_path": "images/834d9516f1ee5d3de593a6a113fe5eb5dcb5b564a1caa3ddf0cfdb3e7b65ec70.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\lambda _ { \\mathrm { L } } ^ { \\uparrow } = \\frac { 1 } { 2 ( B + 1 ) \\mu ^ { * } } \\left( ( B - 2 ) \\left[ 1 - { \\bf J } _ { 1 } \\left( \\frac { - 1 } { B - 1 } \\right) \\right] + \\frac { B } { B - 1 } { \\bf J } _ { 1 } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) ~ \\nearrow ~ 1 } } \\\\ { { \\displaystyle \\lambda _ { \\mathrm { M } } ^ { \\uparrow } = \\frac { B } { 2 ( B + 1 ) \\mu ^ { * } } { \\bf J } _ { 1 } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) ~ \\nearrow ~ \\frac { \\pi } { 2 ( \\pi - 1 ) } \\approx 0 . 7 3 3 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 823, + 769, + 896 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { \\mu ^ { * } = q ^ { * } ( 1 - c ^ { * } ) = \\frac { 1 } { 2 } \\left( 1 - \\operatorname { J } _ { 1 } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } \\end{array}$ , and $\\nearrow$ denotes increasing limit as $B \\to \\infty$ ", + "bbox": [ + 171, + 902, + 774, + 928 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "More generally, we can evaluate them for general nonlinearity using spherical integration (Appendix F.3.1) and, more enlightening, using the Gegenbauer method, is the following ", + "bbox": [ + 171, + 103, + 823, + 133 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Theorem 3.8. If φ(√B − 1x) has Gegenbauer expansion P∞l=0 al 1cB−1,l C $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) , } \\end{array}$ , then ", + "bbox": [ + 174, + 136, + 769, + 161 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/bf6797394f2f710950a084db12d4e3dd01cee025709e563a14971cd08eb9c8f0.jpg", + "text": "$$\n\\lambda _ { \\mathbb { L } } ^ { \\uparrow } = \\frac { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } w _ { B - 1 , l } + a _ { l } a _ { l + 2 } u _ { B - 1 , l } } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } v _ { B - 1 , l } } , \\qquad \\lambda _ { \\mathbb { M } } ^ { \\uparrow } = \\frac { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\tilde { w } _ { B - 1 , l } + a _ { l } a _ { l + 2 } \\tilde { u } _ { B - 1 , l } } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } v _ { B - 1 , l } }\n$$", + "text_format": "latex", + "bbox": [ + 204, + 166, + 764, + 204 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where the coefficients $w _ { B - 1 , l } , u _ { B - 1 , l } , \\tilde { w } _ { B - 1 , l } , \\tilde { u } _ { B - 1 , l } , v _ { B - 1 , l }$ are given in Thms $\\cdot$ and F.24. ", + "bbox": [ + 171, + 209, + 794, + 226 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "A BSB1 fixed point is not locally attracting if $\\lambda _ { \\mathbb { L } } ^ { \\uparrow } > 1$ or $\\lambda _ { \\mathbb { M } } ^ { \\uparrow } > 1$ . Thus $\\mathrm { T h m } 3 . 8$ yields insight on the stability of the BSB1 fixed point, which we can interpret heuristically as follows. The specific forms of the coefficients $w _ { B - 1 , l } , u _ { B - 1 , l } , \\tilde { w } _ { B - 1 , l } , \\tilde { u } _ { B - 1 , l } , v _ { B - 1 , l }$ show that $\\lambda _ { \\mathbb { M } } ^ { \\uparrow }$ is typically much smaller than $\\lambda _ { \\mathbb { L } } ^ { \\uparrow }$ (but there are exceptions like $\\phi = \\sin )$ , and $w _ { B - 1 , 1 } < v _ { B - 1 , 1 }$ but $w _ { B - 1 , l } \\geq v _ { B - 1 , l }$ for all $l \\geq 2$ . Thus one expects that, the larger $a _ { 1 }$ is, i.e. the “more linear” and less explosive $\\phi$ is, the smaller $\\lambda _ { \\mathbb { L } } ^ { \\uparrow }$ is and the more likely that Eq. (8) converges to a BSB1 fixed point. This is consistent with the “winner-take-all” intuition for the emergence of BSB2 fixed point explained above. See Appendix F.3.2 for more discussion. ", + "bbox": [ + 173, + 236, + 826, + 358 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.1.2 GRADIENT BACKPROPAGATION ", + "text_level": 1, + "bbox": [ + 176, + 372, + 446, + 387 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "With a mean field theory of the pre-activations of feed-forward networks with batch normalization having been developed, we turn our attention to the backpropagation of gradients. In contrast to the case of networks without batch normalization, we will see that exploding gradients at initialization are a severe problem here. To this end, one of the main results from this section will be to show that fully-connected networks with batch normalization feature exploding gradients for any choice of nonlinearity such that $\\Sigma ^ { l }$ converges to a BSB1 fixed point. Below, by rate of gradient explosion we mean the $\\beta$ such that the gradient norm squared grows as $\\beta ^ { L + o ( L ) }$ with depth $L$ . As before, all computations below assumes gradient independence (see Appendix B for a discussion). ", + "bbox": [ + 173, + 396, + 825, + 511 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As a starting point we seek an analog of Eq. (6) in the case of batch normalization. However, because the activation functions no longer act point-wise on the pre-activations, the backpropagation equation becomes, ", + "bbox": [ + 174, + 516, + 821, + 558 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/62a47d5fe107454a9a7efc5b3917e32fd26a68428f95a6078dafb5054a41cfc1.jpg", + "text": "$$\n\\delta _ { \\alpha i } ^ { l } = \\sum _ { \\beta j } \\frac { \\partial B _ { \\phi } ( \\pmb { h } _ { \\alpha } ^ { l } ) _ { j } } { \\partial h _ { \\alpha i } ^ { l } } W _ { \\beta \\alpha } ^ { l + 1 } \\delta _ { \\beta j } ^ { l + 1 }\n$$", + "text_format": "latex", + "bbox": [ + 392, + 556, + 606, + 598 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where $\\pmb { h } _ { \\alpha } ^ { l } = ( h _ { \\alpha 1 } ^ { l } , \\dots , h _ { \\alpha B } ^ { l } )$ and we observe the additional sum over the batch. Computing the resulting covariance matrix $\\Pi ^ { l }$ , we arrive at the recurrence relation, ", + "bbox": [ + 171, + 602, + 825, + 633 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/512c1a3b20d0d025b9c9e2f9374181696204e905de4b0be70832e426b61eefa3.jpg", + "text": "$$\n\\Pi ^ { l } = \\mathbb { E } \\left[ \\left( \\frac { \\mathrm { d } \\mathcal { B } _ { \\phi } ( h ) } { \\mathrm { d } h } \\right) ^ { T } \\Pi ^ { l + 1 } \\frac { \\mathrm { d } \\mathcal { B _ { \\phi } } ( h ) } { \\mathrm { d } h } : h \\sim \\mathcal { N } ( 0 , \\Sigma ^ { l } ) \\right] = : \\mathrm { V } _ { \\mathcal { B _ { \\phi } ^ { \\prime } } } ( \\Sigma ^ { l } ) ^ { \\dagger } \\{ \\Pi ^ { l + 1 } \\}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 638, + 750, + 681 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "where we have defined the linear operator $\\mathrm { V } _ { F } ( \\Sigma ) ^ { \\dagger } \\{ \\cdot \\}$ such that $\\mathrm { V } _ { F } ( \\Sigma ) ^ { \\dagger } \\{ \\Pi \\} = \\mathbb { E } [ F _ { h } ^ { T } \\Pi F _ { h } : h \\sim$ $\\mathcal { N } ( 0 , \\Sigma ) ]$ for any vector-indexed linear operator $F _ { h }$ . As in the case of vanilla feed-forward networks, here we will be concerned with the behavior of gradients when $\\Sigma ^ { l }$ is close to its fixed point. We therefore study the asymptotic approximation to Eq. (17) given by $\\Pi ^ { l } = \\mathrm { V } _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ^ { * } ) ^ { \\dagger } \\{ \\hat { \\Pi } ^ { l + 1 } \\}$ . In this case the dynamics of $\\Pi$ are linear and are therefore naturally determined by the eigenvalues of $\\mathrm { V } _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ^ { * } ) ^ { \\dag }$ . ", + "bbox": [ + 173, + 688, + 826, + 777 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As in the forward case, batch normalization is the composition of three operations $B _ { \\phi } = \\phi \\circ \\mathsf { n o G }$ . Applying the chain rule, Eq. (17) can be rewritten as, ", + "bbox": [ + 171, + 784, + 823, + 814 + ], + "page_idx": 7 + }, + { + "type": "equation", + "img_path": "images/8e01058c87dc979621f29380955e93a3cab18206f008c6ac2a78ead62fd576b7.jpg", + "text": "$$\n\\nabla _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ) ^ { \\dagger } = G ^ { \\otimes 2 } \\circ \\mathbb { E } \\left[ \\left( \\frac { \\mathrm { d } ( \\phi \\circ \\boldsymbol { \\mathsf { n } } ) ( z ) } { \\mathrm { d } z } \\bigg | _ { z = G h } ^ { r } \\right) ^ { \\otimes 2 } : h \\sim \\mathcal { N } ( 0 , \\Sigma ) \\right] = : G ^ { \\otimes 2 } \\circ F ( \\Sigma )\n$$", + "text_format": "latex", + "bbox": [ + 227, + 819, + 769, + 869 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "with $F ( \\Sigma )$ appropriately defined. Note that since $G ^ { \\otimes 2 }$ is an idempotent operator, $( \\mathrm { V } _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ) ^ { \\dag } ) ^ { n } =$ $( G ^ { \\otimes 2 } \\circ F ( \\Sigma ) ) ^ { n } = ( G ^ { \\otimes 2 } \\circ F ( \\Sigma ) \\circ G ^ { \\otimes 2 } ) ^ { n - 1 } \\circ G ^ { \\otimes 2 } \\circ F ( \\Sigma ) .$ , so that it suffices to study the eigendecomposition of $G ^ { \\otimes 2 } \\circ F ( \\Sigma ^ { * } ) \\circ G ^ { \\otimes 2 }$ . Due to the symmetry of $\\Sigma ^ { * }$ , $F ( \\Sigma ^ { * } )$ is ultrasymmetric, so that ", + "bbox": [ + 173, + 876, + 825, + 925 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "$G ^ { \\otimes 2 } \\circ F ( \\Sigma ^ { * } ) \\circ G ^ { \\otimes 2 }$ has eigenspaces $\\mathbb { R } G , \\mathbb { L } , \\mathbb { M }$ and we can compute its eigenvalues via $\\cdot$ . More illuminating, however, is the Gegenbauer expansion (see Thm G.5). It requires a new identity $\\cdot$ involving Gegenbauer polynomials integrated over a sphere, which may be of independent interest. ", + "bbox": [ + 173, + 102, + 825, + 160 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Theorem 3.9 (Batchnorm causes gradient explosion). Suppose $\\phi ( \\sqrt { B - 1 } x ) , \\phi ^ { \\prime } ( \\sqrt { B - 1 } x ) \\in$ L2((1 − x2) B−32 ). If φ(√B − 1x) has Gegenbauer expansion P∞l=0 al 1cB−1,l $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ , then gradients explode at the rate of ", + "bbox": [ + 174, + 161, + 825, + 215 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/9fe4b50a12d598f014b241d1983868b30669d2c2aab8b562d23d743e2e4ff250.jpg", + "text": "$$\n\\lambda _ { G } ^ { \\downarrow } = \\frac { \\sum _ { l = 0 } ^ { \\infty } \\left( \\frac { l + B - 3 } { B - 3 } \\cdot l \\right) a _ { l } ^ { 2 } r _ { l } } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } r _ { l } }\n$$", + "text_format": "latex", + "bbox": [ + 395, + 218, + 601, + 263 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "where $r _ { l } = c _ { B - 1 , l } ^ { - 1 } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( 1 \\right) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) > 0$ . Consequently, for any non-constant $\\phi$ 6 G in which case gradients explode at the rate of (i.e. there is a $j > 0$ such that $a _ { j } \\neq 0 ,$ ), $\\lambda _ { G } ^ { \\downarrow } > 1$ $\\frac { B - 2 } { B - 3 }$ ; $\\phi$ minimizes . $\\lambda _ { G } ^ { \\downarrow }$ iff it is linear (i.e. $a _ { i } = 0 , \\forall i \\geq 2$ ), ", + "bbox": [ + 173, + 268, + 825, + 329 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "This contrasts starkly with the case of non-normalized fully-connected networks, which can use the weight and bias variances to control its mean field network dynamics (Poole et al., 2016; Schoenholz et al., 2016). As a corollary, we disprove the conjecture of the original batchnorm paper (Ioffe & Szegedy, 2015) that “Batch Normalization may lead the layer Jacobians to have singular values close to $1 ^ { \\circ }$ in the initialization setting, and in fact prove the exact opposite, that batchnorm forces the layer Jacobian singular values away from $^ { l }$ . ", + "bbox": [ + 173, + 338, + 825, + 424 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Appendix G.1 discusses the numerical evaluation of all eigenvalues, and as usual, the Laplace method yields closed forms for positive homogeneous $\\phi$ $\\begin{array} { c c c c c c c c c c c c c c c c } { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { { } } & { } & { { } } & { } & { { } } & { } & { { } } & { } & { } { } & { } & { } & { } { } & { } & { } & { } { } & { } & { } & { } & \\end{array}$ . We highlight the result for ReLU. ", + "bbox": [ + 176, + 429, + 823, + 472 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Theorem 3.10. In a ReLU-batchnorm network, the gradient norm explodes exponentially at the rate of ", + "bbox": [ + 173, + 476, + 823, + 505 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/706aa2a26b2bfdcad89433f6422b37d62feb2d018679c81690af476b5e7e9746.jpg", + "text": "$$\n{ \\frac { 1 } { B - 3 } } \\left( { \\frac { B - 1 + \\mathrm { J } _ { 1 } ^ { \\prime } \\left( { \\frac { - 1 } { B - 1 } } \\right) } { 1 - \\mathrm { J } _ { 1 } \\left( { \\frac { - 1 } { B - 1 } } \\right) } } - 1 \\right)\n$$", + "text_format": "latex", + "bbox": [ + 377, + 508, + 620, + 561 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "which decreases to $\\textstyle { \\frac { \\pi } { \\pi - 1 } } \\approx 1 . 4 6 7$ as $B \\infty$ . In contrast, for a linear batchnorm network, the gradient norm explodes exponentially at the rate of $\\frac { B - 2 } { B - 3 }$ , which goes to $^ { l }$ as $B \\infty$ . ", + "bbox": [ + 174, + 566, + 825, + 601 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Fig. 1 shows theory and simulation for ReLU gradient dynamics. ", + "bbox": [ + 173, + 611, + 599, + 626 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Weight Gradient While all of the above only study the gradient with respect to the hidden preactivations, Appendix L shows that the weight gradient norms at layer $l$ is just $\\langle \\Pi ^ { l } , \\mu ^ { \\ast } G \\rangle = \\mu ^ { \\ast } \\mathrm { t r } \\Pi ^ { l }$ , and thus by $\\mathrm { T h m } 3 . 9$ , the weight gradients explode as well at the same rate $\\lambda _ { G } ^ { \\downarrow }$ . ", + "bbox": [ + 173, + 640, + 825, + 686 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Effect of $\\epsilon$ as a hyperparameter In practice, $\\epsilon$ is usually treated as small constant and is not regarded as a hyperparameter to be tuned. Nevertheless, we can investigate its effect on gradient explosion. A straightforward generalization of the analysis presented above to the case of $\\epsilon > 0$ suggests somewhat larger $\\epsilon$ values than typically used can ameliorate (but not eliminate) gradient explosion problems. See Fig. 5(c,d). ", + "bbox": [ + 173, + 700, + 825, + 771 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.2 CROSS-BATCH DYNAMICS ", + "text_level": 1, + "bbox": [ + 174, + 786, + 398, + 801 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Forward In addition to analyzing the correlation between preactivations of samples in a batch, we also study the correlation between those of different batches. The dynamics Eq. (8) can be generalized to simultaneous propagation of $k$ batches (see Eq. (54) and Appendix H). ", + "bbox": [ + 174, + 813, + 825, + 856 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/37bebe19867e435f33d294412372bbaafd58befbdb2d69073c984dfb377b4a49.jpg", + "text": "$$\n\\widetilde { \\Sigma } ^ { l } = \\mathrm { V } _ { B _ { \\phi } ^ { \\oplus k } } ( \\widetilde { \\Sigma } ^ { l - 1 } ) = \\underset { ( h _ { 1 } , \\ldots , h _ { k } ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { l - 1 } ) } { \\mathbb { E } } ( \\mathcal { B } _ { \\phi } ( h _ { 1 } ) , \\ldots , \\mathcal { B } _ { \\phi } ( h _ { k } ) ) ^ { \\otimes 2 } , \\quad \\mathrm { w i t h ~ } \\widetilde { \\Sigma } ^ { l } \\in \\mathbb { R } ^ { k B \\times k B } .\n$$", + "text_format": "latex", + "bbox": [ + 186, + 861, + 784, + 890 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Here the domain of the dynamics is the space of block matrices, with diagonal blocks and offdiagonal blocks resp. representing within-batch and cross-batch covariance. We observe empirically ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/efb897665ab6775072dfc47255054a708efc5abdb3c4da6c9fe4aac78d8cc35b.jpg", + "image_caption": [ + "Figure 1: Numerical confirmation of theoretical predictions. (a,b) Comparison between theoretical prediction (dashed lines) and Monte Carlo simulations (solid lines) for the eigenvalues of the backwards Jacobian (see Thms 3.10 and G.12) as a function of batch size and the magnitude of gradients as a function of depth respectively for rectified linear networks. In each case Monte Carlo simulations are averaged over 200 sample networks of width 1000 and shaded regions denote 1 standard deviation. Dashed lines are shifted slightly for easier comparison. (c,d) Demonstration of the existence of a BSB1 to BSB2 symmetry breaking transition as a function of $\\alpha$ for $\\alpha$ -ReLU (i.e. the αth power of ReLU) activations. In (c) we plot the empirical variance of the diagonal and off-diagonal entries of the covariance matrix which clearly shows a jump at the transition. In (d) we conference paper at ICLR 2019plot representative covariance matrices for the two phases (BSB1 bottom, BSB2 top). " + ], + "image_footnote": [], + "bbox": [ + 176, + 104, + 812, + 219 + ], + "page_idx": 9 + }, + { + "type": "image", + "img_path": "images/6c04ca6ffc5dda281f31ee174cfcc17477c0b86d7ecdb3211affcd060b8c7d38.jpg", + "image_caption": [ + "Figure A.1: Batch norm leads to a chaotic input-output map with increasingFigure 2: Batch norm leads to a chaotic input-output map with increasing depth. A linear network with batch norm is shown acting on two minibatches of size 64 after ranetwork with batch norm is shown acting on two minibatches of size 64 after random orthogonal initialization. The datapoints in the minibatch are chosen to form a 2d circle in ininitialization. The datapoints in the minibatch are chosen to form a 2d circle in input space, except for one datapoint that is perturbed separately in each minibatch (leftmost datapointfor one datapoint that is perturbed separately in each minibatch (leftmost datapoint at input layer 0). c input-output map with increasing depth. A linearBecause the network is linear, for a given minibatch it performs an affine transformaBecause the network is linear, for a given minibatch it performs an affine transformation on its inputs n two minibatches of size 64 after random orthogonal– a circle in input space remains an ellipse throughout the network. However, d– a circle in input space remains an ellipse throughout the network. However, due to batch norm h are chosen to form a 2d circle in input space, exceptthe coefficients of that affine transformation change nonlinearly as the datapointsthe coefficients of that affine transformation change nonlinearly as the datapoints in the minibatch in each minibatch (leftmost datapoint at input layer 0).are changed. (a) Each pane shows a scatterplot of activations at a given layer fare changed. (a) Each pane shows a scatterplot of activations at a given layer for all datapoints throughout the network. However, due to batch normin the minibatch, projected onto the top two PCA directions. PCA directions arein the minibatch, projected onto the top two PCA directions. PCA directions are computed using change nonlinearly as the datapoints in the minibatchthe concatenation of the two minibatches. Due to the batch norm nonlinearity,the concatenation of the two minibatches. Due to the batch norm nonlinearity, minibatches that rplot of activations at a given layer for all datapointsare nearly identical in input space grow increasingly dissimilar with depth. Intuitiare nearly identical in input space grow increasingly dissimilar with depth. Intuitively, this chaotic PCA directions. PCA directions are computed usinginput-output map can be understood as the source of exploding gradients when batcinput-output map can be understood as the source of exploding gradients when batch norm is applied (a)Due to the batch norm nonlinearity, minibatches thatto very deep networks, since very small changes in an input correspond to very larto very deep networks, since very small changes in an input correspond to very large movements in network outputs. (b) The correlation between the two minibatches, as a functionnetwork outputs. (b) The correlation between the two minibatches, as a function of layer, for the nges in an input correspond to very large movements insame network. Despite having a correlation near one at the input layer, the two misame network. Despite having a correlation near one at the input layer, the two minibatches rapidly en the two minibatches, as a function of layer, for thedecorrelate with depth. See for a theoretical treadecorrelate with depth. See Appendix H for a theoretical treatment. " + ], + "image_footnote": [], + "bbox": [ + 196, + 409, + 808, + 517 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "A.1 VGG19 WITH BATCHNORM ON CIFAR100that for most nonlinearities used in practice like ReLU, or even for the identity function, ie no (b)ON CIFAR100pointwise nonlinearity, the global fixed point of this dynamics is cross-batch BSB1, with diagonal atch norm leads to a chaotic input-output map with increasing depth. A lineam causes gradient explosion, after the first few epochs,Even though at initialization time batchnorm causeBSB1 blocks, and off-diagonal entries all equal to the same constant $c _ { \\mathrm { C B } } ^ { * }$ ad: ", + "bbox": [ + 174, + 799, + 825, + 843 + ], + "page_idx": 9 + }, + { + "type": "equation", + "img_path": "images/23f17e5e3c1bcbbd76af590b931e5b470cd6ea49c0cc18d6df99ecf59d8ebb07.jpg", + "text": "$$\n\\widetilde { \\Sigma } ^ { * } = \\left( { \\begin{array} { c c c c } { \\Sigma ^ { * } } & { c _ { \\mathrm { C B } } ^ { * } } & { c _ { \\mathrm { C B } } ^ { * } } & { \\cdot \\cdot } \\\\ { c _ { \\mathrm { C B } } ^ { * } } & { \\Sigma ^ { * } } & { c _ { \\mathrm { C B } } ^ { * } } & { \\cdot \\cdot } \\\\ { c _ { \\mathrm { C B } } ^ { * } } & { c _ { \\mathrm { C B } } ^ { * } } & { \\Sigma ^ { * } } & { \\cdot \\cdot } \\\\ { \\vdots } & { \\vdots } & { \\vdots } & { \\ddots } \\end{array} } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 383, + 859, + 612, + 928 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "To interpret this phenomenon, note that, after mean centering of the preactivations, this covariance matrix becomes a multiple of identity. Thus, deep embedding of batches loses the mutual information between them in the input space. Qualitatively, this implies that two batches that are similar at the input to the network will become increasingly dissimilar — i.e. chaotic — as the signal propagates deep into the network. This loss in fact happens exponentially fast, as illustrated in Fig. 2, and as shown theoretically by Thm H.13, at a rate of $\\bar { \\lambda } _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } < 1$ . Formally, the linearized dynamics of Eq. (20) has an eigenspace $\\smash { \\widetilde { \\mathbb { M } } }$ given by all block matrices with zero diagonal blocks. The associated eigenvalue is $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow }$ . We shall return to $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow }$ shortly and give its Gegenbauer expansion $( \\mathrm { T h m } 3 . 1 1 )$ ). ", + "bbox": [ + 173, + 103, + 825, + 229 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Backward The gradients of two batches of input are correlated throughout the course of backpropagation. We hence also study the generalization of Eq. (17) to the $k$ -batch setting (see Appendix I): ", + "bbox": [ + 171, + 242, + 821, + 272 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/c963d60cdd78f2d39dd438bdeb581ec29df9e920046c508a41c11a258e1706df.jpg", + "text": "$$\n\\begin{array}{c} \\begin{array} { r l } & { \\tilde { \\Pi } ^ { l } = \\mathrm { V } _ { B _ { \\phi } ^ { \\Phi 2 } } ( \\widetilde \\Sigma ^ { l } ) ^ { \\dagger } \\{ \\tilde { \\Pi } ^ { l + 1 } \\} } \\\\ { \\mathrm { V } _ { B _ { \\phi } ^ { \\Phi 2 } } / \\widetilde \\Sigma ^ { * } ) ^ { \\dagger } \\{ ( \\begin{array} { l l } { \\Pi _ { 1 } } & { \\Xi } \\\\ { \\Xi ^ { T } } & { \\Pi _ { 2 } } \\end{array} ) \\} = \\underset { ( x , y ) \\sim \\mathcal { N } ( 0 , \\widetilde \\Sigma ^ { * } ) } { \\mathbb { E } } [ \\binom { \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) ^ { T } \\Pi _ { 1 } \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) } { \\mathcal { B } _ { \\phi } ^ { \\prime } ( y ) ^ { T } \\Xi ^ { T } \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) } } & { \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) ^ { T } \\Xi \\mathcal { B } _ { \\phi } ^ { \\prime } ( y ) } \\end{array} ) ] \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 192, + 277, + 779, + 343 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "where for simplicity of exposition and without loss of generality, we take $k = 2$ . Like in Eq. (17), we have assumed that $\\widetilde { \\Sigma } ^ { l }$ has converged to its limit $\\widetilde { \\Sigma } ^ { * }$ . It is not hard to see that each block of $\\tilde { \\Pi } ^ { l }$ evolves independently, with the diagonal blocks in particular evolving according to Eq. (17). Analyzing the off-diagonal blocks might seem unwiby scalar multiplication by a constant $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow }$ at first, but it turns out that its dynamics is given simply(Thm I.4). Even more surprisingly, ", + "bbox": [ + 173, + 347, + 825, + 425 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Theorem 3.11 (Batchnorm causes information loss). The convergence rate of cross-batch preactivation covariance $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow }$ is equal to the decay rate of the cross-batch gradient covariance $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow }$ . If $\\phi ( { \\sqrt { B - 1 } } x )$ has Gegenbauer expansion $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) , } \\end{array}$ , then both are given by ", + "bbox": [ + 174, + 428, + 825, + 487 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/3b0ac34b419867e262f6c16fc6849f2808a385518855a29ad7e05d2b9d4b6ff0.jpg", + "text": "$$\n\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } = \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow } = \\frac { \\frac { 1 } { 2 \\pi } B ( B - 1 ) a _ { 1 } ^ { 2 } \\operatorname { B e t a } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { 2 } } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 493, + 707, + 542 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "This quantity is always $< 1$ , and for any fixed $B$ , is maximized by $\\phi = \\mathrm { i d } .$ . Furthermore, for $\\phi = \\mathrm { i d } ,$ , $\\begin{array} { r } { \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } = \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow } = \\frac { B - 1 } { 2 } \\mathrm { P } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { - 2 } } \\end{array}$ and increase to $^ { l }$ from below as $B \\infty$ . ", + "bbox": [ + 173, + 547, + 825, + 583 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Thus, a deep batchnorm network loses correlation information between two input batches, exponentially fast in depth, no matter what nonlinearity (that induces fixed point of the form given by Eq. (21)). This again contrasts with the case for vanilla networks which can control this rate of information loss by tweaking the initialization variances for weights and biases Poole et al. (2016); Schoenholz et al. (2016). The absence of coordinatewise nonlinearity, i.e. $\\phi = \\mathrm { i d }$ , maximally suppresses both this loss of information as well as the gradient explosion, and in the ideal, infinite batch scenario, can cure both problems. ", + "bbox": [ + 173, + 593, + 825, + 693 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 713, + 326, + 728 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Having developed a theory for neural networks with batch normalization at initialization, we now explore the relationship between the properties of these random networks and their learning dynamics. We will see that the trainability of networks with batch normalization is controlled by gradient explosion. We quantify the depth scale over which gradients explode by $\\xi = 1 / \\log \\lambda _ { G } ^ { \\downarrow }$ where, as above, $\\lambda _ { G } ^ { \\downarrow }$ is the largest eigenvalue of the jacobian. Across many different experiments we will see strong agreement between $\\xi$ and the maximum trainable depth. ", + "bbox": [ + 174, + 743, + 825, + 833 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "We first investigate the relationship between trainability and initialization for rectified linear networks as a function of batch size. The results of these experiments are shown in Fig. 3 where in each case we plot the test accuracy after training as a function of the depth and the batch size and overlay $1 6 \\xi$ in white dashed lines. In Fig. 3 (a) we consider networks trained using SGD on MNIST where we observe that networks deeper than about 50 layers are untrainable regardless of batch size. In (b) we compare standard batch normalization with a modified version in which the batch size is held fixed but batch statistics are computed over subsets of size $B$ . This removes subtle gradient fluctuation effects noted in Smith & Le (2018). In (c) we do the same experiment with RMSProp and in (d) we train the networks on CIFAR10. In all cases we observe a nearly identical trainable region. ", + "bbox": [ + 174, + 839, + 825, + 924 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/06fb6d785031a6f5279637e6f03b5a8e24426114f3d6c70dee999e09b27a7afe.jpg", + "image_caption": [ + "Figure 3: Batch normalization strongly limits the maximum trainable depth. Colors show test accuracy for rectified linear networks with batch normalization and $\\gamma = 1$ , $\\beta = 0$ , $\\epsilon = 1 0 ^ { - 3 }$ , $N = 3 8 4$ , and $\\eta \\ : = \\ : 1 0 ^ { - 5 } B$ . (a) trained on MNIST for 10 epochs (b) trained with fixed batch size 1000 and batch statistics computed over sub batches of size $B$ . (c) trained using RMSProp. (d) Trained on CIFAR10 for 50 epochs. In each case, White dashed line indicates a theoretical prediction of trainable depth, as discussed in the text. " + ], + "image_footnote": [], + "bbox": [ + 181, + 102, + 812, + 232 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/a5f30b83245a38e0f886a3bb347ed85b4c6ef7d3156584dc8f17e6f2ec118c68.jpg", + "image_caption": [ + "Figure 4: Gradients in networks with batch normalization quickly achieve dynamical equilibrium. Plots of the relative magnitudes of (a) the weights (b) the gradients of the loss with respect to the pre-activations and (c) the gradients of the loss with respect to the weights for rectified linear networks of varying depths during the first 10 steps of training. Colors show step number from 0 (black) to 10 (green). " + ], + "image_footnote": [], + "bbox": [ + 181, + 348, + 818, + 463 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 574, + 825, + 631 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "It is counter intuitive that training can occur at intermediate depths, from 10 to 50 layers, where there is significant gradient explosion. To gain insight into the behavior of the network during learning we record the magnitudes of the weights, the gradients with respect to the pre-activations, and the gradients with respect to the weights for the first 10 steps of training for networks of different depths. The result of this experiment is shown in Fig. 4. Here we see that before learning, as expected, the norm of the weights is constant and independent of layer while the gradients feature exponential explosion. However, we observe that two related phenomena occur after a single step of learning: the weights grow exponentially in the depth and the magnitude of the gradients are stable up to some threshold after which they vanish exponentially in the depth. This is as the result of the scaling property of batchnorm, where $B _ { \\phi } ^ { \\prime } ( \\alpha \\bar { h } ) = \\alpha ^ { - 1 } \\bar { B } _ { \\phi } ^ { \\prime } ( h )$ : The first-step gradients dominate the weights due to gradient explosion, hence the exponential growth in weight norms, and thereafter, the gradients are scaled down commensurately. Thus, it seems that although the gradients of batch normalized networks at initialization are ill-conditioned, the gradients appear to quickly reach a stable dynamical equilibrium. While this appears to be beneficial for shallower networks, in deeper ones, the relative gradient vanishing can in fact be so severe as to cause lower layers to mostly stay constant during training. Aside from numerical issues, this seems to be the primary mechanism through which gradient explosion causes training problems for networks deeper than 50 layers. ", + "bbox": [ + 174, + 638, + 825, + 875 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "As discussed in the theoretical exposition above, batch normalization necessarily features exploding gradients for any nonlinearity that converges to a BSB1 fixed point. We performed a number of experiments exploring different ways of ameliorating this gradient explosion. These experiments are shown in Fig. 5 with theoretical predictions for the maximum trainable depth overlaid; in all cases we see exceptional agreement. In Fig. 5 (a,b) we explore two different ways of tuning the degree to which activation functions in a network are nonlinear. In Fig. 5 (a) we tune $\\gamma \\in [ 0 , 2 ]$ for networks with tanh-activations and note that in the $\\gamma 0$ limit the function is linear. In Fig. 5 (b) we tune $\\beta \\in [ 0 , 2 ]$ for networks with rectified linear activations and we note, similarly, that in the $\\beta \\to \\infty$ limit the function is linear. As expected, we see the maximum trainable depth increase significantly with decreasing $\\gamma$ and increasing $\\beta$ . In Fig. 5 (c,d) we vary $\\epsilon$ for tanh and rectified linear networks respectively. In both cases, we observe a critical point at large $\\epsilon$ where gradients do not explode and very deep networks are trainable. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/719edc40217e94666f7845a907bb79a6913bae0e8aebc61292cfa994eedb0428.jpg", + "image_caption": [ + "Figure 5: Three techniques for counteracting gradient explosion. Test accuracy on MNIST as a function of different hyperparameters along with theoretical predictions (white dashed line) for the maximum trainable depth. (a) tanh network changing the overall scale of the pre-activations, here $\\gamma 0$ corresponds to the linear regime. (b) Rectified linear network changing the mean of the preactivations, here $\\beta \\to \\infty$ corresponds to the linear regime. (c,d) tanh and rectified linear networks respectively as a function of $\\epsilon$ , here we observe a well defined phase transition near $\\epsilon \\sim 1$ . Note that in the case of rectified linear activations we use $\\beta = 2$ so that the function is locally linear about 0. We also find initializing $\\beta$ and/or setting $\\epsilon > 0$ having positive effect on VGG19 with batchnorm. See Figs. 8 and 9. " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 821, + 229 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 397, + 825, + 522 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 544, + 318, + 560 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "In this work we have presented a theory for neural networks with batch normalization at initialization. In the process of doing so, we have uncovered a number of counterintuitive aspects of batch normalization and – in particular – the fact that at initialization it unavoidably causes gradients to explode with depth. We have introduced several methods to reduce the degree of gradient explosion, enabling the training of significantly deeper networks in the presence of batch normalization. Finally, this work paves the way for future work on more advanced, state-of-the-art, network architectures and topologies. 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", + "bbox": [ + 174, + 835, + 825, + 864 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/cfaf38a6f9689f6177902ab53192b119e12ff08a39f90e76ac5c89eb19aa50b8.jpg", + "image_caption": [ + "Figure 6: Batch norm leads to a chaotic input-output map with increasing depth. A linear network with batch norm is shown acting on two minibatches of size 64 after random orthogonal initialization. The datapoints in the minibatch are chosen to form a 2d circle in input space, except for one datapoint that is perturbed separately in each minibatch (leftmost datapoint at input layer 0). Because the network is linear, for a given minibatch it performs an affine transformation on its inputs – a circle in input space remains an ellipse throughout the network. However, due to batch norm the coefficients of that affine transformation change nonlinearly as the datapoints in the minibatch are changed. (a) Each pane shows a scatterplot of activations at a given layer for all datapoints in the minibatch, projected onto the top two PCA directions. PCA directions are computed using the concatenation of the two minibatches. Due to the batch norm nonlinearity, minibatches that are nearly identical in input space grow increasingly dissimilar with depth. Intuitively, this chaotic input-output map can be understood as the source of exploding gradients when batch norm is applied to very deep networks, since very small changes in an input correspond to very large movements in network outputs. $( b )$ The correlation between the two minibatches, as a function of layer, for the same network. Despite having a correlation near one at the input layer, the two minibatches rapidly decorrelate with depth. See Appendix H for a theoretical treatment. " + ], + "image_footnote": [], + "bbox": [ + 285, + 102, + 707, + 587 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "A VGG19 WITH BATCHNORM ON CIFAR100 ", + "text_level": 1, + "bbox": [ + 174, + 849, + 568, + 866 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Even though at initialization time batchnorm causes gradient explosion, after the first few epochs, the relative gradient norms $\\| \\nabla _ { \\theta } L \\| / \\| \\theta \\|$ for weight parameters $\\theta = W$ or BN scale parameter $\\theta = \\gamma$ , equilibrate to about the same magnitude. See Fig. 7. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/1271c37190d4a136d372e3f3a01c0fdc8b9c25f650bc101aedd4505499dee4d8.jpg", + "image_caption": [ + "Figure 7: relative gradient norms of different parameters in layer order (input to output from left to right), with $\\gamma$ and $W$ interleaving. From dark to light blue, each curve is separated by (a) 3, (b) 5, or (c) 10 epochs. We see that after 10 epochs, the relative gradient norms of both $\\gamma$ and $W$ for all layers become approximately equal despite gradient explosion initially. " + ], + "image_footnote": [], + "bbox": [ + 184, + 111, + 795, + 209 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We find acceleration effects, especially in initial training, due to setting $\\epsilon > 0$ and/or initializing $\\beta > 0$ . See Figs. 8 and 9. ", + "bbox": [ + 174, + 308, + 823, + 337 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/053708b11fbf0db9b129fca4bc8269325ed80518835b7c1df3d1b09d39879d2d.jpg", + "image_caption": [ + "Figure 8: We sweep over different values of learning rate, $\\beta$ initialization, and $\\epsilon$ , in training VGG19 with batchnorm on CIFAR100 with data augmentation. We use 8 random seeds for each combination, and assign to each combination the median training/validation accuracy over all runs. We then aggregate these scores here. In the first row we look at training accuracy with different learning rate vs $\\beta$ initialization at different epochs of training, presenting the max over $\\epsilon$ . In the second row we do the same for validation accuracy. In the third row, we look at the matrix of training accuracy for learning rate vs $\\epsilon$ , taking max over $\\beta$ . In the fourth row, we do the same for validation accuracy. " + ], + "image_footnote": [], + "bbox": [ + 176, + 352, + 820, + 686 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "B GRADIENT INDEPENDENCE ASSUMPTION ", + "text_level": 1, + "bbox": [ + 174, + 833, + 553, + 848 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "Following prior literature Schoenholz et al. (2016); Yang & Schoenholz (2017); Xiao et al. (2018), in this paper, in regards to computations involving backprop, we assume \nAssumption 2. During backpropagation, whenever we multiply by $W ^ { T }$ for some weight matrix $W$ , we multiply by an iid copy instead. ", + "bbox": [ + 169, + 864, + 826, + 924 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/8831ac5316669b9a2076ca0903ff531318abb9c8e1e8e0d2529f5e15666bad87.jpg", + "image_caption": [ + "Figure 9: In the same setting as Fig. 8, except we don’t take the max over the unseen hyperparameter but rather set it to 0 (the default value). " + ], + "image_footnote": [], + "bbox": [ + 176, + 99, + 820, + 436 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "As in these previous works, we find excellent agreement between computations made under this assumption and the simulations (see Fig. 1). Yang (2019) in fact recently rigorously justified this assumption as used in the computation of moments, for a wide variety of architectures, like multilayer perceptron (Schoenholz et al., 2016), residual networks (Yang & Schoenholz, 2017), and convolutional networks (Xiao et al., 2018) studied previously, but without batchnorm. The reason that their argument does not extend to batchnorm is because of the singularity in its Jacobian at 0. However, as observed in our experiments, we expect that a proof can be found to extend Yang & Schoenholz (2017)’s work to batchnorm. ", + "bbox": [ + 173, + 511, + 825, + 623 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "C NOTATIONS ", + "text_level": 1, + "bbox": [ + 174, + 650, + 305, + 666 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Definition C.1. Let $S _ { B }$ be the space of PSD matrices of size $B \\times B$ . Given a measurable function $\\Phi : \\mathbb { R } ^ { B } \\mathbb { R } ^ { A }$ , define the integral transform $\\mathrm { V } _ { \\Phi } : { \\mathcal { S } } _ { B } \\to { \\mathcal { S } } _ { A }$ by $\\mathrm { V } _ { \\Phi } ( \\Sigma ) = \\mathbb { E } [ \\Phi ( h ) ^ { \\otimes 2 } : h \\sim$ $\\mathcal { N } ( 0 , \\Sigma ) ]$ .3 When $\\phi : \\mathbb { R } \\mathbb { R }$ and $B$ is clear from context, we also write $\\mathrm { V } _ { \\phi }$ for $\\mathrm { V }$ applied to the function acting coordinatewise by $\\phi$ . ", + "bbox": [ + 173, + 685, + 825, + 742 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Definition C.2. For any $\\phi ~ : ~ \\mathbb { R } ~ \\to ~ \\mathbb { R }$ , let $B _ { \\phi } \\quad : \\quad \\mathbb { R } ^ { B } \\quad \\to \\quad \\mathbb { R } ^ { B }$ be batchnorm (applied to a batch of neuronal activations) followed by coordinatewise applications of $\\phi$ , $B _ { \\phi } ( x ) _ { j } ~ =$ $\\begin{array} { r } { \\phi \\left( \\left( x _ { j } - \\operatorname { A v g } x \\right) / \\sqrt { \\frac { 1 } { B } \\sum _ { i = 1 } ^ { B } \\left( x _ { i } - \\operatorname { A v g } x \\right) ^ { 2 } } \\right) } \\end{array}$ (here $\\begin{array} { r } { \\operatorname { A v g } x = \\frac { 1 } { B } \\sum _ { i = 1 } ^ { B } x _ { i } ) } \\end{array}$ . When $\\phi = \\mathrm { i d }$ we will also write $\\boldsymbol { B } = \\boldsymbol { B } _ { \\mathrm { i d } }$ . ", + "bbox": [ + 173, + 748, + 825, + 824 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Definition C.3. Define the matrix $\\begin{array} { r } { G ^ { B } = I - \\frac { 1 } { B } \\mathbf { 1 } \\mathbf { 1 } ^ { T } } \\end{array}$ . Let $\\mathcal { S } _ { B } ^ { G }$ be the space of PSD matrices of size $B \\times B$ with zero mean across rows and columns, $S _ { B } ^ { G } : = \\{ \\Sigma \\in { \\cal S } _ { B } : G ^ { B } \\Sigma G ^ { B } = \\Sigma \\} = \\{ \\Sigma \\in { \\cal S } _ { B } :$ $\\Sigma \\mathbf { 1 } = 0 \\}$ . ", + "bbox": [ + 173, + 828, + 825, + 876 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "When $B$ is clear from context, we will suppress the subscript/superscript $B$ . In short, for $h \\in \\mathbb { R } ^ { B }$ , $G h$ zeros the sample mean of $h$ . $G$ is a projection matrix to the subspace of vectors $\\boldsymbol { h } \\in \\mathbb { R } ^ { B }$ of zero coordinate sum. With the above definitions, we then have $B _ { \\phi } ( h ) = \\bar { \\phi } ( \\sqrt { B } G h / \\| G h \\| )$ . ", + "bbox": [ + 174, + 103, + 825, + 148 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We use $\\Gamma$ to denote the Gamma function, $\\mathrm { P } \\left( a , b \\right) : = \\Gamma ( a + b ) / \\Gamma ( a )$ to denote the Pochhammer symbol, and $\\begin{array} { r } { \\mathrm { B e t a } ( a , b ) = \\frac { \\Gamma ( a ) \\Gamma ( b ) } { \\Gamma ( a + b ) } } \\end{array}$ to denote the Beta function. We use $\\langle \\cdot , \\cdot \\rangle$ to denote the dot product for vectors and trace inner product for matrices. Sometimes when it makes for simpler notation, we will also use $\\cdot$ for dot product. ", + "bbox": [ + 174, + 154, + 825, + 217 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We adopt the matrix convention that, for a multivariate function $f : \\mathbb { R } ^ { n } \\mathbb { R } ^ { m }$ , the Jacobian is ", + "bbox": [ + 173, + 222, + 794, + 238 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/e9afb55feef1ec6e078b2953f53b666d562e77709e182b4a338f9e1a53c52212.jpg", + "text": "$$\n\\frac { \\mathrm { d } f } { \\mathrm { d } x } = \\left( \\begin{array} { c c c c } { \\frac { \\partial f _ { 1 } } { \\partial x _ { 1 } } } & { \\frac { \\partial f _ { 1 } } { \\partial x _ { 2 } } } & { \\cdot \\cdot \\cdot } & { \\frac { \\partial f _ { 1 } } { \\partial x _ { n } } } \\\\ { \\frac { \\partial f _ { 2 } } { \\partial x _ { 1 } } } & { \\frac { \\partial f _ { 2 } } { \\partial x _ { 2 } } } & { \\cdot \\cdot \\cdot } & { \\frac { \\partial f _ { 2 } } { \\partial x _ { n } } } \\\\ { \\vdots } & { \\vdots } & { \\ddots } & { \\vdots } \\\\ { \\frac { \\partial f _ { m } } { \\partial x _ { 1 } } } & { \\frac { \\partial f _ { m } } { \\partial x _ { 2 } } } & { \\cdot \\cdot \\cdot } & { \\frac { \\partial f _ { m } } { \\partial x _ { n } } } \\end{array} \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 377, + 244, + 617, + 327 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "so that $\\begin{array} { r } { f ( x + \\Delta x ) = \\frac { \\mathrm { d } f } { \\mathrm { d } x } \\Delta x + o ( \\Delta x ) } \\end{array}$ , where $x$ and $\\Delta x$ are treated as column vectors. In what follows we further abbreviate $\\begin{array} { r } { \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) = \\frac { \\mathrm { d } \\mathcal { B } _ { \\phi } ( z ) } { \\mathrm { d } z } \\big | _ { z = x } , \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) \\Delta x = \\frac { \\mathrm { d } \\mathcal { B } _ { \\phi } ( z ) } { \\mathrm { d } z } \\big | _ { z = x } \\Delta x } \\end{array}$ . ", + "bbox": [ + 173, + 332, + 825, + 371 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "D A GUIDE TO THE REST OF THE APPENDIX ", + "text_level": 1, + "bbox": [ + 173, + 387, + 560, + 405 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "As discussed in the main text, we are interested in several closely related dynamics induced by batchnorm in a fully-connected network with random weights. One is the forward propagation equation Eq. (43) ", + "bbox": [ + 173, + 420, + 825, + 463 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/ae1292c6b9ba34cd7ac54160612ba842d73471e225d34c1f2269598445c2c70c.jpg", + "text": "$$\n\\Sigma ^ { l } = \\mathrm { V } _ { \\mathcal { B _ { \\phi } } } ( \\Sigma ^ { l - 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 439, + 469, + 557, + 489 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "studied in Appendix F. Another is the backward propagation equation Eq. (51) ", + "bbox": [ + 173, + 496, + 689, + 511 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/24d4f5fb08d3ad23bd9aebb8af742e4f9843f6514450eed10a20a89208072d24.jpg", + "text": "$$\n\\Pi ^ { l } = \\mathrm { V } _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ^ { * } ) ^ { \\dagger } \\{ \\Pi ^ { l + 1 } \\}\n$$", + "text_format": "latex", + "bbox": [ + 418, + 517, + 580, + 540 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "studied in Appendix G. We will also study their generalizations to the simultaneous propagation of multiple batches in Appendices $\\cdot$ and I. ", + "bbox": [ + 173, + 546, + 825, + 574 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "In general, we discover several ways to go about such analyses, each with their own benefits and drawbacks: ", + "bbox": [ + 173, + 580, + 823, + 609 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• The Laplace Method (Appendix E.1): If the nonlinearity $\\phi$ is positive homogeneous, we can use Schwinger’s parametrization to turn $\\mathrm { V } _ { B _ { \\phi } }$ and $\\mathrm { V } _ { B _ { \\phi } ^ { \\prime } }$ into nicer forms involving only 1 or 2 integrals, as mentioned in the main text. All relevant quantities, such as eigenvalues of the backward dynamics, can be further simplified to closed forms. Lemma E.2 gives the master equation for the Laplace method. Because the Laplace method requires $\\phi$ to be positive homogeneous to apply, we review relevant results of such functions in Appendices E.1.1 and E.1.2. ", + "bbox": [ + 215, + 621, + 825, + 722 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "The Fourier Method (Appendix E.2): For polynomially bounded and continuous nonlinearity $\\phi$ (for the most general set of conditions, see Thm E.25), we can obtain similar simplifications using the Fourier expansion of the delta function, with the penalty of an additional complex integral. ", + "bbox": [ + 217, + 727, + 823, + 784 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• Spherical Integration (Appendix E.3): Batchnorm can naturally be interpreted as a linear projection followed by projection to a sphere of radius $\\sqrt { B }$ . Assuming that the forward dynamics converges to a BSB1 fixed point, one can express $\\mathrm { V } _ { B _ { \\phi } }$ and $\\mathrm { V } _ { B _ { \\phi } ^ { \\prime } }$ as spherical integrals, for all measurable $\\phi$ . One can reduce this $( B - 1 )$ -dimensional integral, in spherical angular coordinates, into 1 or 2 dimensions by noting that the integrand depending on $\\phi$ only depends on 1 or 2 angles. Thus this method is very suitable for numerical evaluation of quantities of interest for general $\\phi$ . ", + "bbox": [ + 215, + 787, + 826, + 890 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "• The Gegenbauer Method (Appendix E.4): Gegenbauer polynomials are orthogonal polynomials under the weight $( 1 - x ^ { 2 } ) ^ { \\alpha }$ for some $\\alpha$ . They correspond to a special type of spherical harmonics called zonal harmonics (Defn E.44) that has the very useful reproducing property (Fact E.46). By expressing the nonlinearity $\\phi$ in this basis, we can see easily that the relevant eigenvalues of the forward and backward dynamics are ratios of two quadratic forms of $\\phi$ ’s Gegenbauer coefficients. In particular, for the largest eigenvalue of the backward dynamics, the two quadratic forms are both diagonal, and in a way that makes apparent the necessity of gradient explosion (under the BSB1 fixed point assumption). In regards to numerical computations, the Gegenbauer method has the benefit of only requiring 1-dimensional integrals to obtain the coefficients, but if $\\phi$ has slowlydecaying coefficients, then a large number of such integrals may be required to get an accurate answer. ", + "bbox": [ + 217, + 895, + 823, + 924 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 232, + 103, + 823, + 242 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In each of the following sections, we will attempt to conduct an analysis with each method when it is feasible. ", + "bbox": [ + 174, + 262, + 823, + 290 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "When studying the convergence rate of the forward dynamics and the gradient explosion of the backward dynamics, we encounter linear operators on matrices that satisfy an abundance of symmetries, called ultrasymmetric operators. We study these operators in Appendix E.5 and contribute several structural results Thms E.61, E.62, E.72 and E.72 on their eigendecomposition that are crucial to deriving the asymptotics of the dynamics. ", + "bbox": [ + 174, + 297, + 825, + 368 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "All of the above will be explained in full in the technique section Appendix E. We then proceed to study the forward and backward dynamics in detail (Appendices F to I), now armed with tools we need. Up to this point, we only consider the dynamics assuming Eq. (43) converges to a BSB1 fixed point, and $B \\geq 4$ . In Appendix J we discuss what happens outside this regime, and in Appendix K we show our current understanding of BSB2 fixed point dynamics. ", + "bbox": [ + 174, + 375, + 825, + 444 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Note that the backward dynamics studied in these sections is that of the gradient with respect to the hidden preactivations. Nevertheless, we can compute the moments of the weight gradients from this, which is of course what is eventually used in gradient descent. We do so in the final section Appendix L. ", + "bbox": [ + 174, + 450, + 825, + 507 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Main Technical Results Corollary F.3 establishes the global convergence of Eq. (43) to BSB1 fixed point for $\\phi = \\mathrm { i d }$ (as long as the initial covariance is nondegenerate), but we are not able to prove such a result for more general $\\phi$ . We thus resort to studying the local convergence properties of BSB1 fixed points. ", + "bbox": [ + 174, + 541, + 823, + 597 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "First, Thms F.5, F.8 and F.13 respectively compute the BSB1 fixed point from the perspectives of spherical integration, the Laplace method, and the Gegenbauer method. Thms F.22 and F.24 give the Gegenbauer expansion of the BSB1 local convergence rate, and Thm F.33 gives a more succinct form of the same for positive homogeneous $\\phi$ . Appendix F.3.1 discusses how to do this computation via spherical integration. ", + "bbox": [ + 174, + 604, + 825, + 674 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Next, we study the gradient dynamics, and Thms G.5 and G.12 yield the gradient explosion rates with the Gegenbauer method and the Laplace method (for positive homogeneous $\\phi ,$ , respectively. Appendix G.1.1 discusses how to compute this for general $\\phi$ using spherical integration. ", + "bbox": [ + 174, + 680, + 823, + 723 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We then turn to the dynamics of cross batch covariances. Thm H.3 gives the form of the cross batch fixed point via spherical integration and the Gegenbauer method, and Thm H.6 yields a more specific form for positive homogeneous $\\phi$ via the Laplace method. The local convergence rate to this fixed point is given by Thms H.13 and H.14, respectively using the Gegenbauer method and the Laplace method. Finally, the correlation between gradients of the two batches is shown to decrease exponentially in Corollary I.6 using the Gegenbauer method, and the decay rate is given succinctly for positive homogeneous $\\phi$ in Thm I.3. ", + "bbox": [ + 174, + 729, + 825, + 828 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "E TECHNIQUES ", + "text_level": 1, + "bbox": [ + 174, + 866, + 316, + 882 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "We introduce the key techniques and tools in our analysis in this section. ", + "bbox": [ + 174, + 909, + 648, + 924 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "As discussed above, the Laplace method is useful for deriving closed form expressions for positive homogeneous $\\phi$ . The key insight here is to apply Schwinger parametrization to deal with normalization. ", + "bbox": [ + 173, + 128, + 826, + 172 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Lemma E.1 (Schwinger parametrization). For $z > 0$ and $c > 0$ , ", + "bbox": [ + 171, + 175, + 599, + 190 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/5aa218a4b5a47faf61f58f810422641a6f4af5d538ca8bd7f4908e100b97d2f3.jpg", + "text": "$$\nc ^ { - z } = \\Gamma ( z ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } x ^ { z - 1 } e ^ { - c x } \\mathrm { d } x\n$$", + "text_format": "latex", + "bbox": [ + 388, + 196, + 611, + 231 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "The following is the key lemma in the Laplace method. ", + "bbox": [ + 173, + 244, + 537, + 260 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Lemma E.2 (The Laplace Method Master Equation). For $A , A ^ { \\prime } \\in \\mathbb { N } ,$ , let $f : \\mathbb { R } ^ { A } \\to \\mathbb { R } ^ { A ^ { \\prime } }$ and let $k \\geq 0$ . Suppose $\\| f ( \\bar { y } ) \\| \\leq h ( \\| y \\| )$ for some nondecreasing function $h : \\mathbb { R } ^ { \\geq 0 } \\mathbb { R } ^ { \\geq 0 }$ such that $\\mathbb { E } [ h ( r \\| y \\| ) : y \\in \\dot { \\mathcal { N } } ( 0 , \\dot { I } _ { A } ) ]$ exists for every $r \\geq 0$ . Define $\\wp ( \\bar { \\Sigma } ) : = \\mathbb { E } [ \\| y \\| ^ { - 2 k } f ( y ) : y \\sim \\mathcal { N } ( 0 , \\Sigma ) ]$ . Then on $\\left\\{ \\Sigma \\in { \\mathcal { S } } _ { A } : { \\mathrm { r a n k } } \\Sigma > 2 k \\right\\}$ , $\\wp ( \\Sigma )$ is well-defined and continuous, and furthermore satisfies ", + "bbox": [ + 173, + 263, + 825, + 323 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/bcefee99076f32346a3d1a1ba665febc5e8f2459c7a501e12638b89ec0f63e2e.jpg", + "text": "$$\n\\wp ( \\Sigma ) = \\Gamma ( k ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { k - 1 } \\operatorname* { d e t } ( I + 2 s \\Sigma ) ^ { - 1 / 2 } \\mathbb { E } [ f ( y ) : y \\sim { \\mathcal { N } } ( 0 , \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } ) ]\n$$", + "text_format": "latex", + "bbox": [ + 204, + 328, + 766, + 362 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof. $\\wp ( \\Sigma )$ is well-defined for full rank $\\Sigma$ because the $\\| y \\| ^ { - 2 k }$ singularity at $y = 0$ is Lebesgueintegrable in a neighborhood of 0 in dimension $A > 2 k$ . ", + "bbox": [ + 173, + 376, + 823, + 406 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "We prove Eq. (25) in the case when $\\Sigma$ is full rank and then apply a continuity argument. ", + "bbox": [ + 173, + 412, + 753, + 428 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Proof of Eq. (25) for full rank $\\Sigma$ . First, we will show that we can exchange the order of integration ", + "bbox": [ + 171, + 434, + 813, + 449 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/39b0eb1b73e05e90ad91d0ebdb0bf01dcee52ffc84752cec800036c5ad73c52f.jpg", + "text": "$$\n\\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ \\int _ { \\mathbb { R } ^ { A } } \\mathrm { d } y \\ s ^ { k - 1 } f ( y ) e ^ { - { \\frac { 1 } { 2 } } y ^ { T } ( \\Sigma ^ { - 1 } + 2 s I ) y } = \\int _ { \\mathbb { R } ^ { A } } \\mathrm { d } y \\ \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { k - 1 } f ( y ) e ^ { - { \\frac { 1 } { 2 } } y ^ { T } ( \\Sigma ^ { - 1 } + 2 s I ) y }\n$$", + "text_format": "latex", + "bbox": [ + 197, + 455, + 797, + 489 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "by Fubini-Tonelli’s theorem. Observe that ", + "bbox": [ + 173, + 494, + 450, + 510 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/fb1d618a9d3198a701d29a5d29b0f9a1b649854ef84b4fb6cb41bb59894b974f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad ( 2 \\pi ) ^ { - A / 2 } \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } \\displaystyle \\int _ { 0 } ^ { \\infty } \\operatorname { d } s ~ s ^ { k - 1 } \\int _ { \\mathbb R ^ { A } } \\mathrm { d } y ~ \\| f ( y ) \\| e ^ { - \\frac { 1 } { 2 } y ^ { T } ( \\Sigma ^ { - 1 } + 2 s I ) y } } \\\\ & { = \\displaystyle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s ~ s ^ { k - 1 } \\operatorname* { d e t } ( \\Sigma ( \\Sigma ^ { - 1 } + 2 s I ) ) ^ { - 1 / 2 } \\mathbb { E } [ \\| f ( y ) \\| : y \\sim \\mathcal { N } ( 0 , ( \\Sigma ^ { - 1 } + 2 s I ) ^ { - 1 } ) ] } \\\\ & { = \\displaystyle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s ~ s ^ { k - 1 } \\operatorname* { d e t } ( I + 2 s \\Sigma ) ^ { - 1 / 2 } \\mathbb { E } [ \\| f ( y ) \\| : y \\sim \\mathcal { N } ( 0 , \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 515, + 769, + 619 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "For $\\lambda = \\| \\Sigma \\| _ { 2 }$ ", + "bbox": [ + 173, + 625, + 272, + 640 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/54f98c63cc25735417e99549f5082ae4a5e02383ef31e66b8a997150fa96d48d.jpg", + "text": "$$\n\\| f ( { \\sqrt { \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } } } y ) \\| \\leq h ( \\| { \\sqrt { \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } } } y \\| \\leq h \\left( \\left\\| { \\sqrt { \\frac { \\lambda } { 1 + 2 s \\lambda } } } y \\right\\| \\right) \\leq h ( { \\sqrt { \\lambda } } \\| y \\| )\n$$", + "text_format": "latex", + "bbox": [ + 212, + 646, + 805, + 689 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Because $\\mathbb { E } [ h ( { \\sqrt { \\lambda } } \\| y \\| ) : y \\in { \\mathcal { N } } ( 0 , I ) ]$ exists by assumption, ", + "bbox": [ + 173, + 695, + 563, + 713 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/3fbab120f242290d600ec168bb6ab1ecc9efbc36533ac969965eb087d4098ce1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathbb { E } [ \\| f ( y ) \\| : y \\sim \\mathcal { N } ( 0 , \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } ) ] } \\\\ & { = \\mathbb { E } [ \\| f ( \\sqrt { \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } } y ) \\| : y \\sim \\mathcal { N } ( 0 , I ) ] } \\\\ & { \\mathbb { E } [ \\| f ( 0 ) \\| : y \\sim \\mathcal { N } ( 0 , I ) ] = \\| f ( 0 ) \\| } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 718, + 651, + 779 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "as $s \\to \\infty$ , by dominated convergence with dominating function $h ( { \\sqrt { \\lambda } } \\| y \\| ) e _ { . } ^ { - { \\frac { 1 } { 2 } } \\| y \\| ^ { 2 } } ( 2 \\pi ) ^ { - A / 2 }$ . By the same reasoning, the function $s \\mapsto \\mathbb { E } [ \\| f ( y ) \\| : y \\sim \\mathcal { N } ( 0 , \\Sigma ( \\bar { I } + 2 s \\dot { \\Sigma } ) ^ { - 1 } ) ]$ is continuous. In particular this implies that $\\begin{array} { r } { \\operatorname* { s u p } _ { 0 < s < \\infty } \\mathbb { E } [ \\| f ( y ) \\| : y \\sim \\mathcal { N } ( 0 , \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } ) ] < \\propto } \\end{array}$ . Combined with the fact that $\\operatorname* { d e t } ( I + 2 s \\Sigma ) ^ { - 1 / 2 } = \\mathbf { \\bar { \\Theta } } ( s ^ { - A / 2 } )$ as $s \\to \\infty$ , ", + "bbox": [ + 174, + 786, + 825, + 848 + ], + "page_idx": 20 + }, + { + "type": "equation", + "img_path": "images/f897649cc81627636657012dd43c160d16d5e0a87751f301961d7ca2fa11b2e4.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\displaystyle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { k - 1 } \\operatorname* { d e t } ( I + 2 s \\Sigma ) ^ { - 1 / 2 } \\mathbb { E } [ \\| f ( y ) \\| : y \\sim \\mathcal { N } ( 0 , \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } ) ] } \\\\ & { \\le \\displaystyle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ \\Theta ( s ^ { k - 1 - A / 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 854, + 746, + 922 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "which is bounded by our assumption that $A / 2 > k$ . This shows that we can apply Fubini-Tonelli’s theorem to allow exchanging order of integration. ", + "bbox": [ + 173, + 103, + 825, + 133 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Thus, ", + "bbox": [ + 173, + 140, + 212, + 154 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/b67ab0caa20baf57c62f4e7366943772304838d94311d39a713ed1dc6f24324c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathbb { E } [ \\| y \\| ^ { - 2 k } f ( y ) \\cdot y \\cdot \\mathcal { N } ( 0 , \\Sigma ) ] } \\\\ & { = \\mathbb { E } [ f ( y ) \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ \\Gamma ( k ) ^ { - 1 } s ^ { k - 1 } e ^ { - \\| y \\| ^ { 2 s } } : y \\sim \\mathcal { N } ( 0 , \\Sigma ) ] } \\\\ & { = ( 2 \\pi ) ^ { - A / 2 } \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } \\int \\mathrm { d } y \\ e ^ { - \\frac { 1 } { 2 } g ^ { T } \\Sigma ^ { - 1 } s } f ( y ) \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ \\Gamma ( k ) ^ { - 1 } s ^ { k - 1 } e ^ { - \\| y \\| ^ { 2 s } } } \\\\ & { = ( 2 \\pi ) ^ { - A / 2 } \\Gamma ( k ) ^ { - 1 } \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { k - 1 } \\int \\mathrm { d } y \\ f ( y ) e ^ { - \\frac { 1 } { 2 } g ^ { T } ( \\Sigma ^ { - 1 } + 2 z / 3 ) } } \\\\ & { \\qquad \\mathrm { ( b y ~ F r u b i n i - I o r e l i ) } } \\\\ & { = \\Gamma ( k ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { k - 1 } \\operatorname* { d e t } [ \\Sigma ( \\Sigma ^ { - 1 } + 2 s T ) ] ^ { - 1 / 2 } \\mathbb { E } [ f ( y ) : y \\sim \\mathcal { N } ( 0 , ( \\Sigma ^ { - 1 } + 2 s T ) ^ { - 1 } ) ] } \\\\ & { = \\Gamma ( k ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { k - 1 } \\operatorname* { d e t } ( I + 2 s \\Sigma ) ^ { - 1 / 2 } \\mathbb { E } [ f ( y ) : y \\sim \\mathcal { N } ( 0 , \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 157, + 787, + 371 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Domain and continuity of $\\wp ( \\Sigma )$ . The LHS of Eq. (25), $\\wp ( \\Sigma )$ , is defined and continuous on rank $\\Sigma / 2 > k$ . Indeed, if $\\begin{array} { r } { \\begin{array} { r c l } { \\dot { \\Sigma } } & { = } & { M I _ { C } M ^ { T } } \\end{array} } \\end{array}$ , where $M$ is a full rank $A \\times C$ matrix with rank $\\Sigma = C \\leq A$ , then ", + "bbox": [ + 173, + 383, + 825, + 428 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/c8502bf587aac9ef518738a6ccc61fda9f196ab74b5bd85a1ff5f2c73ad44fb7.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } [ \\| y \\| ^ { - 2 k } \\| f ( y ) \\| : y \\sim { \\mathcal { N } } ( 0 , \\Sigma ) ] } \\\\ & { = \\mathbb { E } [ \\| M z \\| ^ { - 2 k } \\| f ( M z ) \\| : z \\sim { \\mathcal { N } } ( 0 , I _ { C } ) ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 356, + 433, + 642, + 474 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "This is integrable in a neighborhood of 0 iff $C > 2 k$ , while it’s always integrable outside a ball around 0 because $\\| f \\|$ by itself already is. So $\\wp ( \\Sigma )$ is defined whenever rank $\\Sigma > 2 k$ . Its continuity can be established by dominated convergence. ", + "bbox": [ + 173, + 479, + 825, + 523 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Proof of Eq. (25) for rank $\\Sigma > 2 k$ . Observe that $\\operatorname* { d e t } ( I + 2 s \\Sigma ) ^ { - 1 / 2 }$ is continuous in $\\Sigma$ and, by an application of dominated convergence as in the above, $\\mathbb { E } [ f ( y ) : y \\sim \\mathcal { N } ( 0 , \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } ) ]$ is continuous in $\\Sigma$ . So the RHS of Eq. (25) is continuous in $\\Sigma$ whenever the integral exists. By the reasoning above, $\\mathbb { E } [ f ( y ) : y \\sim \\mathcal { N } ( 0 , \\Sigma ( I + 2 s \\Sigma ) ^ { - 1 } ) ]$ is bounded in $s$ and $\\mathrm { d e t } \\bar { ( } I + 2 s \\Sigma ) ^ { - 1 } \\bar { / } 2 =$ $\\Theta ( s ^ { - \\mathrm { r a n k } \\Sigma / 2 } )$ , so that the integral exists iff rank $\\Sigma / 2 > k$ . ", + "bbox": [ + 173, + 536, + 825, + 612 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "To summarize, we have proved that both sides of Eq. (25) are defined and continous for rank $\\Sigma >$ $2 k$ . Because the full rank matrices are dense in this set, by continuity Eq. (25) holds for all rank $\\Sigma >$ $2 k$ . □ ", + "bbox": [ + 173, + 617, + 825, + 660 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "If $\\phi$ is degree- $\\alpha$ positive homogeneous, i.e. $\\phi ( r u ) = r ^ { \\alpha } \\phi ( u )$ for any $u \\in \\mathbb { R } , r \\in \\mathbb { R } ^ { + }$ , we can apply Lemma E.2 with $k = \\alpha$ , ", + "bbox": [ + 173, + 678, + 823, + 708 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/0c9cd61cb079c44830d7737b134900f924ccb7156b3480efeb354e19bee4d86f.jpg", + "text": "$$\n\\begin{array} { r l } & { { \\mathrm { V } _ { B _ { \\phi } } ( \\Sigma ) } = \\mathbb { E } [ \\phi ( \\sqrt { B } G h / \\| G h \\| ) ^ { \\otimes 2 } : h \\sim \\mathcal { N } ( 0 , \\Sigma ) ] = B ^ { \\alpha } \\mathbb { E } [ \\phi ( G h ) ^ { \\otimes 2 } / \\| G h \\| ^ { 2 \\alpha } : h \\sim \\mathcal { N } ( 0 , \\Sigma ) ] } \\\\ & { \\qquad = B ^ { \\alpha } \\mathbb { E } [ \\phi ( z ) ^ { \\otimes 2 } / \\| z \\| ^ { 2 \\alpha } : z \\sim \\mathcal { N } ( 0 , G \\Sigma G ) ] } \\\\ & { \\qquad = B ^ { \\alpha } \\Gamma ( \\alpha ) ^ { - 1 } \\displaystyle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s s ^ { \\alpha - 1 } \\operatorname* { d e t } ( I + 2 s \\Sigma G ) ^ { - 1 / 2 } \\mathrm { V } _ { \\phi } ( G ( I + 2 s \\Sigma G ) ^ { - 1 } \\Sigma G ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 733, + 812, + 810 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "If $\\phi = \\mathrm { r e l u }$ , then ", + "bbox": [ + 173, + 824, + 287, + 838 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/edd64d7bf4f71d7ad8831131f55edaee0ecfb3b2f50fa178a76a3a09abef6899.jpg", + "text": "$$\n\\mathrm { V } _ { \\phi } ( \\Sigma ) _ { i j } = { \\left\\{ \\begin{array} { l l } { { { \\frac { 1 } { 2 } } \\Sigma _ { i i } } } & { { \\mathrm { i f ~ } } i = j } \\\\ { { \\frac { 1 } { 2 } } \\mathrm { J } _ { 1 } \\left( \\Sigma _ { i j } / { \\sqrt { \\Sigma _ { i i } \\Sigma _ { j j } } } \\right) { \\sqrt { \\Sigma _ { i i } \\Sigma _ { j j } } } } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 312, + 847, + 683, + 883 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "and, more succinctly, $\\mathrm { V } _ { \\phi } ( \\Sigma ) ~ = ~ D ^ { 1 / 2 } \\mathrm { V } _ { \\phi } ( D ^ { - 1 / 2 } \\Sigma D ^ { - 1 / 2 } ) D ^ { 1 / 2 }$ where $D \\ = \\ \\operatorname { D i a g } ( \\Sigma )$ . Here $\\begin{array} { r } { \\mathrm { J _ { 1 } } ( c ) : = \\frac { 1 } { \\pi } ( \\sqrt { 1 - c ^ { 2 } } + ( \\pi - \\operatorname { a r c c o s } ( c ) ) c ) } \\end{array}$ (Cho & Saul, 2009). ", + "bbox": [ + 173, + 890, + 823, + 926 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Matrix simplification. We can simplify the expression $G ( I + 2 s \\Sigma G ) ^ { - 1 } \\Sigma G$ , leveraging the fact that $G$ is a projection matrix. ", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition E.3. Let $\\mathbb { P }$ be an $B \\times ( B - 1 )$ matrix whose columns form an orthonormal basis of $\\begin{array} { r } { \\operatorname { i m } G : = \\{ G v : v \\in \\mathbb { R } ^ { B } \\} = \\{ w \\in \\mathbb { R } ^ { B } : \\sum _ { i } w _ { i } = 0 \\} } \\end{array}$ . Then the $B \\times B$ matrix $\\tilde { \\mathbb { e } } = \\left( \\mathbb { e } | B ^ { - 1 / 2 } \\mathbf { 1 } \\right)$ is an orthogonal matrix. For much of this paper $\\mathbb { P }$ can be any such basis, but at certain sections we will consider specific realizations of $\\mathbb { P }$ for explicit computation. ", + "bbox": [ + 173, + 136, + 825, + 194 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "From easy computations it can be seen that $G = \\tilde { \\mathbb { e } } \\left( \\begin{array} { c c } { I _ { B - 1 } } & { 0 } \\\\ { 0 } & { 0 } \\end{array} \\right) \\tilde { \\mathbb { e } } ^ { T }$ . Suppose $\\Sigma = \\tilde { \\mathbb { P } } \\left( \\stackrel { \\Sigma } { v ^ { T } } \\stackrel { v } { a } \\right) \\tilde { \\mathbb { P } } ^ { T }$ where $\\Sigma _ { \\square }$ is $( B - 1 ) \\times ( B - 1 )$ , $v$ is a column vector and $a$ is a scalar. Then $\\Sigma _ { \\bigstar } = \\ L _ { \\mathbb { e } } ^ { T } \\Sigma _ { \\mathbb { e } }$ and $\\Sigma G = \\tilde { \\mathbb { e } } \\left( \\stackrel { \\Sigma } { v ^ { T } } \\stackrel { 0 } { 0 } \\right) \\tilde { \\mathbb { e } } ^ { T }$ is block lower triangular, and ", + "bbox": [ + 173, + 205, + 825, + 284 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/51883f21acb65dcf344b69b127a9f92bc7e04c5389a2bde8980844cb162c4282.jpg", + "text": "$$\n\\begin{array} { r l } & { \\left( I + 2 s \\Sigma G \\right) ^ { - 1 } = \\tilde { \\Phi } \\left( \\begin{array} { c c } { ( I + 2 s \\Sigma _ { \\perp } ) ^ { - 1 } } & { 0 } \\\\ { \\ast } & { 1 } \\end{array} \\right) \\tilde { \\Phi } ^ { T } } \\\\ & { ~ } \\\\ & { ( I + 2 s \\Sigma G ) ^ { - 1 } \\Sigma G = \\tilde { \\Phi } \\left( \\begin{array} { c c } { ( I + 2 s \\Sigma _ { \\perp } ) ^ { - 1 } \\Sigma _ { \\perp } } & { 0 } \\\\ { \\ast } & { 0 } \\end{array} \\right) \\tilde { \\Phi } ^ { T } } \\\\ & { G ( I + 2 s \\Sigma G ) ^ { - 1 } \\Sigma G = \\tilde { \\Phi } \\left( \\begin{array} { c c } { ( I + 2 s \\Sigma _ { \\perp } ) ^ { - 1 } \\Sigma _ { \\perp } } & { 0 } \\\\ { 0 } & { 0 } \\end{array} \\right) \\tilde { \\Phi } ^ { T } } \\\\ & { ~ } \\\\ & { ~ = \\varphi ( I + 2 s \\Sigma _ { \\perp } ) ^ { - 1 } \\Sigma _ { \\perp } \\tilde { \\Phi } ^ { T } } \\\\ & { ~ = G \\Sigma G ( I + 2 s G \\Sigma G ) ^ { - 1 } } \\\\ & { ~ = : \\Sigma ^ { G } ( I + 2 s \\Sigma ^ { G } ) ^ { - 1 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 315, + 290, + 681, + 457 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 174, + 462, + 217, + 474 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition E.4. For any matrix $\\Sigma$ , write $\\Sigma ^ { G } : = G \\Sigma G$ . ", + "bbox": [ + 173, + 478, + 534, + 494 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Similarly, $\\operatorname* { d e t } ( I _ { B } + 2 s \\Sigma G ) = \\operatorname* { d e t } ( I _ { B - 1 } + 2 s \\Sigma _ { \\Sigma } ) = \\operatorname* { d e t } ( I _ { B } + 2 s \\Sigma ^ { G } )$ . So, altogether, we have Theorem E.5. Suppose $\\phi : \\mathbb { R } \\mathbb { R }$ is degree- $\\alpha$ positive homogeneous. Then for any $B \\times ( B - 1 )$ matrix $\\mathbb { P }$ whose columns form an orthonormal basis of $\\operatorname { i m } G : = \\{ G v : v \\in { \\bar { \\mathbb { R } } } ^ { B } \\} \\ = \\{ w \\in \\mathbb { R } ^ { B } ^ { * }$ : $\\begin{array} { r } { \\sum _ { i } w _ { i } = 0 \\} } \\end{array}$ , with $\\Sigma _ { \\perp } = \\mathrm { \\bar { e } } ^ { T } \\Sigma _ { \\Phi }$ , ", + "bbox": [ + 174, + 505, + 826, + 568 + ], + "page_idx": 22 + }, + { + "type": "equation", + "img_path": "images/af1c3338968a4f6f3315b26956fa6b82a1f5e7e24910a4955589bc12791c2985.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( \\Sigma ) = B ^ { \\alpha } \\Gamma ( \\alpha ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s ~ s ^ { \\alpha - 1 } \\operatorname * { d e t } ( I + 2 s \\Sigma _ { \\boxdot } ) ^ { - 1 / 2 } \\mathrm { V } _ { \\phi } ( \\mathtt { e } ( I + 2 s \\Sigma _ { \\boxdot } ) ^ { - 1 } \\Sigma _ { \\boxdot } ) ^ { - 1 } \\mathrm { e } ^ { T } ) } } \\\\ { { = B ^ { \\alpha } \\Gamma ( \\alpha ) ^ { - 1 } \\displaystyle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s ~ s ^ { \\alpha - 1 } \\operatorname * { d e t } ( I + 2 s \\Sigma ^ { G } ) ^ { - 1 / 2 } \\mathrm { V } _ { \\phi } ( ( I + 2 s \\Sigma ^ { G } ) ^ { - 1 } \\Sigma ^ { G } ) ^ { - 1 } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 192, + 574, + 779, + 643 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Now, in order to use Laplace’s method, we require $\\phi$ to be positive homogeneous. What do these functions look like? The most familiar example to a machine learning audience is most likely ReLU. It turns out that 1-dimensional positive homogeneous functions can always be described as a linear combination of powers of ReLU and its reflection across the y-axis. Below, we review known facts about these functions and their integral transforms, starting with the powers of ReLU in Appendix E.1.1 and then onto the general case in Appendix E.1.2. ", + "bbox": [ + 173, + 672, + 825, + 757 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "E.1.1 $\\alpha$ -RELU ", + "text_level": 1, + "bbox": [ + 174, + 772, + 292, + 786 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Recall that $\\alpha$ -ReLUs (Yang & Schoenholz, 2017) are, roughly speaking, the $\\alpha$ th power of ReLU. ", + "bbox": [ + 173, + 796, + 808, + 811 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition E.6. The $\\alpha$ -ReLU function $\\rho _ { \\alpha } : \\mathbb { R } \\to \\mathbb { R }$ sends $x \\mapsto x ^ { \\alpha }$ when $x > 0$ and $x \\mapsto 0$ otherwise. ", + "bbox": [ + 173, + 814, + 823, + 844 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "This is a continuous function for $\\alpha > 0$ but discontinuous at 0 for all other $\\alpha$ . ", + "bbox": [ + 176, + 854, + 681, + 869 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "We briefly review what is currently known about the $\\mathrm { V }$ and W transforms of $\\rho _ { \\alpha }$ (Cho & Saul, 2009; \nYang & Schoenholz, 2017). ", + "bbox": [ + 176, + 876, + 825, + 905 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "Definition E.7. For any $\\alpha > - \\frac { 1 } { 2 }$ , define $\\begin{array} { r } { \\mathsf { c } _ { \\alpha } = \\frac { 1 } { \\sqrt { \\pi } } 2 ^ { \\alpha - 1 } \\Gamma ( \\alpha + \\frac { 1 } { 2 } ) } \\end{array}$ ", + "bbox": [ + 173, + 906, + 609, + 928 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "When considering only 1-dimensional Gaussians, $\\mathrm { V } _ { \\rho _ { \\alpha } }$ is very simple. ", + "bbox": [ + 173, + 102, + 632, + 119 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proposition E.8. If $\\alpha > - \\frac { 1 } { 2 }$ , then for any $q \\in S _ { 1 } = \\mathbb { R } ^ { \\geq 0 }$ , $\\mathrm { V } _ { \\rho _ { \\alpha } } ( q ) = \\mathsf { c } _ { \\alpha } q ^ { \\alpha }$ ", + "bbox": [ + 174, + 121, + 663, + 140 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "To express results of $\\mathrm { V } _ { \\rho _ { \\alpha } }$ on $\\boldsymbol { S } _ { B }$ for higher $B$ , we first need the following Definition E.9. Define ", + "bbox": [ + 171, + 146, + 660, + 179 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/eeab9c6001c9fa3539d4fc7dcb25a11b487ee7f26dcf051029acd843a37abf87.jpg", + "text": "$$\nJ _ { \\alpha } ( \\theta ) : = \\frac { 1 } { 2 \\pi \\mathsf { c } _ { \\alpha } } ( \\sin \\theta ) ^ { 2 \\alpha + 1 } \\Gamma ( \\alpha + 1 ) \\int _ { 0 } ^ { \\pi / 2 } \\frac { \\mathrm { d } \\eta \\cos ^ { \\alpha } \\eta } { ( 1 - \\cos \\theta \\cos \\eta ) ^ { 1 + \\alpha } }\n$$", + "text_format": "latex", + "bbox": [ + 284, + 179, + 714, + 215 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "and $\\mathrm { J } _ { \\alpha } ( c ) = J _ { \\alpha } ( \\operatorname { a r c c o s } c )$ for $\\alpha > - 1 / 2$ . ", + "bbox": [ + 174, + 218, + 449, + 233 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Then ", + "bbox": [ + 173, + 242, + 210, + 256 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proposition E.10. For any $\\Sigma \\in { \\cal S } _ { B }$ , let $D$ be the diagonal matrix with the same diagonal as $\\Sigma$ . Then ", + "bbox": [ + 171, + 258, + 823, + 287 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/76e7b1bc8e0c6dc921fc45cfa730fdb84fde86ca681eccca023a8095e320c3ac.jpg", + "text": "$$\n\\mathrm { V } _ { \\rho _ { \\alpha } } ( \\Sigma ) = \\mathsf { c } _ { \\alpha } D ^ { \\alpha / 2 } \\mathrm { J } _ { \\alpha } ( D ^ { - 1 / 2 } \\Sigma D ^ { - 1 / 2 } ) D ^ { \\alpha / 2 }\n$$", + "text_format": "latex", + "bbox": [ + 348, + 289, + 647, + 309 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "where $\\mathrm { J } _ { \\alpha }$ is applied entrywise. ", + "bbox": [ + 174, + 310, + 375, + 325 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "For example, $J _ { \\alpha }$ and $\\mathrm { J } _ { \\alpha }$ for the first few integral $\\alpha$ are ", + "bbox": [ + 173, + 334, + 531, + 351 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/df903b4e8b1d04c0f49132d0196f0d353eec3dd18722686dfdace887ed34b21f.jpg", + "text": "$$\n\\begin{array} { l } { { J _ { 0 } ( \\theta ) = \\displaystyle \\frac { \\pi - \\theta } { \\pi } } } \\\\ { { J _ { 1 } ( \\theta ) = \\displaystyle \\frac { \\sin \\theta + ( \\pi - \\theta ) \\cos \\theta } { \\pi } } } \\\\ { { J _ { 2 } ( \\theta ) = \\displaystyle \\frac { 3 \\sin \\theta \\cos \\theta + ( \\pi - \\theta ) ( 1 + 2 \\cos ^ { 2 } \\theta ) } { 3 \\pi } } } \\\\ { { J _ { 0 } ( c ) = \\displaystyle \\frac { \\pi - \\mathrm { a r c c o s c } } { \\pi } } } \\\\ { { J _ { 1 } ( c ) = \\displaystyle \\frac { \\sqrt { 1 - c ^ { 2 } + ( \\pi - \\mathrm { a r c c o s c } ) c } } { \\pi } } } \\\\ { { { \\bf J } _ { 2 } ( c ) = \\displaystyle \\frac { 3 c \\sqrt { 1 - c ^ { 2 } + ( \\pi - \\mathrm { a r c c o s c } ) ( 1 + 2 c ^ { 2 } ) } } { 3 \\pi } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 339, + 371, + 655, + 564 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "One can observe very easily that (Daniely et al., 2016; Yang & Schoenholz, 2017) ", + "bbox": [ + 171, + 570, + 712, + 587 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Proposition E.11. For each $\\alpha > - 1 / 2 , \\mathrm { J } _ { \\alpha } ( c )$ is an increasing and convex function on $c \\in [ 0 , 1 ]$ , and is continous on $c \\in [ 0 , 1 ]$ and smooth on $c \\in ( 0 , 1 )$ . $\\mathrm { J } _ { \\alpha } ( 1 ) = 1$ , $\\begin{array} { r } { \\mathrm { J } _ { \\alpha } ( 0 ) = \\frac { 1 } { 2 \\sqrt { \\pi } } \\frac { \\Gamma ( \\frac { \\alpha } { 2 } + \\frac { 1 } { 2 } ) ^ { 2 } } { \\Gamma ( \\alpha + \\frac { 1 } { 2 } ) } } \\end{array}$ , and ${ \\mathrm J } _ { \\alpha } ( - 1 ) = 0$ . ", + "bbox": [ + 173, + 588, + 826, + 642 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Yang & Schoenholz (2017) also showed the following fixed point structure ", + "bbox": [ + 176, + 650, + 666, + 666 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Theorem E.12. For $\\alpha \\in [ 1 / 2 , 1 )$ , $\\mathrm { J } _ { \\alpha } ( c ) = c$ has two solutions: an unstable solution at $I$ (”unstable” meaning $\\mathrm { J } _ { \\alpha } ^ { \\prime } ( 1 ) > 1$ ) and a stable solution in $c ^ { * } \\in ( 0 , 1 )$ (”stable” meaning $\\mathrm { J } _ { \\alpha } ^ { \\prime } ( c ^ { * } ) < 1 \\}$ ). ", + "bbox": [ + 174, + 667, + 823, + 698 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "The $\\alpha$ -ReLUs satisfy very interesting relations amongst themselves. For example, ", + "bbox": [ + 173, + 707, + 710, + 722 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Lemma E.13. Suppose $\\alpha > 1$ . Then ", + "bbox": [ + 174, + 724, + 419, + 738 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/d2c2d05690c687f03e4ff369ad2954a1ee062e68a5bd02e54aa1050815929a0c.jpg", + "text": "$$\n\\begin{array} { l } { { J _ { \\alpha } ( \\theta ) = \\cos \\theta J _ { \\alpha - 1 } ( \\theta ) + ( \\alpha - 1 ) ^ { 2 } ( 2 \\alpha - 1 ) ^ { - 1 } ( 2 \\alpha - 3 ) ^ { - 1 } \\sin ^ { 2 } \\theta J _ { \\alpha - 2 } ( \\theta ) } } \\\\ { { \\mathrm J _ { \\alpha } ( c ) = c \\mathrm J _ { \\alpha - 1 } ( c ) + ( \\alpha - 1 ) ^ { 2 } ( 2 \\alpha - 1 ) ^ { - 1 } ( 2 \\alpha - 3 ) ^ { - 1 } ( 1 - c ^ { 2 } ) \\mathrm J _ { \\alpha - 2 } ( c ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 258, + 739, + 740, + 781 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "In additional, surprisingly, one can use differentiation to go from $\\alpha$ to $\\alpha + 1$ and from $\\alpha$ to $\\alpha - 1$ ! Proposition E.14 (Yang & Schoenholz (2017)). Suppose $\\alpha > 1 / 2$ . Then ", + "bbox": [ + 173, + 787, + 815, + 820 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/ee65200c8f7e64f03700417d8a4c6b6928b2f3853ffe1d9078093d402ea30b50.jpg", + "text": "$$\n\\begin{array} { r l } & { J _ { \\alpha } ^ { \\prime } ( \\theta ) = - \\alpha ^ { 2 } ( 2 \\alpha - 1 ) ^ { - 1 } J _ { \\alpha - 1 } ( \\theta ) \\sin \\theta } \\\\ & { \\mathrm { J } _ { \\alpha } ^ { \\prime } ( c ) = \\alpha ^ { 2 } ( 2 \\alpha - 1 ) ^ { - 1 } \\mathrm { J } _ { \\alpha - 1 } ( c ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 823, + 630, + 864 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "so that ", + "bbox": [ + 173, + 864, + 222, + 878 + ], + "page_idx": 23 + }, + { + "type": "equation", + "img_path": "images/e38fe9d4c6ed69aa936cdbdcf525d1e258d1d7df6adfeee32a811b3cbf2654de.jpg", + "text": "$$\n\\operatorname { J } _ { \\alpha - n } ( c ) = { \\Bigg [ } \\prod _ { \\beta = \\alpha - n + 1 } ^ { \\alpha } \\beta ^ { - 2 } ( 2 \\beta - 1 ) { \\Bigg ] } ( \\partial / \\partial c ) ^ { n } \\mathrm { J } _ { \\alpha } ( c )\n$$", + "text_format": "latex", + "bbox": [ + 323, + 878, + 673, + 929 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "We have the following from Cho & Saul (2009) ", + "bbox": [ + 174, + 103, + 485, + 118 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proposition E.15 (Cho & Saul (2009)). For all $\\alpha \\geq 0$ and integer $n \\geq 1$ ", + "bbox": [ + 174, + 119, + 653, + 135 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/c22d9de697df1aa4011e0863f31ecb2c8a99c95e3255fbac442c8588a6446019.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathrm { J } _ { n + \\alpha } ( c ) = \\frac { \\mathrm { c } _ { \\alpha } } { \\mathrm { c } _ { n + \\alpha } } ( 1 - c ^ { 2 } ) ^ { n + \\alpha + \\frac { 1 } { 2 } } ( \\partial / \\partial c ) ^ { n } \\mathrm { ( J } _ { \\alpha } ( c ) / ( 1 - c ^ { 2 } ) ^ { \\alpha + \\frac { 1 } { 2 } } ) } \\\\ { \\displaystyle \\qquad = \\left[ \\prod _ { \\beta = \\alpha } ^ { \\alpha + n - 1 } ( 2 \\beta + 1 ) \\right] ^ { - 1 } ( 1 - c ^ { 2 } ) ^ { n + \\alpha + \\frac { 1 } { 2 } } ( \\partial / \\partial c ) ^ { n } \\mathrm { ( J } _ { \\alpha } ( c ) / ( 1 - c ^ { 2 } ) ^ { \\alpha + \\frac { 1 } { 2 } } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 235, + 135, + 761, + 219 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "This implies in particular that we can obtain $\\mathrm { J } _ { \\alpha } ^ { \\prime }$ from $\\mathrm { J } _ { \\alpha }$ and $\\mathrm { J } _ { \\alpha + 1 }$ . ", + "bbox": [ + 173, + 226, + 612, + 242 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proposition E.16. For all $\\alpha \\geq 0$ , ", + "bbox": [ + 174, + 243, + 395, + 257 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/321396f875ebde4c2b333c8916296e96f80e576ca0569b260028fa441326a268.jpg", + "text": "$$\n\\mathrm { J } _ { \\alpha } ^ { \\prime } ( c ) = ( 2 \\alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( \\mathrm { J } _ { 1 + \\alpha } ( c ) - c \\mathrm { J } _ { \\alpha } ( c ) ) )\n$$", + "text_format": "latex", + "bbox": [ + 331, + 257, + 665, + 275 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proof. ", + "bbox": [ + 173, + 289, + 217, + 303 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/0c5b26e963d42f6b1b8a12a9ced1072d2ed244739abd414f40d9ba0d41d65db3.jpg", + "text": "$$\n{ \\begin{array} { r l } & { \\operatorname { J } _ { 1 + \\alpha } ( c ) = { \\frac { c _ { \\alpha } } { c _ { 1 + \\alpha } } } ( 1 - c ^ { 2 } ) ^ { 1 + \\alpha + { \\frac { 1 } { 2 } } } ( \\partial / \\partial c ) ( \\operatorname { J } _ { \\alpha } ( c ) / ( 1 - c ^ { 2 } ) ^ { \\alpha + { \\frac { 1 } { 2 } } } ) } \\\\ & { \\qquad = ( 2 \\alpha + 1 ) ^ { - 1 } ( 1 - c ^ { 2 } ) ^ { \\alpha + 3 / 2 } ( \\operatorname { J } _ { \\alpha } ^ { \\prime } ( c ) / ( 1 - c ^ { 2 } ) ^ { \\alpha + 1 / 2 } + 2 c ( \\alpha + 1 / 2 ) \\operatorname { J } _ { \\alpha } ( c ) / ( 1 - c ^ { 2 } ) ^ { \\alpha + 3 / 2 } ) } \\\\ & { \\qquad = ( 2 \\alpha + 1 ) ^ { - 1 } \\operatorname { J } _ { \\alpha } ^ { \\prime } ( c ) ( 1 - c ^ { 2 } ) + c \\operatorname { J } _ { \\alpha } ( c ) } \\\\ & { \\qquad \\operatorname { J } _ { \\alpha } ^ { \\prime } ( c ) = ( 2 \\alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( \\operatorname { J } _ { 1 + \\alpha } ( c ) - c \\operatorname { J } _ { \\alpha } ( c ) ) } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 304, + 823, + 396 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Note that we can also obtain this via Lemma E.13 and Proposition E.14. ", + "bbox": [ + 174, + 422, + 647, + 439 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "E.1.2 POSITIVE-HOMOGENEOUS FUNCTIONS IN 1 DIMENSION ", + "text_level": 1, + "bbox": [ + 173, + 452, + 622, + 467 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Suppose for some $\\alpha \\in \\mathbb { R } , \\phi : \\mathbb { R } \\to \\mathbb { R }$ is degree $\\alpha$ positive-homogeneous, i.e. $\\phi ( r x ) = r ^ { \\alpha } \\phi ( x )$ for any $x \\in R , r > 0$ . The following simple lemma says that we can always express $\\phi$ as linear combination of powers of $\\alpha$ -ReLUs. ", + "bbox": [ + 176, + 477, + 823, + 518 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proposition E.17. Any degree α positive-homogeneous function $\\phi : \\mathbb { R } \\mathbb { R }$ with $\\phi ( 0 ) = 0$ can be written as $x \\mapsto a \\rho _ { \\alpha } ( x ) - b \\rho _ { \\alpha } ( - x )$ . ", + "bbox": [ + 173, + 521, + 823, + 551 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proof. Take $a = \\phi ( 1 )$ and $b = \\phi ( - 1 )$ . Then positive-homogeneity determines the value of $\\phi$ on $\\mathbb { R } \\backslash \\{ 0 \\}$ and it coincides with $x \\mapsto a \\rho _ { \\alpha } ( x ) - b \\rho _ { \\alpha } ( - x )$ . □ ", + "bbox": [ + 171, + 564, + 825, + 594 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "As a result we can express the $\\mathrm { V }$ and W transforms of any positive-homogeneous function in terms of those of $\\alpha$ -ReLUs. ", + "bbox": [ + 176, + 608, + 823, + 636 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proposition E.18. Suppose $\\phi : \\mathbb { R } \\mathbb { R }$ is degree α positive-homogeneous. By Proposition E.17, $\\phi$ restricted to $\\mathbb { R } \\backslash \\{ 0 \\}$ can be written as $x \\mapsto a \\rho _ { \\alpha } ( x ) - b \\rho _ { \\alpha } ( - x ) .$ for some a and $b$ . Then for any PSD $2 \\times 2$ matrix $M$ , ", + "bbox": [ + 176, + 637, + 825, + 680 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/44d4ecb14ab269e81a4254e32af0809399b3aa3e74c512fb6d922c23df4b40bc.jpg", + "text": "$$\n\\begin{array} { l } { { \\mathrm { V } _ { \\phi } ( M ) _ { 1 1 } = ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { V } _ { \\rho _ { \\alpha } } ( M ) _ { 1 1 } = \\mathrm { c } _ { \\alpha } ( a ^ { 2 } + b ^ { 2 } ) M _ { 1 1 } ^ { \\alpha } } } \\\\ { { \\mathrm { V } _ { \\phi } ( M ) _ { 2 2 } = ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { V } _ { \\rho _ { \\alpha } } ( M ) _ { 2 2 } = \\mathrm { c } _ { \\alpha } ( a ^ { 2 } + b ^ { 2 } ) M _ { 2 2 } ^ { \\alpha } } } \\\\ { { \\mathrm { } / _ { \\phi } ( M ) _ { 1 2 } = \\mathrm { V } _ { \\phi } ( M ) _ { 2 1 } = ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { V } _ { \\rho _ { \\alpha } } ( M ) _ { 1 2 } - 2 a b \\mathrm { V } _ { \\rho _ { \\alpha } } ( M ^ { \\prime } ) _ { 1 2 } } } \\\\ { { = \\mathrm { c } _ { \\alpha } ( M _ { 1 1 } M _ { 2 2 } ) ^ { \\alpha / 2 } ( ( a ^ { 2 } + b ^ { 2 } ) ] _ { \\alpha } ( M _ { 1 2 } / \\sqrt { M _ { 1 1 } M _ { 2 2 } } ) - 2 a b \\mathrm { J } _ { \\alpha } ( - M _ { 1 2 } / \\sqrt { M _ { 1 1 } M _ { 2 2 } } ) } } \\\\ { { \\mathrm { } } } \\\\ { \\mathrm { } \\nu h e r e \\ M ^ { \\prime } : = \\left( { 1 0 \\ \\begin{array} { l l } { { \\ } & { { 0 } } \\\\ { 0 \\ } & { { - 1 } \\end{array} } } M \\left( { 0 } & { - 1 } \\right) . } } \\end{\\right)array} \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 174, + 678, + 839, + 796 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Proof. We directly compute, using the expansion of $\\phi$ into $\\rho _ { \\alpha }$ s: ", + "bbox": [ + 174, + 808, + 593, + 823 + ], + "page_idx": 24 + }, + { + "type": "equation", + "img_path": "images/9972a18844e70a2691dccab9a3c03df44eb62722d432a0805a5f411690bcac51.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { V } _ { \\phi } ( M ) _ { 1 1 } = \\mathbb { E } [ \\phi ( x ) ^ { 2 } : x \\sim \\mathcal { N } ( 0 , M _ { 1 1 } ) ] } \\\\ & { \\qquad = \\mathbb { E } [ a ^ { 2 } \\rho _ { \\alpha } ( x ) ^ { 2 } + b ^ { 2 } \\rho _ { \\alpha } ( - x ) ^ { 2 } + 2 a b \\rho _ { \\alpha } ( x ) \\rho _ { \\alpha } ( - x ) ] } \\\\ & { \\qquad = a ^ { 2 } \\mathbb { E } [ \\rho _ { \\alpha } ( x ) ^ { 2 } ] + b ^ { 2 } \\mathbb { E } [ \\rho _ { \\alpha } ( - x ) ^ { 2 } ] } \\\\ & { \\qquad = ( a ^ { 2 } + b ^ { 2 } ) \\mathbb { E } [ \\rho _ { \\alpha } ( x ) ^ { 2 } : x \\sim \\mathcal { N } ( 0 , M _ { 1 1 } ) ] } \\\\ & { \\qquad = \\mathsf { c } _ { \\alpha } ( a ^ { 2 } + b ^ { 2 } ) M _ { 1 1 } ^ { \\alpha } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 821, + 699, + 924 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "where in Eq. (28) we used negation symmetry of centered Gaussians. The case of $\\mathrm { V } _ { \\phi } ( M ) _ { 2 2 }$ is similar. ", + "bbox": [ + 171, + 102, + 825, + 133 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/4fdcd3e8de9065d770df5b1f4b013e4575951a8afbf0a29ff456670b64c2fdc3.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { V } _ { \\phi } ( M ) _ { 1 2 } = \\mathbb { E } [ \\phi ( x ) \\phi ( y ) : ( x , y ) \\sim \\mathcal { N } ( 0 , M ) ] } \\\\ & { \\qquad = \\mathbb { E } [ a ^ { 2 } \\rho _ { \\alpha } ( x ) \\rho _ { \\alpha } ( y ) + b ^ { 2 } \\rho _ { \\alpha } ( - x ) \\rho _ { \\alpha } ( - y ) - a b \\rho _ { \\alpha } ( x ) \\rho _ { \\alpha } ( - y ) - a b \\rho _ { \\alpha } ( - x ) \\rho _ { \\alpha } ( y ) ] } \\\\ & { \\qquad = ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { V } _ { \\rho _ { \\alpha } } ( M ) _ { 1 2 } - 2 a b \\mathrm { V } _ { \\rho _ { \\alpha } } ( M ^ { \\prime } ) _ { 1 2 } } \\\\ & { \\qquad = \\mathrm { c } _ { \\alpha } ( M _ { 1 1 } M _ { 2 2 } ) ^ { \\alpha / 2 } ( ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { J } _ { \\alpha } ( M _ { 1 2 } / \\sqrt { M _ { 1 1 } M _ { 2 2 } } ) - 2 a b \\mathrm { J } _ { \\alpha } ( - M _ { 1 2 } / \\sqrt { M _ { 1 1 } M _ { 2 2 } } ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 186, + 137, + 810, + 220 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "where in the last equation we have applied Proposition E.10. ", + "bbox": [ + 173, + 224, + 571, + 239 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "This then easily generalizes to PSD matrices of arbitrary dimension: ", + "bbox": [ + 174, + 255, + 625, + 270 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Corollary E.19. Suppose $\\phi : \\mathbb { R } \\mathbb { R }$ is degree $\\alpha$ positive-homogeneous. By Proposition E.17, $\\phi$ restricted to $\\mathbb { R } \\backslash \\{ 0 \\}$ can be written as $x \\mapsto a \\rho _ { \\alpha } ( x ) - b \\rho _ { \\alpha } ( - x )$ for some a and $b$ . Let $\\Sigma \\in S _ { B }$ . Then ", + "bbox": [ + 173, + 273, + 825, + 304 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/cf99e8a23fc923ebfa03cde0ee1d32f38cb503cc702736f5b01e6b024548355c.jpg", + "text": "$$\n\\mathrm { V } _ { \\phi } ( \\Sigma ) = \\mathsf { c } _ { \\alpha } D ^ { \\alpha / 2 } \\mathrm { J } _ { \\phi } ( D ^ { - 1 / 2 } \\Sigma D ^ { - 1 / 2 } ) D ^ { \\alpha / 2 }\n$$", + "text_format": "latex", + "bbox": [ + 352, + 310, + 643, + 329 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "where $\\mathrm { J } _ { \\phi }$ is defined below and is applied entrywise. Explicitly, this means that for all $i ,$ ", + "bbox": [ + 173, + 335, + 741, + 352 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/84db548ea262490da25cc1cc29705274e60450bfe54569e81c7eb3f406fae992.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { V } _ { \\phi } ( \\Sigma ) _ { i i } = \\mathsf { c } _ { \\alpha } \\Sigma _ { i i } ^ { \\alpha } \\mathrm { J } _ { \\phi } ( 1 ) } \\\\ & { \\mathrm { V } _ { \\phi } ( \\Sigma ) _ { i j } = \\mathsf { c } _ { \\alpha } \\mathrm { J } _ { \\phi } ( \\Sigma _ { i j } / \\sqrt { \\Sigma _ { i i } \\Sigma _ { j j } } ) \\Sigma _ { i i } ^ { \\alpha / 2 } \\Sigma _ { j j } ^ { \\alpha / 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 356, + 643, + 400 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Definition E.20. Suppose $\\phi : \\mathbb { R } \\mathbb { R }$ is degree $\\alpha$ positive-homogeneous. By Proposition E.17, $\\phi$ restricted to $\\mathbb { R } \\ \\backslash \\ \\{ 0 \\}$ can be written as $x \\mapsto a \\rho _ { \\alpha } ( x ) - b \\rho _ { \\alpha } ( - x )$ for some $a$ and $b$ . Define $\\mathrm { J } _ { \\phi } ( c ) : = ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { \\bar { J } } _ { \\alpha } ( \\bar { c } ) - 2 a b \\mathrm { J } _ { \\alpha } ( - c )$ . ", + "bbox": [ + 173, + 405, + 825, + 449 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Let us immediately make the following easy but important observations. ", + "bbox": [ + 174, + 458, + 647, + 474 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/99a9ff39c724bbc57e134e08e88562bc01e72bfa75e5e01abb8dcd2f69dde58f.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { J } _ { \\phi } ( 0 ) = ( a ^ { 2 } + b ^ { 2 } - 2 a b ) \\mathrm { J } _ { \\alpha } ( 0 ) = ( a - b ) ^ { 2 } \\frac { 1 } { 2 \\sqrt { \\pi } } \\frac { \\Gamma ( \\frac { \\alpha } { 2 } + \\frac { 1 } { 2 } ) ^ { 2 } } { \\Gamma ( \\alpha + \\frac { 1 } { 2 } ) } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 442, + 478, + 818, + 503 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. Use the fact that $\\mathrm { J } _ { \\alpha } ( - 1 ) = 0$ and $\\begin{array} { r } { \\mathrm { J } _ { \\alpha } ( 0 ) = \\frac { 1 } { 2 \\sqrt { \\pi } } \\frac { \\Gamma ( \\frac { \\alpha } { 2 } + \\frac { 1 } { 2 } ) ^ { 2 } } { \\Gamma ( \\alpha + \\frac { 1 } { 2 } ) } } \\end{array}$ by Proposition E.11. ", + "bbox": [ + 174, + 518, + 733, + 545 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "As a sanity check, we can easily compute that $\\mathrm { J _ { i d } } ( c ) = 2 \\mathrm { J _ { 1 } } ( c ) - 2 \\mathrm { J _ { 1 } } ( - c ) = 2 c$ because $\\operatorname { i d } ( x ) =$ $\\mathrm { r e l u } ( x ) { - } \\mathrm { r e l u } ( - x )$ . By Corollary E.19 and $\\mathsf { c } _ { 1 } = \\textstyle { \\frac { 1 } { 2 } }$ , this recovers the obvious fact that $\\mathrm { V } _ { \\mathrm { ( i d ) } } ( \\Sigma ) = \\Sigma$ . ", + "bbox": [ + 174, + 558, + 825, + 589 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "We record the partial derivatives of $\\mathrm { V } _ { \\phi }$ ", + "bbox": [ + 174, + 594, + 433, + 609 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proposition E.22. Let $\\phi$ be positive homogeneous of degree $\\alpha$ . Then for all $i$ with $\\Sigma _ { i i } \\neq 0$ , ", + "bbox": [ + 171, + 613, + 771, + 628 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/a040eeb229ac0e88a317dc732f878ce0d1fc05783dde9cabce238a465fa4aad6.jpg", + "text": "$$\n\\frac { \\partial \\mathrm { V } _ { \\phi } ( \\Sigma ) _ { i i } } { \\partial \\Sigma _ { i i } } = \\mathsf { c } _ { \\alpha } \\alpha \\Sigma _ { i i } ^ { \\alpha - 1 } \\mathrm { J } _ { \\phi } ( 1 )\n$$", + "text_format": "latex", + "bbox": [ + 400, + 633, + 598, + 667 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "For all $i \\neq j$ with $\\Sigma _ { i i } , \\Sigma _ { j j } \\neq 0$ , ", + "bbox": [ + 173, + 672, + 383, + 689 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/c91378482671fe86c084f0069f7aef0be8f087148d98e9f879d9ef1e98310572.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { \\partial \\nabla _ { \\phi } \\left( \\Sigma \\right) _ { i i } } { \\partial \\Sigma _ { i j } } = 0 } \\\\ & { \\frac { \\partial \\nabla _ { \\phi } \\left( \\Sigma \\right) _ { i j } } { \\partial \\Sigma _ { i j } } = \\mathsf { c } _ { \\alpha } \\Sigma _ { i i } ^ { ( \\alpha - 1 ) / 2 } \\Sigma _ { j j } ^ { ( \\alpha - 1 ) / 2 } \\mathbf { J } _ { \\phi } ^ { \\prime } ( { c } _ { i j } ) } \\\\ & { \\frac { \\partial \\nabla _ { \\phi } \\left( \\Sigma \\right) _ { i j } } { \\partial \\Sigma _ { i i } } = \\frac { 1 } { 2 } \\mathsf { c } _ { \\alpha } \\Sigma _ { i i } ^ { \\frac { 1 } { 2 } ( \\alpha - 2 ) } \\Sigma _ { j j } ^ { \\frac { 1 } { 2 } \\alpha } ( \\alpha \\mathbf { J } _ { \\phi } ( { c } _ { i j } ) - { c } _ { i j } \\mathbf { J } _ { \\phi } ^ { \\prime } ( { c } _ { i j } ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 695, + 686, + 800 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "where $c _ { i j } = \\Sigma _ { i j } / \\sqrt { \\Sigma _ { i i } \\Sigma _ { j j } }$ and $\\mathrm { J } _ { \\phi } ^ { \\prime }$ denotes its derivative. ", + "bbox": [ + 173, + 806, + 552, + 824 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proposition E.23. If $\\phi ( c ) = a \\rho _ { \\alpha } ( c ) - b \\rho _ { \\alpha } ( - c )$ with $\\alpha > 1 / 2$ on $\\mathbb { R } \\backslash \\{ 0 \\}$ , then ", + "bbox": [ + 173, + 828, + 699, + 844 + ], + "page_idx": 25 + }, + { + "type": "equation", + "img_path": "images/db0e677d7135cb02cc8b99889fb103058495984e1b92992c31aa4dbe37524454.jpg", + "text": "$$\n\\begin{array} { l } { { { \\bf J } _ { \\phi } ^ { \\prime } ( c ) = ( 2 \\alpha - 1 ) ^ { - 1 } { \\bf J } _ { \\phi ^ { \\prime } } ( c ) } } \\\\ { { \\mathrm { ~ \\alpha ~ } = ( 2 \\alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( ( a ^ { 2 } + b ^ { 2 } ) { \\bf J } _ { \\alpha + 1 } ( c ) + 2 a b { \\bf J } _ { \\alpha + 1 } ( - c ) ) - c { \\bf J } _ { \\phi } ( c ) } } \\\\ { { { \\bf J } _ { \\phi } ^ { \\prime } ( 0 ) = ( a + b ) ^ { 2 } \\displaystyle \\frac { 1 } { \\sqrt { \\pi } } \\displaystyle \\frac { \\Gamma ( \\frac { \\alpha } { 2 } + 1 ) ^ { 2 } } { \\Gamma ( \\alpha + \\frac { 1 } { 2 } ) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 248, + 851, + 748, + 929 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "Proof. We have $\\phi ^ { \\prime } ( c ) = a \\alpha \\rho _ { \\alpha - 1 } ( c ) + b \\alpha \\rho _ { \\alpha - 1 } ( - c )$ . On the other hand, $\\mathrm { J } _ { \\phi } ^ { \\prime } = ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { J } _ { \\alpha } ^ { \\prime } ( c ) +$ $2 a b \\mathbf { J } _ { \\alpha } ^ { \\prime } ( - c )$ which by Proposition E.14 is $\\alpha ^ { 2 } ( 2 \\alpha - 1 ) ^ { - 1 } ( ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { J } _ { \\alpha - 1 } ( \\dot { c } ) + 2 a b \\mathrm { J } _ { \\alpha - 1 } ( - c ) ) =$ $( 2 \\alpha - \\mathrm { i } ) ^ { - 1 } \\mathrm { J } _ { \\phi ^ { \\prime } } ( c )$ . This proves the first equation. ", + "bbox": [ + 174, + 102, + 825, + 150 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "With Proposition E.16, ", + "bbox": [ + 174, + 155, + 326, + 170 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/1340b9d7789b80552d9923ec1b652735eb1728548ccf1f9f7d22815f7d0ea5de.jpg", + "text": "$$\n\\begin{array} { l } { \\mathrm { J } _ { \\phi } ^ { \\prime } ( c ) = ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { J } _ { \\alpha } ^ { \\prime } ( c ) + 2 a b \\mathrm { J } _ { \\alpha } ^ { \\prime } ( - c ) } \\\\ { = ( a ^ { 2 } + b ^ { 2 } ) ( 2 \\alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( \\mathrm { J } _ { \\alpha + 1 } ( c ) - c \\mathrm { J } _ { \\alpha } ( c ) ) } \\\\ { \\qquad + 2 a b ( 2 \\alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( \\mathrm { J } _ { \\alpha + 1 } ( - c ) + c \\mathrm { J } _ { \\alpha } ( - c ) ) } \\\\ { \\qquad = ( 2 \\alpha + 1 ) ( 1 - c ^ { 2 } ) ^ { - 1 } ( ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { J } _ { \\alpha + 1 } ( c ) + 2 a b \\mathrm { J } _ { \\alpha + 1 } ( - c ) ) - c \\mathrm { J } _ { \\phi } ( c ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 178, + 748, + 262 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "This gives the second equation. ", + "bbox": [ + 174, + 268, + 380, + 284 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Expanding $\\mathrm { J } _ { \\alpha + 1 } ( 0 )$ With Proposition E.11, we get ", + "bbox": [ + 173, + 290, + 508, + 306 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/d604a08488be84b0afa4c608e2174c94c62bfe9f0189208dd25ee99f7c4eb05b.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathrm { J } _ { \\phi } ^ { \\prime } ( 0 ) = ( 2 \\alpha + 1 ) ( ( a ^ { 2 } + b ^ { 2 } ) \\mathrm { J } _ { \\alpha + 1 } ( 0 ) + 2 a b \\mathrm { J } _ { \\alpha + 1 } ( 0 ) ) } \\\\ { = ( 2 \\alpha + 1 ) ( a + b ) ^ { 2 } \\displaystyle \\frac { 1 } { 2 \\sqrt { \\pi } } \\frac { \\Gamma ( \\frac { \\alpha } { 2 } + 1 ) ^ { 2 } } { \\Gamma ( \\alpha + \\frac { 3 } { 2 } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 321, + 314, + 676, + 376 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Unpacking the definition of $\\Gamma ( \\alpha + \\frac { 3 } { 2 } )$ then yields the third equation. ", + "bbox": [ + 173, + 383, + 620, + 401 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "In general, we can factor diagonal matrices out of $\\mathrm { V } _ { \\phi }$ . ", + "bbox": [ + 173, + 426, + 529, + 443 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proposition E.24. For any $\\Sigma \\in { \\cal S } _ { B }$ , $D$ any diagonal matrix, and $\\phi$ positive-homogeneous with degree $\\alpha$ , ", + "bbox": [ + 173, + 448, + 825, + 477 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/e574e51762024e15c3ba47eece5b89da5e9101b25f3bb3de2743d3cfeeec612b.jpg", + "text": "$$\n\\mathrm { V } _ { \\phi } ( D \\Sigma D ) = D ^ { \\alpha } \\mathrm { V } _ { \\phi } ( \\Sigma ) D ^ { \\alpha }\n$$", + "text_format": "latex", + "bbox": [ + 403, + 479, + 594, + 498 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proof. For any $i$ ", + "bbox": [ + 173, + 523, + 289, + 539 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/85816d587fc138843fc507037c02d315a8a1a72ba1941d8227f7240689937909.jpg", + "text": "$$\n\\begin{array} { r l } & { { \\mathbf V } _ { \\phi } ( D \\Sigma D ) _ { i i } = \\mathbb E [ \\phi ( x ) ^ { 2 } : x \\sim { \\mathcal N } ( 0 , D _ { i i } ^ { 2 } \\Sigma _ { i i } ) ] } \\\\ & { \\qquad = \\mathbb E [ \\phi ( D _ { i i } x ) ^ { 2 } : x \\sim { \\mathcal N } ( 0 , \\Sigma _ { i i } ) ] } \\\\ & { \\qquad = \\mathbb E [ D _ { i i } ^ { 2 \\alpha } \\phi ( x ) ^ { 2 } : x \\sim { \\mathcal N } ( 0 , \\Sigma _ { i i } ) ] } \\\\ & { \\qquad = D _ { i i } ^ { 2 \\alpha } { \\mathbf V } _ { \\phi } ( \\Sigma ) _ { i i } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 546, + 651, + 630 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "For any $i \\neq j$ ", + "bbox": [ + 173, + 655, + 269, + 670 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/0026682f39bfccecab109fc297ed4a51b730a8bfb38e2f66b82c65a65aadff29.jpg", + "text": "$$\n\\begin{array} { r l } & { { \\mathrm { V } _ { \\phi } } ( D \\Sigma D ) _ { i j } = \\mathbb { E } [ \\phi ( x ) \\phi ( y ) : ( x , y ) \\sim \\mathcal { N } ( 0 , \\left( \\begin{array} { c c } { D _ { i i } ^ { 2 } \\Sigma _ { i i } } & { D _ { i i } D _ { j j } \\Sigma _ { i j } } \\\\ { D _ { i i } D _ { j j } \\Sigma _ { j i } } & { D _ { j j } ^ { 2 } \\Sigma _ { j j } } \\end{array} \\right) ) } \\\\ & { \\quad \\quad \\quad \\quad = \\mathbb { E } [ \\phi ( D _ { i i } x ) \\phi ( D _ { j j } y ) : ( x , y ) \\sim \\mathcal { N } ( 0 , \\left( \\begin{array} { c c } { \\Sigma _ { i i } } & { \\Sigma _ { i j } } \\\\ { \\Sigma _ { j i } } & { \\Sigma _ { j j } } \\end{array} \\right) ) ] } \\\\ & { \\quad \\quad \\quad = D _ { i i } ^ { \\alpha } D _ { j j } ^ { \\alpha } \\mathbb { E } [ \\phi ( x ) \\phi ( y ) : ( x , y ) \\sim \\mathcal { N } ( 0 , \\left( \\begin{array} { c c } { \\Sigma _ { i i } } & { \\Sigma _ { i j } } \\\\ { \\Sigma _ { j i } } & { \\Sigma _ { j j } } \\end{array} \\right) ) ] } \\\\ & { \\quad \\quad \\quad = D _ { i i } ^ { \\alpha } D _ { j j } ^ { \\alpha } \\nabla _ { \\phi } ( \\Sigma ) _ { i j } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 681, + 745, + 806 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "E.2 FOURIER METHOD ", + "text_level": 1, + "bbox": [ + 174, + 867, + 348, + 882 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "The Laplace method crucially used the fact that we can pull out the norm factor $\\| G h \\|$ out of $\\phi$ , so that we can apply Schwinger parametrization. For general $\\phi$ this is not possible, but we can apply ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "some ", + "bbox": [ + 174, + 103, + 475, + 118 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/6aa7b856497603082fe39e73d666df2bb30c3bb73f8f0a773801a04388f2b28e.jpg", + "text": "$$\n\\begin{array}{c} \\begin{array} { r l } & { \\mathrm { ( d e r a t ) ~ t o ~ E q r i e n ) } } \\\\ & { \\mathrm { ( b y ~ t o ~ E q r i e n ) } } \\\\ & { \\mathrm { ( b y ~ t o ~ E q r i e n ) } } \\\\ & { = \\mathrm { ( d e r a t ) } } \\\\ & { \\mathrm { ( b y ~ t o ~ E q r i e n ) } } \\\\ & { = \\mathrm { ( d e r a t ) } } \\\\ & { = \\mathrm { ( d e r a t ) } } \\\\ & { \\mathrm { ( b y ~ t o ~ E q r i e n ) } } \\\\ & { = \\mathrm { ( d e r a t ) } } \\\\ & { \\mathrm { ( b y ~ t o ~ E q r i e n ) } } \\\\ & { = \\mathrm { ( d e r a t ) } } \\\\ & { \\mathrm { ( c r a t e n ) } } \\\\ & { = \\mathrm { ( d e r a t ) } } \\\\ & { \\mathrm { ( c r a t e n ) } } \\end{array} \\quad \\begin{array} { r l } { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\end{array} \\\\ & { \\quad } \\\\ & { \\quad + \\mathrm { ( d ) } } \\\\ & { \\quad { ( d ) } } \\\\ & { = \\mathrm { ( d e r a t ) } } \\\\ & { \\quad + \\mathrm { ( d ) } } \\\\ & { \\quad { ( d ) } } \\end{array} \\\\ & { \\quad = \\begin{array} { r l } { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\end{array} } \\\\ & { \\quad = \\begin{array} { r l } { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\end{array} } \\\\ & \\quad = \\begin{array} { r l } { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\\\ { \\mathrm { ( d ) } } & { \\mathrm { ( d ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 106, + 802, + 518 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Here all steps are legal except for possibly Eq. (29), Eq. (30), and Eq. (31). In Eq. (29), we “Fourier expanded” the delta function — this could possibly be justified as a principal value integral, but this cannot work in combination with Eq. (30) where we have switched the order of integration, integrating over $h$ first. Fubini’s theorem would not apply here because $e ^ { i \\lambda ( r ^ { 2 } - \\| G h \\| ^ { 2 } / B ) }$ has norm 1 and is not integrable. Finally, in Eq. (31), we need to extend the definition of Gaussian to complex covariance matrices, via complex Gaussian integration. This last point is no problem, and we will just define ", + "bbox": [ + 173, + 561, + 826, + 662 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/b141636dd0c6db3e7efb76e70199ee4b327362850a22a1bff7c76d49f945eae0.jpg", + "text": "$$\n\\underset { h \\sim \\mathcal { N } ( 0 , \\Sigma ) } { \\mathbb { E } } f ( h ) : = \\operatorname* { d e t } ( 2 \\pi \\Sigma ) ^ { - 1 / 2 } \\int _ { \\mathbb { R } ^ { B } } \\mathrm { d } h f ( h ) e ^ { - \\frac { 1 } { 2 } h ^ { T } \\Sigma ^ { - 1 } h }\n$$", + "text_format": "latex", + "bbox": [ + 305, + 669, + 691, + 702 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "for general complex $\\Sigma$ , whenever $\\Sigma$ is nonsingular and this integral is exists, and ", + "bbox": [ + 171, + 708, + 702, + 723 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/9e3320cdce33e24a9ac8713c24fb098669492647afeaa8ae80f16b228911e143.jpg", + "text": "$$\n\\underset { h \\sim \\mathcal { N } ( 0 , G \\Sigma G ) } { \\mathbb { E } } f ( h ) : = \\operatorname* { d e t } ( 2 \\pi ( \\mathfrak { e } ^ { T } \\Sigma \\mathfrak { e } ) ) ^ { - 1 / 2 } \\int _ { \\mathbb { R } ^ { B - 1 } } \\mathrm { d } z f ( \\mathfrak { e } z ) e ^ { - \\frac { 1 } { 2 } z ^ { T } ( \\mathfrak { e } ^ { T } \\Sigma \\mathfrak { e } ) ^ { - 1 } z }\n$$", + "text_format": "latex", + "bbox": [ + 248, + 729, + 746, + 762 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "if $\\Phi ^ { T } \\Sigma \\Phi$ is nonsingular (as in the case above). The definition of $\\mathrm { V } _ { \\phi }$ stays the same given the above extension of the definition of Gaussian expectation. ", + "bbox": [ + 173, + 771, + 821, + 800 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Nevertheless, the above derivation is not correct mathematically due to the other points. However, its end result can be justified rigorously by carefully expressing the delta function as the limit of mollifiers. ", + "bbox": [ + 174, + 806, + 825, + 848 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "Theorem E.25. Let $\\Sigma$ be a positive-definite $D \\times D$ matrix. Let $\\phi : \\mathbb { R } \\mathbb { R }$ . Suppose for any ${ \\mathsf b } > { \\mathsf a } > 0$ , ", + "bbox": [ + 169, + 852, + 823, + 883 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "1. $\\begin{array} { r } { \\int _ { \\mathsf { a } } ^ { \\mathsf { b } } \\mathrm { d } ( r ^ { 2 } ) | \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) | } \\end{array}$ exists and is finite for any $z _ { a } , z _ { b } \\in \\mathbb { R }$ (i.e. $\\phi ( z _ { a } / r ) \\phi ( z _ { b } / r )$ is locally integrable in $r$ ). ", + "bbox": [ + 209, + 893, + 825, + 924 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "2. there exists an $\\epsilon > 0$ such that any $\\gamma \\in [ - \\epsilon , \\epsilon ]$ and for each $a , b \\in [ D ]$ , ", + "bbox": [ + 209, + 102, + 696, + 119 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/98d0b82c9520a9380476ad22743043bdb767f2d017a06a642f29476238f60a33.jpg", + "text": "$$\n\\int _ { \\| z \\| ^ { 2 } \\in [ \\mathbf { a } , \\mathbf { b } ] } \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } | \\phi ( z _ { a } / \\sqrt { \\| z \\| ^ { 2 } + \\gamma } ) \\phi ( z _ { b } / \\sqrt { \\| z \\| ^ { 2 } + \\gamma } ) |\n$$", + "text_format": "latex", + "bbox": [ + 313, + 123, + 741, + 157 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "exists and is uniformly bounded by some number possibly depending on a and b. ", + "bbox": [ + 230, + 161, + 759, + 178 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "3. for each $\\begin{array} { r } { a , b \\in [ D ] , \\int _ { \\mathbb { R } ^ { D } } \\mathrm { d } z \\int _ { \\mathsf { a } } ^ { \\mathsf { b } } \\mathrm { d } ( r ^ { 2 } ) e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } | \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) | } \\end{array}$ exists and is finite. ", + "bbox": [ + 210, + 185, + 787, + 207 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "4. $\\begin{array} { r } { \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\left| \\operatorname* { d e t } ( I + 2 i \\lambda \\Sigma ) ^ { - 1 / 2 } \\mathrm { V } _ { \\phi } ( ( \\Sigma ^ { - 1 } + 2 i \\lambda I ) ^ { - 1 } r ^ { - 2 } ) \\right| } \\end{array}$ exists and is finite for each $r > 0$ ", + "bbox": [ + 187, + 213, + 821, + 233 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "$I f e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\phi ( z / \\| z \\| ) ^ { \\otimes 2 }$ is integrable over $z \\in \\mathbb { R } ^ { D }$ , then ", + "bbox": [ + 171, + 252, + 558, + 271 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/07e9ca6e758676b6b2343f22a9c65fd71cdcdd6722e842e4f6884aea80086d6c.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle ( 2 \\pi ) ^ { - D / 2 } \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } \\int \\mathrm { d } z e ^ { - \\frac 1 2 z ^ { T } \\Sigma ^ { - 1 } z } \\phi ( z / \\| z \\| ) ^ { \\otimes 2 } } } \\\\ { { \\displaystyle = \\frac 1 { 2 \\pi } \\operatorname* { l i m } _ { a \\searrow 0 , \\flat \\ne \\infty } \\int _ { a } ^ { \\flat } \\mathrm { d } s \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda e ^ { i \\lambda s } \\operatorname* { d e t } ( I + 2 i \\lambda \\Sigma ) ^ { - 1 / 2 } \\nabla _ { \\phi } \\big ( ( \\Sigma ^ { - 1 } + 2 i \\lambda I ) ^ { - 1 } s ^ { - 1 } \\big ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 273, + 776, + 347 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Similarly, ", + "bbox": [ + 173, + 348, + 238, + 363 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/ac2bf9c312c16ea59270cc9b71cb94e95aa1e33d472fd86a2848f2a52ecb613f.jpg", + "text": "$$\n\\begin{array} { l } { { ( 2 \\pi ) ^ { - D / 2 } \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } \\displaystyle \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\phi ( \\sqrt { D } z / \\| z \\| ) ^ { \\otimes 2 } } } \\\\ { { = \\displaystyle \\frac { 1 } { 2 \\pi } \\operatorname* { l i m } _ { a \\searrow 0 , \\flat , \\ne \\infty } \\int _ { \\mathrm { a } } ^ { \\flat } \\mathrm { d } s \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda e ^ { i \\lambda s } \\operatorname* { d e t } ( I + 2 i \\lambda \\Sigma ) ^ { - 1 / 2 } \\nabla _ { \\phi } ( ( \\Sigma ^ { - 1 } + 2 i \\lambda I / D ) ^ { - 1 } s ^ { - 1 } ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 367, + 787, + 439 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "If $G$ is the mean-centering projection matrix and $\\Sigma ^ { G } = G \\Sigma G$ , then by the same reasoning as in Thm E.5, ", + "bbox": [ + 173, + 449, + 825, + 481 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/ccc5a752e0addc91fd546c15881377742d3d6efb59b7bcb3bf5efdf811189269.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\gamma _ { B _ { \\phi } } ( \\Sigma ) = ( 2 \\pi ) ^ { - D / 2 } \\operatorname * { d e t } \\Sigma ^ { - 1 / 2 } \\int \\operatorname * { d } z e ^ { - \\frac 1 2 z ^ { T } \\Sigma ^ { - 1 } z } \\phi ( \\sqrt { D } G z / \\| G z \\| ) ^ { \\otimes 2 } } } \\\\ { { \\displaystyle \\qquad = \\frac 1 { 2 \\pi } \\operatorname* { l i m } _ { a \\searrow 0 , 6 , 7 \\infty } \\int _ { \\mathbb { a } } ^ { \\mathbb { b } } \\mathrm { d } s \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda e ^ { i \\lambda s } \\operatorname * { d e t } ( I + 2 i \\lambda \\Sigma ^ { G } / D ) ^ { - 1 / 2 } { \\mathrm { V } } _ { \\phi } ( \\Sigma ^ { G } ( I + 2 i \\lambda \\Sigma ^ { G } / D ) ^ { - 1 } s ^ { - 1 / 2 } ) , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 482, + 839, + 555 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Note that assumption (1) is satisfied if $\\phi$ is continuous; assumption (4) is satisfied if $D \\geq 3$ and for all $\\Pi$ , $\\| \\mathrm { V } _ { \\phi } ( \\bar { \\Pi } ) \\| \\leq \\| \\Pi \\| ^ { \\alpha }$ for some $\\alpha \\geq 0$ ; this latter condition, as well as assumptions (2) and (3), will be satisfied if $\\phi$ is polynomially bounded. Thus the common coordinatewise nonlinearities ReLU, identity, tanh, etc all satisfy these assumptions. ", + "bbox": [ + 173, + 579, + 826, + 636 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Warning: in general, we cannot swap the order of integration as $\\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\int _ { - \\infty } ^ { \\infty }$ ds. For example, if $\\phi = \\mathrm { i d }$ , then ", + "bbox": [ + 171, + 641, + 825, + 672 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/1bbfc1ce19b35b3670c9fdcd27544789346891f02fcbe60ec5aa154253127ac7.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad ( 2 \\pi ) ^ { - D / 2 } \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } \\displaystyle \\int \\mathrm { d } z \\ e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\phi ( z / \\| z \\| ) ^ { \\otimes 2 } } \\\\ & { = \\displaystyle \\frac { 1 } { 2 \\pi } \\operatorname* { l i m } _ { \\mathrm { a } \\searrow 0 , \\mathrm { b } \\ne \\infty } \\int _ { \\mathrm { a } } ^ { \\mathrm { b } } \\mathrm { d } s \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda e ^ { i \\lambda s } \\operatorname* { d e t } ( I + 2 i \\lambda \\Sigma ) ^ { - 1 / 2 } ( \\Sigma ^ { - 1 } + 2 i \\lambda I ) ^ { - 1 } s ^ { - 1 } } \\\\ & { \\ne \\displaystyle \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\int _ { 0 } ^ { \\infty } \\mathrm { d } s e ^ { i \\lambda s } \\operatorname* { d e t } ( I + 2 i \\lambda \\Sigma ) ^ { - 1 / 2 } ( \\Sigma ^ { - 1 } + 2 i \\lambda I ) ^ { - 1 } s ^ { - 1 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 676, + 756, + 784 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "because the $s$ -integral in the latter diverges (in a neighborhood of 0). ", + "bbox": [ + 173, + 785, + 622, + 801 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Proof. We will only prove the first equation; the others follow similarly. ", + "bbox": [ + 173, + 815, + 647, + 832 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "By dominated convergence, ", + "bbox": [ + 173, + 837, + 357, + 852 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/696b07919a2407a41792f2489d4afef9d8c81e429b1d52b3e5f3c6679986aa62.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\phi ( z / \\| z \\| ) ^ { \\otimes 2 } } \\\\ & { = \\displaystyle \\operatorname* { l i m } _ { \\mathfrak { a } \\searrow 0 , \\mathfrak { b } , \\nearrow \\infty } \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\phi ( z / \\| z \\| ) ^ { \\otimes 2 } \\mathbb { I } ( \\| z \\| ^ { 2 } \\in [ \\mathsf { a } , \\mathfrak { b } ] ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 856, + 694, + 922 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Let $\\eta : \\mathbb { R } \\mathbb { R }$ be a nonnegative bump function (i.e. compactly supported and smooth) with support $[ - 1 , 1 ]$ and integral 1 such that $\\eta ^ { \\prime } ( 0 \\bar { ) } = \\eta ^ { \\prime \\prime } ( 0 ) = 0$ . Then its Fourier transform $\\hat { \\eta } ( t )$ decays like $\\bar { \\boldsymbol { O } } ( t ^ { - 2 } )$ . Furthermore, $\\eta _ { \\epsilon } ( x ) : = \\epsilon ^ { - 1 } \\eta ( x / \\epsilon )$ is a mollifier, i.e. for all $f \\in L ^ { 1 } ( \\mathbb { R } )$ , $f * \\eta _ { \\epsilon } \\to f$ in $L ^ { 1 }$ and pointwise almost everywhere. ", + "bbox": [ + 173, + 103, + 825, + 160 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Now, we will show that ", + "bbox": [ + 174, + 166, + 331, + 181 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/3ff38307eafa3d46b3c258575b9d8a63b0a197263bf47169839cdb4329a1390c.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\mathbb { I } ( | | z | | ^ { 2 } \\in [ \\mathbf { a } , \\mathbf { b } ] ) } } \\\\ { { \\displaystyle \\qquad = \\operatorname* { l i m } _ { \\epsilon \\searrow 0 } \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\int _ { \\mathbf { a } } ^ { \\mathbf { b } } \\mathrm { d } ( r ^ { 2 } ) \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\eta _ { \\epsilon } ( | | z | | ^ { 2 } - r ^ { 2 } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 198, + 746, + 268 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "by dominated convergence. Pointwise convergence is immediate, because ", + "bbox": [ + 173, + 284, + 656, + 300 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/4de25cc84defea0591427b88d7083e8b8e0b0a8b5c0a340b6ea462b91929c697.jpg", + "text": "$$\n\\begin{array} { r l } { \\displaystyle \\int _ { a } ^ { \\mathsf { b } } \\mathrm { d } ( r ^ { 2 } ) \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\eta _ { \\epsilon } ( r ^ { 2 } - \\| z \\| ^ { 2 } ) = \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } ( r ^ { 2 } ) \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\mathbb { I } ( r ^ { 2 } \\in [ \\mathsf { a } , \\mathsf { b } ] ) \\eta _ { \\epsilon } ( \\| z \\| ^ { 2 } - r ^ { 2 } ) } & { } \\\\ { = [ ( r ^ { 2 } \\mapsto \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\mathbb { I } ( r ^ { 2 } \\in [ \\mathsf { a } , \\mathsf { b } ] ) ) * \\eta _ { \\epsilon } ] ( \\| z \\| ^ { 2 } ) } & { } \\\\ { \\to \\phi ( z _ { a } / \\| z \\| ) \\phi ( z _ { b } / \\| z \\| ) \\mathbb { I } ( \\| z \\| ^ { 2 } \\in [ \\mathsf { a } , \\mathsf { b } ] ) } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 318, + 825, + 396 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "as $\\epsilon 0$ (where we used assumption (1) that $\\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\mathbb { I } ( r ^ { 2 } \\in [ \\mathsf { a } , \\mathsf { b } ] ) \\mathrm { i s } L ^ { 1 } )$ . ", + "bbox": [ + 173, + 411, + 714, + 429 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Finally, we construct a dominating integrable function. Observe ", + "bbox": [ + 173, + 434, + 594, + 450 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/a33c644bc5629200bcb2aac85045df8bc37ef99c959477efb7788736e722512e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\int \\mathrm { d } s e ^ { - \\frac { 1 } { 2 } s ^ { 2 } \\tau ^ { 2 } s ^ { 3 } } \\int _ { s } ^ { s } \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\omega _ { s } \\gamma ) \\mathrm { d } s ( | \\gamma | \\geq s _ { 1 } \\gamma ) } \\\\ & { = \\int \\mathrm { d } s ( e ^ { - \\frac { 1 } { 2 } s ^ { 2 } \\tau ^ { 3 } } ) ^ { s } \\int _ { s } ^ { s } \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( | \\gamma | \\geq - \\gamma ^ { 2 } ) } \\\\ & { = \\int \\mathrm { d } s \\mathrm { e } ^ { - \\frac { 1 } { 2 } s ^ { 2 } \\tau ^ { 3 } } \\int _ { s } ^ { s } \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma - \\gamma ) } \\\\ & { = \\int \\mathrm { d } s \\mathrm { e } ^ { - \\frac { 1 } { 2 } s ^ { 2 } \\tau ^ { 3 } } \\int _ { s } ^ { s } \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma - \\gamma ) } \\\\ & { = \\int \\mathrm { d } s \\mathrm { e } ^ { - \\frac { 1 } { 2 } s ^ { 2 } \\tau ^ { 3 } } \\int _ { s } ^ { s } \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma - \\gamma ) } \\\\ & { = \\int \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) \\mathrm { d } s ( \\gamma ) } \\\\ & \\qquad \\mathrm { d } s \\mathrm { e } ^ { - \\frac { 1 } { 2 } s ^ { 2 } } \\mathrm { e } ^ { \\frac { 1 } { 2 } s ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 196, + 467, + 810, + 755 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "For small enough $\\epsilon$ then, this is integrable by assumption (2), and yields a dominating integrable function for our application of dominated convergence. ", + "bbox": [ + 174, + 785, + 826, + 814 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "In summary, we have just proven that ", + "bbox": [ + 174, + 820, + 421, + 835 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/12acb27fe0287ab7e14c395d40347666cf95a906f504ed45887e976f5830d055.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\phi ( z / \\| z \\| ) ^ { \\otimes 2 } } } \\\\ { { \\displaystyle = \\operatorname* { l i m } _ { \\mathrm { a \\setminus { } 0 , b \\setminus { \\infty } } } \\operatorname* { l i m } _ { \\epsilon \\searrow 0 } \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\int _ { \\mathrm { a } } ^ { \\mathrm { b } } \\mathrm { d } ( r ^ { 2 } ) \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\eta _ { \\epsilon } ( \\| z \\| ^ { 2 } - r ^ { 2 } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 852, + 746, + 922 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Now, ", + "bbox": [ + 173, + 103, + 210, + 118 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/4bdc2b12d716aff3b455907cc70615f1dbee015b669347c4a6c2be526c66736a.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\int \\mathrm { d } z e ^ { - \\frac 1 2 z ^ { T } \\Sigma ^ { - 1 } z } \\int _ { 0 } ^ { \\mathbf { b } } \\mathrm { d } ( r ^ { 2 } ) \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\eta _ { \\epsilon } ( \\| z \\| ^ { 2 } - r ^ { 2 } ) } } \\\\ { { \\displaystyle = \\int \\mathrm { d } z e ^ { - \\frac 1 2 z ^ { T } \\Sigma ^ { - 1 } z } \\int _ { \\mathbf { a } } ^ { \\mathbf { b } } \\mathrm { d } ( r ^ { 2 } ) \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\frac 1 { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\hat { \\eta } _ { \\epsilon } ( \\lambda ) e ^ { - i \\lambda ( \\| z \\| ^ { 2 } - r ^ { 2 } ) } } } \\\\ { { \\displaystyle = \\frac 1 { 2 \\pi } \\int \\mathrm { d } z e ^ { - \\frac 1 2 z ^ { T } \\Sigma ^ { - 1 } z } \\int _ { \\mathbf { a } } ^ { \\mathbf { b } } \\mathrm { d } ( r ^ { 2 } ) \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\hat { \\eta } ( \\epsilon \\lambda ) e ^ { - i \\lambda ( \\| z \\| ^ { 2 } - r ^ { 2 } ) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 241, + 127, + 759, + 239 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Note that the absolute value of the integral is bounded above by ", + "bbox": [ + 173, + 247, + 591, + 262 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/e5c3af3be969e400ffbefc46991be209b6fd7e66e5e762975a4769259e132911.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\frac 1 { 2 \\pi } \\int \\mathrm { d } z e ^ { - \\frac 1 2 z ^ { T } \\Sigma ^ { - 1 } z } \\int _ { \\mathrm { a } } ^ { \\mathrm { b } } \\mathrm { d } ( r ^ { 2 } ) | \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) | \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda | \\hat { \\eta } ( \\epsilon \\lambda ) e ^ { - i \\lambda ( \\| z \\| ^ { 2 } - r ^ { 2 } ) } | } } \\\\ { { \\displaystyle = \\frac 1 { 2 \\pi } \\int \\mathrm { d } z e ^ { - \\frac 1 2 z ^ { T } \\Sigma ^ { - 1 } z } \\int _ { \\mathrm { a } } ^ { \\mathrm { b } } \\mathrm { d } ( r ^ { 2 } ) | \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) | \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda | \\hat { \\eta } ( \\epsilon \\lambda ) | } } \\\\ { { \\displaystyle \\le C \\int \\mathrm { d } z e ^ { - \\frac 1 2 z ^ { T } \\Sigma ^ { - 1 } z } \\int _ { \\mathrm { a } } ^ { \\mathrm { b } } \\mathrm { d } ( r ^ { 2 } ) | \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) | } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 271, + 769, + 385 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "for some $C$ , by our construction of $\\eta$ that $\\hat { \\eta } ( t ) = O ( t ^ { - 2 } )$ for large $| t |$ . By assumption (3), this integral exists. Therefore we can apply the Fubini-Tonelli theorem and swap order of integration. ", + "bbox": [ + 174, + 392, + 825, + 422 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/1e91861764a9beba2810317fde021d6566fb15961c5e74472f61bff561df695a.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\frac 1 { 2 \\pi } \\int \\mathrm { d } z e ^ { - \\frac 1 2 z ^ { 2 } \\sum ^ { - 1 } z } \\int _ { a } ^ { \\mathbf b } \\mathrm { d } ( r ^ { 2 } ) \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\hat { \\eta } ( c ) \\ : e ^ { - i \\lambda ( | z | ^ { 2 } - r ^ { 2 } ) } } } \\\\ { { \\displaystyle = \\frac 1 { 2 \\pi } \\int _ { a } ^ { \\mathbf b } \\mathrm { d } ( r ^ { 2 } ) \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\hat { \\eta } ( c \\lambda ) e ^ { - i \\lambda ( | z | ^ { 2 } - r ^ { 2 } ) } \\int \\mathrm { d } z \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) e ^ { - \\frac 1 2 z ^ { 2 } \\tau \\sum ^ { - 1 } z } } } \\\\ { { \\displaystyle = \\frac 1 { 2 \\pi } \\int _ { a } ^ { \\mathbf b } \\mathrm { d } ( r ^ { 2 } ) \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\hat { \\eta } ( c ) \\ : e ^ { i \\lambda r ^ { 2 } } \\int \\mathrm { d } z \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) e ^ { - \\frac 1 2 z ^ { 2 } \\tau ( \\mathbb S ^ { - 1 } + 2 i \\lambda I ) z } } } \\\\ { { \\displaystyle = \\frac 1 { 2 \\pi } \\int _ { a } ^ { \\mathbf b } \\mathrm { d } ( r ^ { 2 } ) r ^ { D } \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\hat { \\eta } ( c \\lambda ) e ^ { i \\lambda r ^ { 2 } } \\int \\mathrm { d } z \\phi ( z _ { a } ) \\phi ( z _ { a } ) \\ : e ^ { - \\frac 1 2 r ^ { 2 } z ^ { 2 } \\tau ( \\mathbb S ^ { - 1 } + 2 i \\lambda I ) z } } } \\\\ { \\displaystyle = \\frac 1 { 2 \\pi } \\int _ { a } ^ { \\mathbf b } \\mathrm { d } ( r ^ { 2 } ) r ^ { 2 } \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\hat { \\eta } ( c ) \\ : e ^ { i \\lambda r ^ { 2 } } \\int \\mathrm { d } z \\phi ( z _ { a } ) \\ : \\phi ( z _ { b } ) e ^ { - \\frac 1 2 r ^ { 2 } z ^ { 2 } \\tau \\left( \\mathbb S ^ { - 1 } + 2 i \\lambda I \\right) z } } \\\\ \\displaystyle = ( 2 \\pi ) ^ { D / 2 - 1 } \\mathrm { d } \\mathbf e \\mathrm { t } \\ : \\Sigma ^ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 431, + 833, + 622 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "By assumption (4), ", + "bbox": [ + 173, + 628, + 302, + 643 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/8ddfdf70b02fb51dd9aca400f986a4725836bcabbd6755fdee7379a2c3651189.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\displaystyle \\int _ { \\ b { \\mathrm { \\ b { a } } } } ^ { \\ b } \\mathrm { d } ( r ^ { 2 } ) \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\ \\left| \\hat { \\eta } ( \\epsilon \\lambda ) e ^ { i \\lambda r ^ { 2 } } \\operatorname* { d e t } ( I + 2 i \\lambda \\Sigma ) ^ { - 1 / 2 } \\nabla _ { \\phi } ( ( \\Sigma ^ { - 1 } + 2 i \\lambda I ) ^ { - 1 } r ^ { - 2 } ) \\right| } \\\\ & { \\leq C \\int _ { \\ b { \\mathrm { \\ b { a } } } } ^ { \\ b } \\mathrm { d } ( r ^ { 2 } ) \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda \\ \\left| \\operatorname* { d e t } ( I + 2 i \\lambda \\Sigma ) ^ { - 1 / 2 } \\nabla _ { \\phi } ( ( \\Sigma ^ { - 1 } + 2 i \\lambda I ) ^ { - 1 } r ^ { - 2 } ) \\right| } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 654, + 769, + 729 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "is integrable (where we used the fact that $\\hat { \\eta }$ is bounded — which is true for all $\\eta \\in C _ { 0 } ^ { \\infty } ( { \\mathbb { R } } ) )$ , so by dominated convergence (applied to $\\epsilon \\searrow 0$ ), ", + "bbox": [ + 173, + 737, + 826, + 767 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/d99b725930df5d77f3774b02487637219d9dad3135b61a4260eeacf39c830079.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\displaystyle \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\phi ( z / \\| z \\| ) ^ { \\otimes 2 } } \\\\ & { = \\underset { \\mathrm { \\tiny ~ a \\sim \\sim \\Theta , b , \\sim \\infty } } { \\operatorname* { l i m } } \\underset { \\epsilon \\searrow } { \\operatorname* { l i m } } \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\int _ { \\mathrm { a } } ^ { \\mathbf { b } } \\mathrm { d } ( r ^ { 2 } ) \\phi ( z _ { a } / r ) \\phi ( z _ { b } / r ) \\eta _ { \\epsilon } ( \\| z \\| ^ { 2 } - r ^ { 2 } ) } \\\\ & { = ( 2 \\pi ) ^ { D / 2 - 1 } \\operatorname* { d e t } \\Sigma ^ { 1 / 2 } \\underset { \\mathrm { \\tiny ~ a \\sim \\Theta , b , \\sim \\infty } } { \\operatorname* { l i m } } \\int _ { \\mathrm { a } } ^ { \\mathbf { b } } \\mathrm { d } ( r ^ { 2 } ) \\int _ { - \\infty } ^ { \\infty } \\mathrm { d } \\lambda e ^ { i \\lambda r ^ { 2 } } \\operatorname* { d e t } ( I + 2 i \\lambda \\Sigma ) ^ { - 1 / 2 } \\mathrm { V } _ { \\phi } ( ( \\Sigma ^ { - 1 } + 2 i \\lambda I ) ^ { - 1 } r ^ { - 2 } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 775, + 839, + 883 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "where we used the fact that $\\begin{array} { r } { \\hat { \\eta } ( 0 ) = \\int _ { \\mathbb { R } } \\eta ( x ) \\mathrm { d } x = 1 } \\end{array}$ by construction. This gives us the desired result after putting back in some constants and changing $r ^ { 2 } \\mapsto s$ . □ ", + "bbox": [ + 173, + 892, + 828, + 925 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "E.3 SPHERICAL INTEGRATION ", + "text_level": 1, + "bbox": [ + 174, + 103, + 398, + 118 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "As mentioned above, the scale invariance of batchnorm combined with the BSB1 fixed point assumption will naturally lead us to consider spherical integration. Here we review several basic facts. ", + "bbox": [ + 173, + 128, + 823, + 159 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Definition E.26. We define the spherical angular coordinates in $\\mathbb { R } ^ { B }$ , $( r , \\theta _ { 1 } , \\ldots , \\theta _ { B - 2 } ) \\in [ 0 , \\infty ) \\times$ $\\mathsf { A } = : [ 0 , \\pi ] ^ { B - 3 } \\times [ 0 , 2 \\pi ]$ , by ", + "bbox": [ + 173, + 161, + 823, + 190 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/d5b6860fc1722c0a2cfebfab20fd183ee0a3d6e8a2e10f1bbafbc179e0891f87.jpg", + "text": "$$\n\\begin{array} { c } { x _ { 1 } = r \\cos \\theta _ { 1 } } \\\\ { x _ { 2 } = r \\sin \\theta _ { 1 } \\cos \\theta _ { 2 } } \\\\ { x _ { 3 } = r \\sin \\theta _ { 1 } \\sin \\theta _ { 2 } \\cos \\theta _ { 3 } } \\\\ { \\vdots } \\\\ { x _ { B - 2 } = r \\sin \\theta _ { 1 } \\cdot \\cdot \\cdot \\sin \\theta _ { B - 3 } \\cos \\theta _ { B - 2 } } \\\\ { x _ { B - 1 } = r \\sin \\theta _ { 1 } \\cdot \\cdot \\cdot \\sin \\theta _ { B - 3 } \\sin \\theta _ { B - 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 194, + 629, + 313 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Then we set ", + "bbox": [ + 173, + 315, + 256, + 330 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/7e3261c0a2d7ae4b3b5cb945e831c88eff8f79ffe5dd39699909b9f8efbf0e64.jpg", + "text": "$$\n\\begin{array} { r } { v : = \\left( \\begin{array} { c } { \\cos \\theta _ { 1 } } \\\\ { \\sin \\theta _ { 1 } \\cos \\theta _ { 2 } } \\\\ { \\sin \\theta _ { 1 } \\sin \\theta _ { 2 } \\cos \\theta _ { 3 } } \\\\ { \\vdots } \\\\ { \\sin \\theta _ { 1 } \\cdot \\cdot \\cdot \\sin \\theta _ { B - 3 } \\cos \\theta _ { B - 2 } } \\\\ { \\sin \\theta _ { 1 } \\cdot \\cdot \\cdot \\sin \\theta _ { B - 3 } \\sin \\theta _ { B - 2 } } \\end{array} \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 372, + 333, + 624, + 430 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "as the unit norm version of $x$ . The integration element in this coordinate satisfies ", + "bbox": [ + 171, + 433, + 704, + 449 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/3cb6a00f4e60bcbcc6e26f28ffb826b4f513a132ca4db9a19db1a1a88e91e021.jpg", + "text": "$$\n\\mathrm { d } x _ { 1 } \\dots \\mathrm { d } x _ { B - 1 } = r ^ { B - 2 } \\sin ^ { B - 3 } \\theta _ { 1 } \\sin ^ { B - 4 } \\theta _ { 2 } \\cdot \\cdot \\cdot \\sin \\theta _ { B - 3 } \\mathrm { d } r \\mathrm { d } \\theta _ { 1 } \\cdot \\cdot \\cdot \\mathrm { d } \\theta _ { B - 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 453, + 746, + 472 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Fact E.27. If $F : S ^ { B - 2 } \\to \\mathbb { R }$ , then its expectation on the sphere can be written as ", + "bbox": [ + 171, + 478, + 715, + 494 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/aab4cdb8306ec4e374153c57da1e66bf9f7945a9ea56c748788025a9bebe51ce.jpg", + "text": "$$\n\\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } F ( v ) = \\frac { 1 } { 2 } \\pi ^ { \\frac { 1 - B } { 2 } } \\Gamma \\left( \\frac { B - 1 } { 2 } \\right) \\int _ { \\mathbb { A } } F ( v ) \\sin ^ { B - 3 } \\theta _ { 1 } \\cdot \\cdot \\cdot \\sin \\theta _ { B - 3 } \\mathrm { d } \\theta _ { 1 } \\cdot \\cdot \\cdot \\mathrm { d } \\theta _ { B - 2 }\n$$", + "text_format": "latex", + "bbox": [ + 223, + 498, + 772, + 534 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "where $\\mathsf { A } = [ 0 , \\pi ] ^ { B - 3 } \\times [ 0 , 2 \\pi ]$ and $v$ are as in Defn E.26. ", + "bbox": [ + 174, + 537, + 552, + 555 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "The following integrals will be helpful for simplifying many expressions involving spherical integration using angular coordinates. ", + "bbox": [ + 173, + 564, + 823, + 594 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Lemma E.28. For $j , k \\geq 0$ and $0 \\leq s \\leq t \\leq \\pi / 2$ , ", + "bbox": [ + 174, + 597, + 506, + 613 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/60a0c058f3a1241ae4b74040d69d56ce9d8ace1435f5790dafac9083a860b10d.jpg", + "text": "$$\n\\int _ { s } ^ { t } \\sin ^ { j } \\theta \\cos ^ { k } \\theta \\mathrm { { d } } \\theta = { \\frac { 1 } { 2 } } \\operatorname { B e t a } \\left( \\cos ^ { 2 } t , \\cos ^ { 2 } s ; { \\frac { k + 1 } { 2 } } , { \\frac { j + 1 } { 2 } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 290, + 618, + 704, + 654 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "By antisymmetry of cos with respect to $\\theta \\mapsto \\pi - \\theta , i f \\pi / 2 \\leq s \\leq t \\leq \\pi ,$ , ", + "bbox": [ + 176, + 657, + 643, + 674 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/0829a7aaf173a7708f1efd90c7018cef4f74589f9b70d639e1d6613e4257b7e4.jpg", + "text": "$$\n\\int _ { s } ^ { t } \\sin ^ { j } \\theta \\cos ^ { k } \\theta \\mathrm { d } \\theta = { \\frac { 1 } { 2 } } ( - 1 ) ^ { k } \\operatorname { B e t a } \\left( \\cos ^ { 2 } s , \\cos ^ { 2 } t ; { \\frac { k + 1 } { 2 } } , { \\frac { j + 1 } { 2 } } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 678, + 725, + 714 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proof. Set $x : = \\cos ^ { 2 } \\theta \\implies \\mathrm { d } x = - 2 \\cos \\theta \\sin \\theta \\mathrm { d } \\theta$ . So the integral in question is ", + "bbox": [ + 173, + 727, + 715, + 744 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/6d27ca9b595273edaf08378bff075584d5e661c2740f4436d302b85dd947196c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\displaystyle \\frac { - 1 } { 2 } \\int _ { \\cos ^ { 2 } s } ^ { \\cos ^ { 2 } t } ( 1 - x ) ^ { \\frac { j - 1 } { 2 } } x ^ { \\frac { k - 1 } { 2 } } \\mathrm { d } x } \\\\ & { = \\displaystyle \\frac { 1 } { 2 } ( \\mathrm { B e t a } ( \\cos ^ { 2 } s ; \\frac { k + 1 } { 2 } , \\frac { j + 1 } { 2 } ) - \\mathrm { B e t a } ( \\cos ^ { 2 } t ; \\frac { k + 1 } { 2 } , \\frac { j + 1 } { 2 } ) ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 285, + 748, + 714, + 820 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "As consequences, ", + "bbox": [ + 174, + 852, + 292, + 864 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Lemma E.29. For $j , k \\geq 0$ , ", + "bbox": [ + 173, + 869, + 361, + 885 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/af2b264074ab0d75e5f564b2e861213713ffec0a624f41ed6637098744485616.jpg", + "text": "$$\n\\int _ { 0 } ^ { \\pi } \\sin ^ { j } \\theta \\cos ^ { k } \\theta \\mathrm { d } \\theta = { \\frac { 1 + ( - 1 ) ^ { k } } { 2 } } \\operatorname { B e t a } ( { \\frac { j + 1 } { 2 } } , { \\frac { k + 1 } { 2 } } ) = { \\frac { 1 + ( - 1 ) ^ { k } } { 2 } } { \\frac { \\Gamma ( { \\frac { 1 + j } { 2 } } ) \\Gamma ( { \\frac { 1 + k } { 2 } } ) } { \\Gamma ( { \\frac { 2 + j + k } { 2 } } ) } } .\n$$", + "text_format": "latex", + "bbox": [ + 214, + 890, + 782, + 928 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Lemma E.30. ", + "text_level": 1, + "bbox": [ + 173, + 103, + 269, + 118 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/eb808bf0f4308d294db542a01d2e8baba04e0c62a38843f82ded67e98f65ee8d.jpg", + "text": "$$\n\\int _ { 0 } ^ { \\pi } \\sin \\theta _ { 1 } \\mathrm { d } \\theta _ { 1 } \\int _ { 0 } ^ { \\pi } \\sin ^ { 2 } \\theta _ { 2 } \\mathrm { d } \\theta _ { 2 } \\cdot \\cdot \\cdot \\int _ { 0 } ^ { \\pi } \\sin ^ { k } \\theta _ { k } \\mathrm { d } \\theta _ { k } = \\pi ^ { k / 2 } / \\Gamma ( \\frac { k + 2 } { 2 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 269, + 133, + 725, + 167 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proof. By Lemma E.29, $\\begin{array} { r } { \\int _ { 0 } ^ { \\pi } \\sin ^ { r } \\theta _ { r } \\mathrm { d } \\theta _ { r } = \\mathrm { B e t a } \\big ( \\frac { r + 1 } { 2 } , \\frac { 1 } { 2 } \\big ) = \\frac { \\Gamma ( \\frac { r + 1 } { 2 } ) \\sqrt { \\pi } } { \\Gamma ( \\frac { r + 2 } { 2 } ) } } \\end{array}$ Γ( r+12 ) πr+2 . Thus this product of integrals is equal to ", + "bbox": [ + 174, + 218, + 823, + 258 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/4eae171b6c575d88389894751865bcbca0aab38af931049bfba8c8cbd674d01f.jpg", + "text": "$$\n\\prod _ { r = 1 } ^ { k } \\sqrt { \\pi } \\frac { \\Gamma \\big ( \\frac { r + 1 } { 2 } \\big ) } { \\Gamma \\big ( \\frac { r + 2 } { 2 } \\big ) } = \\pi ^ { k / 2 } \\Gamma ( 1 ) / \\Gamma \\big ( \\frac { k + 2 } { 2 } \\big ) = \\pi ^ { k / 2 } / \\Gamma \\big ( \\frac { k + 2 } { 2 } \\big ) .\n$$", + "text_format": "latex", + "bbox": [ + 302, + 275, + 694, + 319 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "$B > 1 , \\mu > 0$ , $\\begin{array} { r } { \\int _ { 0 } ^ { \\infty } r ^ { B } e ^ { - r ^ { 2 } / 2 \\mu } \\mathrm { d } r = 2 ^ { \\frac { B - 1 } { 2 } } \\mu ^ { \\frac { B + 1 } { 2 } } \\Gamma ( \\frac { B + 1 } { 2 } ) } \\end{array}$ ", + "bbox": [ + 173, + 392, + 671, + 414 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proof. Apply change of coordinates $z = r ^ { 2 } / 2 \\mu$ ", + "bbox": [ + 173, + 465, + 491, + 481 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Definition E.32. Let $K ( v ; \\Sigma , \\beta ) \\ : = \\ ( \\operatorname * { d e t } \\circledast ^ { T } \\Sigma \\circledast ^ { - 1 / 2 } ( v ^ { T } ( \\operatorname { \\mathbb { e } } ^ { T } \\Sigma \\circledast ^ { - 1 } v ) ^ { - \\frac { B - 1 + \\beta } { 2 } }$ and $\\kappa ( v ; \\Sigma ) : =$ $\\kappa ( v ; \\Sigma , 0 )$ . ", + "bbox": [ + 173, + 525, + 825, + 559 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "The following proposition allows us to express integrals involving batchnorm as expectations over the uniform spherical measure. ", + "bbox": [ + 173, + 574, + 821, + 604 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proposition E.33. If $f : \\mathbb { R } ^ { B - 1 } \\mathbb { R } ^ { B ^ { \\prime } }$ , $B > 1 - \\beta$ , and $\\Sigma \\in \\mathbb { R } ^ { ( B - 1 ) \\times ( B - 1 ) }$ is nonsingular, then ", + "bbox": [ + 171, + 612, + 813, + 632 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/c482110426d2ba4a868462407f8f118e0e03f472096801b189ca811336d39a92.jpg", + "text": "$$\n\\underset { h \\sim \\bar { N } ( 0 , \\Sigma ) } { \\mathbb { E } } \\Vert h \\Vert ^ { \\beta } f ( h / \\Vert h \\Vert ) = 2 ^ { \\beta / 2 } \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\frac { \\beta } { 2 } \\right) \\underset { v \\sim \\bar { S } ^ { B - 2 } } { \\mathbb { E } } f ( v ) ( \\operatorname* { d e t } \\Sigma ) ^ { - 1 / 2 } ( v ^ { T } \\Sigma ^ { - 1 } v ) ^ { - \\frac { B - 1 + \\beta } { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 187, + 648, + 805, + 683 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "If $f : \\mathbb { R } ^ { B } \\mathbb { R } ^ { B ^ { \\prime } }$ and $\\Sigma \\in \\mathbb { R } ^ { B \\times B }$ is such that $\\Phi ^ { T } \\Sigma \\Phi$ is nonsingular (where e is as in Defn E.3) and $G \\Sigma G = \\Sigma$ , ", + "bbox": [ + 168, + 702, + 826, + 733 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/8bd41e57d888ef50990461fb51e1e16c113f2eba8fe2d18f9492e29cd4572cf0.jpg", + "text": "$$\n\\underset { h \\sim \\mathcal { N } ( 0 , \\Sigma ) } { \\mathbb { E } } \\Vert G h \\Vert ^ { \\beta } f ( G h / \\Vert G h \\Vert ) = 2 ^ { \\beta / 2 } \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\frac { \\beta } { 2 } \\right) \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } f ( \\mathfrak { e } v ) K ( v ; \\Sigma , \\beta ) .\n$$", + "text_format": "latex", + "bbox": [ + 232, + 750, + 764, + 784 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "In particular, ", + "bbox": [ + 173, + 801, + 263, + 816 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/dbff5fc85290b337125825047c09a42e347917be77abd680fdd457dcf29fdb58.jpg", + "text": "$$\n\\mathbb { E } [ f ( B ( h ) ) : h \\sim \\mathcal { N } ( 0 , \\Sigma ) ] = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } f ( \\mathbb { e } v ) K ( v ; \\Sigma )\n$$", + "text_format": "latex", + "bbox": [ + 325, + 834, + 671, + 858 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "Proof. We will prove the second statement. The others follow trivially. ", + "bbox": [ + 173, + 909, + 640, + 925 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/671969c46105434b77ccb10af83ffefec016cb8ee591bec06788e8e5513bbc22.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathbb { E } [ \\| G h \\| ^ { \\beta } f ( \\mathcal { B } ( h ) ) : h \\sim \\mathcal { N } ( 0 , \\Sigma ) ] } \\\\ & { = \\mathbb { E } [ \\| x \\| ^ { \\beta } f ( \\mathbf { e x } / \\| x \\| ) : x \\sim \\mathcal { N } ( 0 , \\mathbf { e } ^ { T } \\Sigma \\mathbf { e } ) ] } \\\\ & { = ( 2 \\pi ) ^ { \\frac { 1 - B } { 2 } } ( \\operatorname* { d e t } \\mathbb { e } ^ { T } \\Sigma \\mathbf { e } ) ^ { - 1 / 2 } \\int _ { \\mathbb { R } ^ { B - 1 } } \\| x \\| ^ { \\beta } f ( \\mathbf { e x } / \\| x \\| ) e ^ { - \\frac { 1 } { 2 } x ^ { T } ( \\mathbf { e } ^ { T } \\Sigma \\mathbf { e } ) ^ { - 1 } x } \\mathrm { d } x } \\\\ & { = ( 2 \\pi ) ^ { \\frac { 1 - B } { 2 } } ( \\operatorname* { d e t } \\mathbb { e } ^ { T } \\Sigma \\mathbf { e } ) ^ { - 1 / 2 } \\int _ { \\mathbb { A } } f ( \\mathbf { e } v ) \\sin ^ { B - 3 } \\theta _ { 1 } \\cdot \\cdot \\cdot \\sin \\theta _ { B - 3 } \\mathrm { d } r \\mathrm { d } \\theta _ { 1 } \\cdot \\cdot \\cdot \\mathrm { d } \\theta _ { B - 2 } \\times } \\\\ & { \\quad \\quad \\quad \\int _ { 0 } ^ { \\infty } r ^ { B - 2 + \\beta } e ^ { - \\frac { 1 } { 2 } r ^ { 2 } \\cdot v ^ { T } ( \\mathbf { e } ^ { T } \\Sigma \\mathbf { e } ) ^ { - 1 } v } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 210, + 125, + 745, + 268 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "(spherical coordinates Defn E.26) ", + "bbox": [ + 565, + 270, + 789, + 285 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/dc744d136ad79fb939a612b88fc391f719b738cf6d076120ff8896296463b72b.jpg", + "text": "$$\n\\begin{array} { r l } & { = ( 2 \\pi ) ^ { \\frac { 1 - 2 } { 2 } } ( \\dot { \\mathrm { f o r } } \\theta ^ { \\top } \\nabla \\mathrm { s } \\mathrm { i } ) ^ { - 1 / 2 } \\times 2 ^ { \\frac { n - 3 } { 2 } } \\Gamma ( \\frac { B - 1 } { 2 } ) \\times \\frac { 2 ^ { \\frac { n - 3 } { 2 } } \\Gamma ( \\frac { B - 1 - \\rho \\xi } { 2 } ) } { 2 ^ { \\frac { n - 3 } { 2 } } \\Gamma ( \\frac { B - 1 } { 2 } ) } } \\\\ & { \\qquad \\times \\int _ { \\mathrm { a } } \\frac { f ( \\mathrm { e v } ) \\sin ^ { \\frac { \\theta - 3 } { 2 } } \\theta _ { 1 } - \\sin \\theta _ { B - 3 } \\sin \\theta _ { 1 } - \\sin \\theta _ { B - 4 } \\sin \\theta _ { B - 2 } } { ( \\sigma ^ { \\prime } ( \\mathrm { e } ^ { \\gamma } \\mathrm { s } \\mathrm { e } ^ { \\gamma } ) ) ^ { \\frac { n - 3 } { 2 } } \\frac { \\mathrm { d } \\theta _ { 1 } - \\cos \\theta _ { B - 2 } } { 1 / \\sigma ^ { \\prime } } } } \\\\ & { = \\frac { 1 } { 2 } \\pi ^ { \\frac { 1 - \\rho } { 2 } } \\Gamma ( \\frac { B - 1 } { 2 } ) \\times 2 ^ { \\rho / 2 } \\Gamma ( \\frac { B - 1 } { 2 } + \\frac { \\beta } { 2 } ) \\Gamma ( \\frac { B - 1 } { 2 } ) ^ { - 1 } } \\\\ & { \\qquad \\times \\int _ { \\mathrm { a } } \\frac { f ( \\mathrm { e v } ) \\sin ^ { \\frac { \\theta - 3 } { 2 } } \\theta _ { 1 } - \\sin \\theta _ { B - 3 } \\sin \\theta _ { 1 } - \\sin \\theta _ { 1 } - \\sin \\theta _ { B - 2 } } { ( \\sigma ^ { \\prime } ( \\mathrm { e } ^ { \\gamma } \\mathrm { e } ^ { \\gamma } ) ) ^ { \\frac { n - 3 } { 2 } - \\frac { n + 3 } { 2 } } } f ( \\mathrm { e } \\mathrm { e } ^ { \\gamma } \\mathrm { e } ^ { \\gamma } ) } \\\\ & = 2 ^ { 3 / 2 } \\Gamma ( \\frac { B - 1 + \\beta } { 2 } ) \\Gamma ( \\frac { B - 1 } { 2 } ) ^ { - 1 } \\frac { \\Gamma } \\mathrm { e } ^ { \\gamma } \\mathrm { e } ^ { \\gamma } \\mathrm { e } ^ { \\gamma } \\mathrm { e } ^ { \\gamma } \\mathrm { f o r } ( \\mathrm { e } ^ { \\gamma } \\mathrm { e } ^ { \\gamma } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 207, + 290, + 787, + 527 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "As a straightforward consequence, ", + "bbox": [ + 174, + 573, + 401, + 588 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proposition E.34. If $\\Phi ^ { T } \\Sigma \\Phi$ is nonsingular, ", + "bbox": [ + 174, + 592, + 459, + 608 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/c5ed8f3d715247160b4ceb9dee52ed91d7a2720e9c3c90d57d23981660978e2d.jpg", + "text": "$$\n\\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( \\Sigma ) = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathbb { e } v ) ^ { \\otimes 2 } \\mathcal { K } ( v ; \\Sigma )\n$$", + "text_format": "latex", + "bbox": [ + 364, + 617, + 633, + 643 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "For numerical calculations, it is useful to realize $\\mathbb { P }$ as the matrix whose columns are $\\begin{array} { l l } { \\Phi _ { B - m } } & { : = } \\end{array}$ $m$ $( m ( m + 1 ) ) ^ { - 1 / 2 } ( 0 , \\ldots , 0 , - m , \\overbrace { 1 , 1 , \\ldots , 1 } ^ { \\cdots } ) ^ { T }$ , for each $m = 1 , \\ldots , B - 1$ . ", + "bbox": [ + 173, + 684, + 825, + 729 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/cb0e24c83c1add9c94c96aaba3bdd1ffe3550d71b22f7df3c7a459adec5f22d4.jpg", + "text": "$$\n\\begin{array} { r } { \\boldsymbol { \\Phi } ^ { T } = \\left( \\begin{array} { l l l l l l } { \\frac { 1 } { \\sqrt { B ( B - 1 ) } } } & & & & & \\\\ & { \\frac { 1 } { \\sqrt { ( B - 1 ) ( B - 2 ) } } } & & & & \\\\ & & { \\ddots } & & & \\\\ & & & & { 6 ^ { - 1 / 2 } } & \\\\ & & & & & { 2 ^ { - 1 / 2 } } \\end{array} \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 738, + 673, + 835 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/57b820a35097179a536c703fad16eceaba8ff27034b5d147afec3e223a518582.jpg", + "text": "$$\n\\times \\left( \\begin{array} { c c c c c c c } { { - ( B - 1 ) } } & { { 1 } } & { { 1 } } & { { \\cdots } } & { { 1 } } & { { 1 } } & { { 1 } } \\\\ { { 0 } } & { { - ( B - 2 ) } } & { { 1 } } & { { \\cdots } } & { { 1 } } & { { 1 } } & { { 1 } } \\\\ { { \\vdots } } & { { } } & { { \\vdots } } & { { \\ddots } } & { { \\vdots } } & { { \\vdots } } & { { \\vdots } } \\\\ { { 0 } } & { { 0 } } & { { 0 } } & { { \\cdots } } & { { - 2 } } & { { 1 } } & { { 1 } } \\\\ { { 0 } } & { { 0 } } & { { 0 } } & { { \\cdots } } & { { 0 } } & { { - 1 } } & { { 1 } } \\end{array} \\right)\n$$", + "text_format": "latex", + "bbox": [ + 357, + 840, + 717, + 921 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "We have ", + "bbox": [ + 173, + 103, + 233, + 118 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/866e3a9841a0676b159f52727dabb45e3bf98bd4295d087d18afc5bfc9834d65.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle ( \\mathtt { e } v ) _ { 1 } = - \\sqrt { \\frac { B - 1 } { B } } \\cos \\theta _ { 1 } } \\\\ { \\displaystyle ( \\mathtt { e } v ) _ { 2 } = \\cos \\theta _ { 1 } - \\sqrt { \\frac { B - 2 } { B - 1 } } \\sin \\theta _ { 1 } \\cos \\theta _ { 2 } } \\\\ { \\displaystyle ( \\mathtt { e } v ) _ { 3 } = \\cos \\theta _ { 1 } + \\sin \\theta _ { 1 } \\cos \\theta _ { 2 } - \\sqrt { \\frac { B - 3 } { B - 2 } } \\sin \\theta _ { 1 } \\sin \\theta _ { 2 } \\cos \\theta _ { 3 } } \\\\ { \\displaystyle ( \\mathtt { e } v ) _ { 4 } = \\cos \\theta _ { 1 } + \\sin \\theta _ { 1 } \\cos \\theta _ { 2 } + \\sin \\theta _ { 1 } \\sin \\theta _ { 2 } \\cos \\theta _ { 3 } - \\sqrt { \\frac { B - 4 } { B - 3 } } \\sin \\theta _ { 1 } \\sin \\theta _ { 2 } \\sin \\theta _ { 3 } \\cos \\theta _ { 4 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 128, + 781, + 275 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "so in particular they only depend on $\\theta _ { 1 } , \\ldots , \\theta _ { 4 }$ . ", + "bbox": [ + 173, + 285, + 485, + 301 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Angular coordinates In many situations, the sparsity of this realization of $\\mathbb { P }$ combined with sufficient symmetry allows us to simplify the high-dimensional spherical integral into only a few dimensions, using one of the following lemmas ", + "bbox": [ + 174, + 324, + 825, + 367 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Lemma E.35. Let $f : \\mathbb { R } ^ { 4 } \\to \\mathbb { R } ^ { A }$ for some $A \\in \\mathbb { N }$ . Suppose $B \\geq 6$ . Then for any $k < B - 1$ ", + "bbox": [ + 166, + 373, + 781, + 390 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/f9dc56709efde69e9b38d4fb6429ef02d44a741a1ec34c2e206c7167fa9a0008.jpg", + "text": "$$\n\\begin{array} { l } { { \\ \\mathbb { E } [ r ^ { - k } f ( v _ { 1 } , \\dots , v _ { 4 } ) : x \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] } } \\\\ { { = { \\frac { \\Gamma ( ( B - 1 - k ) / 2 ) } { \\Gamma ( ( B - 5 ) / 2 ) } } 2 ^ { - k / 2 } \\pi ^ { - 2 } \\displaystyle \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } \\cdots \\displaystyle \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 4 } \\ f ( v _ { 1 } ^ { \\theta } , \\dots , v _ { 4 } ^ { \\theta } ) \\sin ^ { B - 3 } \\theta _ { 1 } \\cdots \\sin ^ { B - 6 } \\theta _ { 4 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 402, + 802, + 460 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "where $v _ { i } = x _ { i } / \\| x \\|$ and $r = \\| x \\|$ , and ", + "bbox": [ + 173, + 472, + 426, + 488 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/993e326cab0f53eb6ef19493bc87552c2c59685a27776d8cd0e60422f1ab644a.jpg", + "text": "$$\n\\begin{array} { r l } & { v _ { 1 } ^ { \\theta } = \\cos \\theta _ { 1 } } \\\\ & { v _ { 2 } ^ { \\theta } = \\sin \\theta _ { 1 } \\cos \\theta _ { 2 } } \\\\ & { v _ { 3 } ^ { \\theta } = \\sin \\theta _ { 1 } \\sin \\theta _ { 2 } \\cos \\theta _ { 3 } } \\\\ & { v _ { 4 } ^ { \\theta } = \\sin \\theta _ { 1 } \\sin \\theta _ { 2 } \\sin \\theta _ { 3 } \\cos \\theta _ { 4 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 395, + 501, + 602, + 583 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Lemma E.36. Let $f : \\mathbb { R } ^ { 2 } \\to \\mathbb { R } ^ { A }$ for some $A \\in \\mathbb { N }$ . Suppose $B \\geq 4$ . Then for any $k < B - 1$ ", + "bbox": [ + 166, + 597, + 781, + 614 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/60334962f0fdcbea43da15d150fc4a1f3ca0f76337439156bb50a114f41777cb.jpg", + "text": "$$\n\\begin{array} { c l } { { } } & { { \\mathbb { E } [ r ^ { - k } f ( v _ { 1 } , v _ { 2 } ) : x \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] } } \\\\ { { } } & { { = \\displaystyle \\frac { \\Gamma ( ( B - 1 - k ) / 2 ) } { \\Gamma ( ( B - 3 ) / 2 ) } 2 ^ { - k / 2 } \\pi ^ { - 1 } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 2 } f ( v _ { 1 } ^ { \\theta } , v _ { 2 } ^ { \\theta } ) \\sin ^ { B - 3 } \\theta _ { 1 } \\sin ^ { B - 4 } \\theta _ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 626, + 764, + 684 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "where $v _ { i } = x _ { i } / \\| x \\|$ and $r = \\| x \\|$ , and ", + "bbox": [ + 173, + 695, + 426, + 712 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/51b5cfc8c3650ada80042263a18514896ccbea1b1114444eb25b96d6b4cca34d.jpg", + "text": "$$\n\\begin{array} { l } { { v _ { 1 } ^ { \\theta } = \\cos \\theta _ { 1 } } } \\\\ { { v _ { 2 } ^ { \\theta } = \\sin \\theta _ { 1 } \\cos \\theta _ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 437, + 724, + 558, + 766 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Lemma E.37. Let $f : \\mathbb { R } \\to \\mathbb { R } ^ { A }$ for some $A \\in \\mathbb { N }$ . Suppose $B \\geq 3 .$ . Then for any $k < B - 1$ ", + "bbox": [ + 166, + 780, + 772, + 797 + ], + "page_idx": 34 + }, + { + "type": "equation", + "img_path": "images/5bc77ebce7c733aa97c02b4cc21ac2ae80096a87a733a70290d9e943ed1c08ae.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathbb { E } [ r ^ { - k } f ( v ) : x \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] } \\\\ & { = \\frac { \\Gamma ( ( B - 1 - k ) / 2 ) } { \\Gamma ( ( B - 2 ) / 2 ) } 2 ^ { - k / 2 } \\pi ^ { - 1 / 2 } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta \\ f ( \\cos \\theta ) \\sin ^ { B - 3 } \\theta } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 809, + 700, + 866 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "where $v _ { i } = x _ { i } / \\| x \\|$ and $r = \\| x \\|$ . ", + "bbox": [ + 173, + 878, + 395, + 895 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "We will prove Lemma E.35; those of Lemma E.36 and Lemma E.37 are similar. ", + "bbox": [ + 171, + 909, + 697, + 924 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof of Lemma E.35. ", + "bbox": [ + 173, + 103, + 318, + 118 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/f489b8282778be5888aaabfdb063fe7beceddea2e3aff660e9305ee5d2a41e9b.jpg", + "text": "$$\n\\begin{array} { l } { { \\mathbb { E } [ r ^ { - k } f ( v _ { 1 } , \\dots , v _ { 4 } ) : x \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] } } \\\\ { { = ( 2 \\pi ) ^ { - ( B - 1 ) / 2 } \\int \\mathrm { d } x \\ r ^ { - k } f ( v _ { 1 } , \\dots , v _ { 4 } ) e ^ { - r ^ { 2 } / 2 } } } \\\\ { { = ( 2 \\pi ) ^ { - ( B - 1 ) / 2 } \\int \\left( r ^ { B - 2 } \\sin ^ { B - 3 } \\theta _ { 1 } \\cdots \\sin \\theta _ { B - 3 } \\mathrm { d } r \\mathrm { d } \\theta _ { 1 } \\cdots \\mathrm { d } \\theta _ { B - 2 } \\right) r ^ { - k } f ( v _ { 1 } ^ { \\theta } , \\dots , v _ { 4 } ^ { \\theta } ) e ^ { - r ^ { 2 } / 2 } } } \\\\ { { = ( 2 \\pi ) ^ { - ( B - 1 ) / 2 } \\int f ( v _ { 1 } ^ { \\theta } , \\dots , v _ { 4 } ^ { \\theta } ) \\sin ^ { B - 3 } \\theta _ { 1 } \\cdots \\sin \\theta _ { B - 3 } \\mathrm { d } \\theta _ { 1 } \\cdots \\mathrm { d } \\theta _ { B - 2 } \\int _ { 0 } ^ { \\infty } r ^ { B - 2 - k } e ^ { - r ^ { 2 } / 2 } \\mathrm { d } r } } \\\\ { { = ( 2 \\pi ) ^ { - ( B - 1 ) / 2 } \\int f ( v _ { 1 } ^ { \\theta } , \\dots , v _ { 4 } ^ { \\theta } ) \\sin ^ { B - 3 } \\theta _ { 1 } \\cdots \\sin \\theta _ { B - 3 } \\mathrm { d } \\theta _ { 1 } \\cdots \\mathrm { d } \\theta _ { B - 2 } ( ^ { B - 3 - k } ) ^ { / 2 } \\Gamma ( ( B - 1 - k ) / 2 ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 118, + 839, + 273 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/4fc06bbd0a9f723031ff2f68989bf016dc4718273aeff8778de47e64cb81262a.jpg", + "text": "$$\n= \\Gamma ( ( B - 1 - k ) / 2 ) 2 ^ { - ( 2 + k ) / 2 } \\pi ^ { - ( B - 1 ) / 2 } \\int f ( v _ { 1 } ^ { \\theta } , \\dots , v _ { 4 } ^ { \\theta } ) \\sin ^ { B - 3 } \\theta _ { 1 } \\cdot \\cdot \\cdot \\sin \\theta _ { B - 3 } \\mathrm { d } \\theta _ { 1 } \\cdot \\cdot \\cdot \\mathrm { d } \\theta _ { B - 2 }\n$$", + "text_format": "latex", + "bbox": [ + 181, + 292, + 816, + 325 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Because $v _ { 1 } ^ { \\theta } , \\ldots , v _ { 4 } ^ { \\theta }$ only depends on $\\theta _ { 1 } , \\ldots , \\theta _ { 4 }$ , we can integrate out $\\theta _ { 5 } , \\ldots , \\theta _ { B - 2 }$ . By applying Lemma E.30 to $\\theta _ { 5 } , \\ldots \\theta _ { B - 3 }$ and noting that $\\begin{array} { r } { \\int _ { 0 } ^ { 2 \\pi } \\mathrm { d } \\theta _ { B - 2 } = 2 \\pi } \\end{array}$ , we get ", + "bbox": [ + 173, + 344, + 823, + 380 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/39cb0e234cb2d59a1b481f01de88f6041b754c0afa1f5704e4bcc7f283d5431f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathbb { E } [ r ^ { - k } f ( v _ { 1 } ^ { \\theta } , \\dots , v _ { d } ^ { \\theta } ) : x \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] } \\\\ & { = \\Gamma ( ( B - 1 - k ) / 2 ) 2 ^ { - ( 2 + k ) / 2 } \\pi ^ { - ( B - 1 ) / 2 } \\int f ( v _ { 1 } ^ { \\theta } , \\dots , v _ { 4 } ^ { \\theta } ) \\sin ^ { B - 3 } \\theta _ { 1 } \\cdots \\sin ^ { B - 6 } \\theta _ { 4 } \\pi ^ { ( B - 7 ) / 2 } \\Gamma ( ( B - 1 , \\dots , I _ { B } ) ) } \\\\ & { = \\frac { \\Gamma ( ( B - 1 - k ) / 2 ) } { \\Gamma ( ( B - 5 ) / 2 ) } 2 ^ { - k / 2 } \\pi ^ { - 2 } \\int f ( v _ { 1 } ^ { \\theta } , \\dots , v _ { 4 } ^ { \\theta } ) \\sin ^ { B - 3 } \\theta _ { 1 } \\cdots \\sin ^ { B - 6 } \\theta _ { 4 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 382, + 839, + 470 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Cartesian coordinates. We can often also simplify the high dimensional spherical integrals without trigonometry, as the next two lemmas show. Both are proved easily using change of coordinates. ", + "bbox": [ + 174, + 497, + 825, + 527 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Lemma E.38. For any $u \\in S ^ { n }$ , ", + "bbox": [ + 173, + 529, + 385, + 542 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/483a7b3ae2ec741798fe3245fd820d98f53c93a2c45796d357ee6e1f3a8ad330.jpg", + "text": "$$\n\\underset { v \\sim S ^ { n } } { \\mathbb { E } } f ( u \\cdot v ) = \\frac { \\omega _ { n - 1 } } { \\omega _ { n } } \\int _ { - 1 } ^ { 1 } \\mathrm { d } h \\left( 1 - h ^ { 2 } \\right) ^ { \\frac { n - 2 } { 2 } } f ( h )\n$$", + "text_format": "latex", + "bbox": [ + 336, + 544, + 661, + 580 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "where $\\omega _ { n - 1 }$ is hypersurface of the $( n - 1 )$ -dimensional unit hypersphere. ", + "bbox": [ + 173, + 582, + 655, + 597 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Lemma E.39. For any $u _ { 1 } , u _ { 2 } \\in S ^ { n }$ ", + "bbox": [ + 173, + 598, + 415, + 613 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/ef95fbf97b6af6c40631687955b19b2279ab55247af2f55a26e8475167eb1692.jpg", + "text": "$$\n\\underbrace { \\mathbb { E } } _ { v \\sim S ^ { n } } f ( u _ { 1 } \\cdot v , u _ { 2 } \\cdot v ) = { \\frac { \\omega _ { n - 2 } } { \\omega _ { n } } } \\int _ { \\underbrace { v \\in \\mathrm { s p a n } ( u _ { 1 } , u _ { 2 } ) } _ { v \\in \\mathrm { s p a n } ( u _ { 1 } , u _ { 2 } ) } } \\mathrm { d } v ( 1 - \\| v \\| ^ { 2 } ) ^ { \\frac { n - 3 } { 2 } } f ( u _ { 1 } \\cdot v , u _ { 2 } \\cdot v )\n$$", + "text_format": "latex", + "bbox": [ + 233, + 613, + 763, + 654 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "where $\\omega _ { n - 1 }$ is hypersurface of the $( n - 1 )$ -dimensional unit hypersphere. ", + "bbox": [ + 174, + 655, + 656, + 670 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "E.4 GEGENBAUER EXPANSION ", + "text_level": 1, + "bbox": [ + 174, + 684, + 400, + 699 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Definition E.40. The Gegenbauer polynomials $\\{ C _ { l } ^ { ( \\alpha ) } ( x ) \\} _ { l = 0 } ^ { \\infty }$ , with $\\deg C _ { l } ^ { ( \\alpha ) } ( x ) ~ = ~ l$ , are the set of orthogonal polynomials with respect to the weight function $( 1 - x ^ { 2 } ) ^ { \\alpha - 1 / 2 }$ (i.e. $\\int _ { - 1 } ^ { 1 } ( 1 ~ -$ $x ^ { 2 } ) ^ { \\alpha - 1 / 2 } C _ { l } ^ { ( \\alpha ) } ( x ) C _ { m } ^ { ( \\alpha ) } ( x ) = 0$ if $l \\neq m$ ). By convention, they are normalized so that ", + "bbox": [ + 173, + 708, + 825, + 765 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/ce4b49dbef2b38dee39289350792faa942ca8e9b3f5c51309296e04179662e15.jpg", + "text": "$$\n\\int _ { - 1 } ^ { 1 } \\left[ C _ { n } ^ { ( \\alpha ) } ( x ) \\right] ^ { 2 } ( 1 - x ^ { 2 } ) ^ { \\alpha - \\frac { 1 } { 2 } } d x = \\frac { \\pi 2 ^ { 1 - 2 \\alpha } \\Gamma ( n + 2 \\alpha ) } { n ! ( n + \\alpha ) \\Gamma ( \\alpha ) ^ { 2 } } .\n$$", + "text_format": "latex", + "bbox": [ + 310, + 766, + 684, + 801 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Here are several examples of low degree Gegenbauer polynomials ", + "bbox": [ + 173, + 809, + 607, + 825 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/3842dfee2e409615b571c9ba7d5f5cbbf038aa630cd87a0b9262fa83e6b39d6c.jpg", + "text": "$$\n\\begin{array} { l } { { C _ { 0 } ^ { ( \\alpha ) } ( x ) = 1 } } \\\\ { { C _ { 1 } ^ { ( \\alpha ) } ( x ) = 2 \\alpha x } } \\\\ { { C _ { 2 } ^ { ( \\alpha ) } ( x ) = - \\alpha + 2 \\alpha ( 1 + \\alpha ) x ^ { 2 } } } \\\\ { { C _ { 3 } ^ { ( \\alpha ) } ( x ) = - 2 \\alpha ( 1 + \\alpha ) x + \\displaystyle \\frac { 4 } { 3 } \\alpha ( 1 + \\alpha ) ( 2 + \\alpha ) x ^ { 3 } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 328, + 825, + 668, + 924 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "They satisfy a few identities summarized below ", + "bbox": [ + 174, + 103, + 486, + 118 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Fact E.41 (Suetin). $C _ { n } ^ { ( \\alpha ) } ( \\pm 1 ) = ( \\pm 1 ) ^ { n } { \\binom { n + 2 \\alpha - 1 } { n } }$ ", + "bbox": [ + 173, + 121, + 504, + 141 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Fact E.42 (Weisstein). ", + "bbox": [ + 174, + 142, + 325, + 156 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/6818b65cecc492888250b18d6f595a743bfb2840328979e9062689cea64376b8.jpg", + "text": "$$\n\\begin{array} { r l } & { \\ C _ { n } ^ { ( \\lambda ) \\prime } ( x ) = 2 \\lambda C _ { n - 1 } ^ { ( \\lambda + 1 ) } ( x ) } \\\\ & { x C _ { n } ^ { ( \\lambda ) \\prime } ( x ) = n C _ { n } ^ { ( \\lambda ) } ( x ) + { C _ { n - 1 } ^ { ( \\lambda ) } } ^ { \\prime } ( x ) } \\\\ & { C _ { n + 1 } ^ { ( \\lambda ) } { ' } ( x ) = ( n + 2 \\lambda ) C _ { n } ^ { ( \\lambda ) } ( x ) + x C _ { n } ^ { ( \\lambda ) \\prime } ( x ) } \\\\ & { n C _ { n } ^ { ( \\lambda ) } ( x ) = 2 ( n - 1 + \\lambda ) x C _ { n - 1 } ^ { ( \\lambda ) } ( x ) - ( n - 2 + 2 \\lambda ) C _ { n - 2 } ^ { ( \\lambda ) } ( x ) , \\forall n \\geq 2 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 157, + 736, + 251 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "By repeated applications of Eq. (38), ", + "bbox": [ + 173, + 257, + 416, + 272 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Proposition E.43. ", + "text_level": 1, + "bbox": [ + 173, + 275, + 295, + 289 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/78555bd1aae424fbf793951bce24585e979ebd25319cf2618eef6e917c3774b3.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle x C _ { n } ^ { ( \\lambda ) } ( x ) = \\frac { 1 } { 2 ( n + \\lambda ) } \\left( ( n + 1 ) C _ { n + 1 } ^ { ( \\lambda ) } ( x ) + ( n - 1 + 2 \\lambda ) C _ { n - 1 } ^ { ( \\lambda ) } ( x ) \\right) , \\forall n \\geq 1 } } \\\\ { { \\displaystyle x ^ { 2 } C _ { n } ^ { ( \\lambda ) } ( x ) = \\kappa _ { 2 } ( n , \\lambda ) C _ { n + 2 } ^ { ( \\lambda ) } ( x ) + \\kappa _ { 0 } ( n , \\lambda ) C _ { n } ^ { ( \\lambda ) } ( x ) + \\kappa _ { - 2 } \\left( n , \\lambda \\right) C _ { n - 2 } ^ { ( \\lambda ) } ( x ) , \\forall n \\geq 2 } } \\\\ { { \\displaystyle x C _ { 0 } ^ { ( \\lambda ) } ( x ) = x = \\frac { 1 } { 2 \\lambda } C _ { 1 } ^ { ( \\lambda ) } ( x ) } } \\\\ { { \\displaystyle x ^ { 2 } C _ { 0 } ^ { ( \\lambda ) } ( x ) = \\frac { 1 } { 4 \\lambda ( 1 + \\lambda ) } \\left( 2 C _ { 2 } ^ { ( \\lambda ) } ( x ) + 2 \\lambda C _ { 0 } ^ { ( \\lambda ) } ( x ) \\right) } } \\\\ { { \\displaystyle \\quad \\quad = \\kappa _ { 2 } ( 0 , \\lambda ) C _ { 2 } ^ { ( \\lambda ) } ( x ) + \\kappa _ { 0 } ( 0 , \\lambda ) C _ { 0 } ^ { ( \\lambda ) } ( x ) } } \\\\ { { \\displaystyle x ^ { 2 } C _ { 1 } ^ { ( \\lambda ) } ( x ) = \\kappa _ { 2 } ( 1 , \\lambda ) C _ { 3 } ^ { ( \\lambda ) } ( x ) + \\kappa _ { 0 } ( 1 , \\lambda ) C _ { 1 } ^ { ( \\lambda ) } ( x ) } } \\\\ { { \\displaystyle \\kappa _ { 0 } ( n , \\lambda ) : = - \\frac { ( n + 1 ) ( n + 2 ) } { 4 \\lambda } , \\kappa _ { 0 } ( n , \\lambda ) : = - \\frac { ( n + 1 ) ( n + 2 \\lambda ) } { 4 } - \\frac { ( n - 1 + 2 \\lambda ) n } { 4 } , \\kappa _ { - } \\rho ( n , \\lambda ) : = - \\frac { 1 } { 4 \\lambda } C _ { 0 } ^ { ( \\lambda ) } ( x ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 223, + 292, + 826, + 474 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "where κ2(n, λ) := (n+1)(n+2)4(n+λ)(n+1+λ) , κ0(n, λ) := 4 (n−1+2λ)n4(n+λ)(n−1+λ) , κ−2(n, λ) := $\\frac { ( n - 1 + 2 \\lambda ) ( n - 2 + 2 \\lambda ) } { 4 ( n + \\lambda ) ( n - 1 + \\lambda ) }$ ", + "bbox": [ + 171, + 458, + 825, + 502 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "This proposition is useful for the Gegenbauer expansion of the local convergence rate of Eq. (43). ", + "bbox": [ + 171, + 510, + 810, + 526 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Zonal harmonics. The Gegenbauer polynomials are closely related to a special type of spherical harmonics called zonal harmonics, which are intuitively those harmonics that only depend on the “height” of the point along some fixed axis. ", + "bbox": [ + 176, + 539, + 821, + 582 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Definition E.44. Let $Z _ { u } ^ { N - 1 , ( l ) }$ denote the degree $l$ zonal spherical harmonics on the $( N - 1 )$ - dimensional sphere with axis $u \\in S ^ { N - 1 }$ , defined by $Z _ { u } ^ { N - 1 , ( l ) } ( v ) = c _ { N , l } ^ { - 1 } C _ { l } ^ { ( \\frac { N - 2 } { 2 } ) } ( u \\cdot v )$ for any u, v ∈ SN−1, where cN,l = N−2N+2l−2 . ", + "bbox": [ + 173, + 584, + 825, + 643 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "The first few zonal spherical harmonics are $Z _ { u } ^ { N - 1 , ( 0 ) } ( v ) = 1 , Z _ { u } ^ { N - 1 , ( 1 ) } ( v ) = N u \\cdot v , \\ldots .$ We note a trivial lemma that is useful for simplying many expressions in Gegenbauer expansions. ", + "bbox": [ + 171, + 652, + 825, + 685 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Lemma E.45. $\\begin{array} { r } { \\frac { c _ { B + 1 , l - 1 } } { c _ { B - 1 , l } } = \\frac { B - 1 } { B - 3 } } \\end{array}$ . ", + "bbox": [ + 174, + 685, + 390, + 707 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Let $\\langle - , - \\rangle _ { S ^ { N } }$ denote the inner product induced by the uniform measure on $S ^ { N }$ . One of the most important properties of zonal harmonics is their ", + "bbox": [ + 173, + 717, + 825, + 746 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Fact E.46 (Reproducing property (Suetin)). ", + "bbox": [ + 174, + 750, + 460, + 763 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/cf5f67ef575017a03df5890b188ec474053fd0eba9a186773cf97855b6621b2d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\langle Z _ { u } ^ { N - 1 , ( l ) } , Z _ { v } ^ { N - 1 , ( l ) } \\rangle _ { S ^ { N - 1 } } = Z _ { u } ^ { N - 1 , ( l ) } ( v ) = c _ { N , l } ^ { - 1 } C _ { l } ^ { ( \\frac { N - 2 } { 2 } ) } ( u \\cdot v ) } \\\\ & { \\langle Z _ { u } ^ { N - 1 , ( l ) } , Z _ { v } ^ { N - 1 , ( m ) } \\rangle _ { S ^ { N - 1 } } = 0 , i f l \\neq m . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 766, + 714, + 814 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "In our applications of the reproducing property, $u$ and $v$ will always be $\\hat { \\mathbb { E } } _ { a , : } : = \\mathbb { e } _ { a , : } / \\Vert \\mathbb { e } _ { a , : } \\Vert \\ =$ $\\mathbb { e } _ { a , : } \\sqrt { \\frac { B } { B - 1 } }$ BB−1 , where ea,: is the ath row of e. ", + "bbox": [ + 173, + 821, + 823, + 858 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Next, we present a new (to the best of our knowledge) identity showing that a certain quadratic form of $\\phi$ , reminiscent of Dirichlet forms, depending only on the derivative $\\phi ^ { \\prime }$ through a spherical integral, is diagonalized in the Gegenbauer basis. This will be crucial to proving the necessity of gradient explosion under the BSB1 fixed point assumption. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Theorem E.47 (Dirichlet-like form of $\\phi$ diagonalizes over Gegenbauer basis). Let $u _ { 1 } , u _ { 2 } \\in S ^ { B - 2 }$ . Suppose $\\phi ( \\sqrt { B - 1 } x ) , \\phi ^ { \\prime } ( \\sqrt { B - 1 } x ) \\in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \\frac { B - 3 } { 2 } } )$ and $\\phi ( \\sqrt { B - 1 } x )$ has Gegenbauer expansion $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ . Then ", + "bbox": [ + 174, + 102, + 823, + 159 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/63aeebc86e5161fcc019019ed0974e729db5b4adce176fc4337f3d18ac55cd0e.jpg", + "text": "$$\n\\begin{array} { r l } { { \\underset { v \\in S ^ { B - 2 } } { \\mathbb { E } } ( u _ { 1 } \\cdot u _ { 2 } - ( u _ { 1 } \\cdot v ) ( u _ { 2 } \\cdot v ) ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } u _ { 1 } \\cdot v ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } u _ { 2 } \\cdot v ) } } \\\\ & { = \\displaystyle \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } ( l + B - 3 ) l c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 254, + 166, + 745, + 237 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "This is proved using the following lemmas. ", + "bbox": [ + 174, + 250, + 457, + 265 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma E.48. ", + "bbox": [ + 173, + 270, + 269, + 285 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/23823aa085344408f68a201958948a6d2e08b48a0ca78b0e1e3035eadda02fbf.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle x C _ { l } ^ { ( \\alpha ) \\prime } ( x ) = l C _ { l } ^ { ( \\alpha ) } ( x ) + \\sum _ { j \\in [ l - 2 ] _ { 2 } } 2 ( j + \\alpha ) C _ { j } ^ { ( \\alpha ) } ( x ) } } \\\\ { { \\displaystyle C _ { l } ^ { ( \\alpha ) \\prime } ( x ) = \\sum _ { j \\in [ l - 1 ] _ { 2 } } 2 ( j + \\alpha ) C _ { j } ^ { ( \\alpha ) } ( x ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 328, + 287, + 669, + 364 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "where $[ n ] _ { 2 }$ is the sequence $\\{ ( n \\% 2 ) , ( n \\% 2 ) + 1 , . . . , n - 2 , n \\}$ and $n \\% 2$ is the remainder of $n / 2$ ", + "bbox": [ + 171, + 369, + 805, + 387 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. Apply Eq. (36) and Eq. (37) alternatingly. ", + "bbox": [ + 174, + 404, + 498, + 420 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma E.49. ", + "bbox": [ + 173, + 431, + 269, + 445 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/ea7ee31545a8b7fe87fc4d4a884f478eba04067aefd704e09d7d9aa9a937059e.jpg", + "text": "$$\nl C _ { l } ^ { ( \\alpha ) } ( t ) + \\sum _ { j \\in [ l - 2 ] _ { 2 } } 2 ( j + \\alpha ) C _ { j } ^ { ( \\alpha ) } ( t ) - t \\sum _ { j \\in [ l - 1 ] _ { 2 } } 2 ( j + \\alpha ) C _ { j } ^ { ( \\alpha ) } ( t ) = 0\n$$", + "text_format": "latex", + "bbox": [ + 266, + 449, + 730, + 487 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. Apply Eq. (38) to $l C _ { l } ^ { ( \\alpha ) } ( t ) , ( l - 2 ) C _ { l - 2 } ^ { ( \\alpha ) } ( t ) , . . . .$ ", + "bbox": [ + 173, + 505, + 535, + 525 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof of Thm E.47. We have ", + "bbox": [ + 174, + 542, + 366, + 558 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/2589e80153a459d74fad8193b13392b9d9cfd1bf832c3f45a67317852d7beea9.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\phi ^ { \\prime } ( \\sqrt { B - 1 } x ) = \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) \\prime } ( x ) } \\\\ { \\displaystyle \\quad = \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } \\sum _ { j \\in [ l - 1 ] _ { 2 } } ( 2 j + B - 3 ) C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\\\ { \\displaystyle \\quad = \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { l = 0 } ^ { \\infty } \\sum _ { j \\in [ l - 1 ] _ { 2 } } a _ { l } \\frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } ( 2 j + B - 3 ) c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\\\ { \\displaystyle \\quad = \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { j = 0 } ^ { \\infty } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) \\sum _ { l \\in [ j + 1 , \\infty ] _ { 2 } } a _ { l } ^ { - } \\frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } ( 2 j + B - 3 ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 225, + 564, + 776, + 742 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "where $[ n , m ] _ { 2 }$ is the sequence $n , n + 2 , \\ldots , m$ . So for any $u _ { 1 } , u _ { 2 } \\in S ^ { B - 2 }$ , ", + "bbox": [ + 176, + 768, + 665, + 785 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/1cc61a598aab1bc14bab7659730bb60ce5b25c8463becf824138a74d37ca4575.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\underset { v \\in S ^ { B - 2 } } { \\mathbb { E } } \\phi ^ { \\prime } ( \\sqrt { B - 1 } u _ { 1 } \\cdot v ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } u _ { 2 } \\cdot v ) } \\\\ & { = \\frac { 1 } { B - 1 } { \\displaystyle \\sum _ { j = 0 } ^ { \\infty } } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) \\left( \\underset { l \\in [ j + 1 , \\infty ] _ { 2 } } { \\sum } a _ { l } \\frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } ( 2 j + B - 3 ) \\right) ^ { 2 } } \\\\ & { = \\frac { 1 } { B - 1 } { \\displaystyle \\sum _ { j = 0 } ^ { \\infty } } c _ { B - 1 , j } ( 2 j + B - 3 ) ^ { 2 } C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) \\left( \\underset { l , m \\in [ j + 1 , \\infty ] _ { 2 } } { \\sum } \\frac { a _ { l } } { c _ { B - 1 , l } } \\frac { a _ { m } } { c _ { B - 1 , m } } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 791, + 789, + 924 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "For $l < m$ and $m - l$ even, the coefficient of $a _ { l } a _ { m }$ is ", + "bbox": [ + 173, + 103, + 522, + 118 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/fe7ec908e53bb6fd1f9abe0727a655c80350519f6ec6b696d091a0d3ed6bfb1e.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\frac { 2 c _ { B - 1 , l } ^ { - 1 } c _ { B - 1 , m } ^ { - 1 } } { B - 1 } \\displaystyle \\sum _ { j \\in [ l - 1 ] _ { 2 } } c _ { B - 1 , j } ( 2 j + B - 3 ) ^ { 2 } C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) } \\\\ & { } & { = \\frac { 2 c _ { B - 1 , l } ^ { - 1 } c _ { B - 1 , m } ^ { - 1 } } { B - 1 } \\displaystyle \\sum _ { j \\in [ l - 1 ] _ { 2 } } ( B - 3 ) ( 2 j + B - 3 ) C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 279, + 119, + 717, + 210 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "For each $l$ , the coefficient of $a _ { l } ^ { 2 }$ is ", + "bbox": [ + 173, + 213, + 397, + 228 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/72e562ed009f8be6b15cae4c5099b604fd18f66c368d960e559c55bfe7526db4.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\frac { c _ { B - 1 , l } ^ { - 2 } } { B - 1 } \\sum _ { j \\in [ l - 1 ] _ { 2 } } c _ { B - 1 , j } ( 2 j + B - 3 ) ^ { 2 } C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) } } \\\\ { { = \\displaystyle \\frac { c _ { B - 1 , l } ^ { - 2 } } { B - 1 } \\sum _ { j \\in [ l - 1 ] _ { 2 } } ( B - 3 ) ( 2 j + B - 3 ) C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 310, + 231, + 687, + 321 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Similarly, if $\\phi ^ { \\prime } ( x ) \\in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \\frac { B - 3 } { 2 } } )$ , we have ", + "bbox": [ + 173, + 332, + 495, + 349 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/46b2cae0e6060d1f9383009c0ef030f81137bfd40017030e5bfe995644034f03.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle x \\phi ^ { \\prime } ( \\sqrt { B - 1 } x ) = \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } x C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) \\prime } ( x ) } \\ ~ } \\\\ { { \\displaystyle ~ = \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } \\left( l C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) + \\sum _ { j \\in [ l - 2 ] _ { 2 } } ( 2 j + B - 3 ) C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) \\right) } } \\\\ { { \\displaystyle ~ = \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { j = 0 } ^ { \\infty } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) \\left( a _ { j } j + \\sum _ { l \\in [ j + 2 , \\infty ] _ { 2 } } a _ { l } \\frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } ( 2 j + B - 3 ) \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 352, + 812, + 496 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Then ", + "bbox": [ + 173, + 496, + 210, + 510 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/73a61ad59a457e9a87ce58fa9f6f3b0fe519d449902d9835a4af8a7ca9d89ed0.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\underset { v \\in S ^ { B - 2 } } { \\mathbb { E } } ( u _ { 1 } \\cdot v ) ( u _ { 2 } \\cdot v ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } u _ { 1 } \\cdot v ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } u _ { 2 } \\cdot v ) } \\\\ & { = \\frac { 1 } { B - 1 } { \\displaystyle \\sum _ { j = 0 } ^ { \\infty } } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) \\left( a _ { j } j + { \\displaystyle \\sum _ { l \\in [ j + 2 , \\infty ] _ { 2 } } } a _ { l } \\frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } ( 2 j + B - 3 ) \\right) ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 223, + 508, + 774, + 590 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "For $l < m$ and $m - l$ even, the coefficient of $a _ { l } a _ { m }$ is ", + "bbox": [ + 176, + 592, + 521, + 606 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/b4d3d262260b1d347631f04363d03af7f875189c8a1ab0159f1c912ab0f75671.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { 1 } { B - 1 } \\left( 2 l ( 2 l + B - 3 ) \\frac { c _ { B - 1 , l } } { c _ { B - 1 , m } } c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) \\right. } \\\\ { \\displaystyle \\qquad \\left. + 2 \\sum _ { j \\in [ l - 2 ] _ { 2 } } \\frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } \\frac { c _ { B - 1 , j } } { c _ { B - 1 , m } } ( 2 j + B - 3 ) ^ { 2 } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) \\right) } \\\\ { \\displaystyle = 2 c _ { B - 1 , l } ^ { - 1 } c _ { B - 1 , m } ^ { - 1 } \\frac { B - 3 } { B - 1 } \\left( l C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) + \\sum _ { j \\in [ l - 2 ] _ { 2 } } ( 2 j + B - 3 ) C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 609, + 781, + 743 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "For each $l$ , the coefficient of $a _ { l } ^ { 2 }$ is ", + "bbox": [ + 173, + 746, + 397, + 761 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/e7531e5e0267545cae226e23394cc334a3da5ca86f040357779b412ce7577b26.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\frac { 1 } { B - 1 } \\left( l ^ { 2 } c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) + \\displaystyle \\sum _ { j \\in [ l - 2 ] _ { 2 } } \\left( \\frac { c _ { B - 1 , j } } { c _ { B - 1 , l } } \\right) ^ { 2 } ( 2 j + B - 3 ) ^ { 2 } c _ { B - 1 , j } ^ { - 1 } C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) \\right. } \\\\ & { } & { \\left. = c _ { B - 1 , l } ^ { - 2 } \\frac { B - 3 } { B - 1 } \\left( \\frac { l ^ { 2 } } { 2 l + B - 3 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) + \\displaystyle \\sum _ { j \\in [ l - 2 ] _ { 2 } } ( 2 j + B - 3 ) C _ { j } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) \\right) ( 4 2 ) \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 763, + 839, + 866 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "By Lemma E.49 and Eqs. (39) and (41), the coefficient of $a _ { l } a _ { m }$ with $m > l$ in $\\mathbb { E } _ { v \\in S ^ { B - 2 } } ( u _ { 1 } \\cdot u _ { 2 } - ( u _ { 1 }$ $v ) ( u _ { 2 } \\cdot v ) ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } u _ { 1 } \\cdot v ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } u _ { 2 } \\cdot v )$ is 0. Similarly, by Lemma E.49 and Eqs. (40) and (42), the coefficient of $a _ { l } ^ { 2 }$ is $c _ { B - 1 , l } ^ { - 2 } \\textstyle { \\frac { { B } - 3 } { { B } - 1 } } \\textstyle { \\frac { l + B - 3 } { 2 l + B - 3 } } l C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } ) = c _ { B - 1 , l } ^ { - 1 } \\textstyle { \\frac { l + B - 3 } { B - 1 } } l C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( u _ { 1 } \\cdot u _ { 2 } )$ . ", + "bbox": [ + 173, + 873, + 825, + 928 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "E.5 SYMMETRY AND ULTRASYMMETRY ", + "text_level": 1, + "bbox": [ + 176, + 103, + 464, + 118 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "As remarked before, all commonly used nonlinearities ReLU, tanh, and so on induce the dynamics Eq. (43) to converge to BSB1 fixed points, which we formally define as follows. ", + "bbox": [ + 171, + 128, + 823, + 159 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Definition E.50. We say a matrix $\\Sigma \\in S _ { B }$ is BSB1 (short for “1-Block Symmetry Breaking”) if $\\Sigma$ has one common entry on the diagonal and one common entry on the off-diagonal, i.e. ", + "bbox": [ + 171, + 161, + 823, + 190 + ], + "page_idx": 39 + }, + { + "type": "equation", + "img_path": "images/6fadd2ea4095aaef01897e636d4e93c0f1d52fe8e4ff318b97d79e17c2596f8a.jpg", + "text": "$$\n\\left( \\begin{array} { c c c c } { { a } } & { { b } } & { { b } } & { { \\ldots } } \\\\ { { b } } & { { a } } & { { b } } & { { \\ldots } } \\\\ { { b } } & { { b } } & { { a } } & { { \\ldots } } \\\\ { { \\vdots } } & { { \\vdots } } & { { \\vdots } } & { { \\ddots } } \\end{array} \\right)\n$$", + "text_format": "latex", + "bbox": [ + 436, + 196, + 562, + 263 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "We will denote such a matrix as $\\mathrm { B S B 1 } ( a , b )$ . Note that $\\mathrm { B S B 1 } ( a , b )$ can be written as $( a - b ) I + b \\mathbf { 1 } \\mathbf { 1 } ^ { T }$ . \nIts spectrum is given below. ", + "bbox": [ + 173, + 277, + 823, + 308 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Lemma E.51. A $B \\times B$ matrix of the form $\\mu I + \\nu \\mathbf { 1 1 } ^ { T }$ has two eigenvalues $\\mu$ and $B \\nu + \\mu$ , each with eigenspaces $\\textstyle \\{ x : \\sum _ { i } x _ { i } = 0 \\}$ and $\\{ x : x _ { 1 } = \\cdot \\cdot \\cdot = x _ { B } \\}$ . Equivalently, if it has a on the diagonal and b on the off-diagonal, then the eigenvalues are $a - b$ and $( B - 1 ) b + a$ . ", + "bbox": [ + 173, + 310, + 825, + 354 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "The following simple lemma will also be very useful ", + "bbox": [ + 173, + 364, + 522, + 380 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Lemma E.52. Let $\\Sigma : = \\mathrm { B S B } 1 _ { B } ( a , b )$ . Then $G \\Sigma G = ( a - b ) G .$ . ", + "bbox": [ + 173, + 382, + 596, + 398 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Proof. $G$ and $\\mathrm { B S B } 1 _ { B } ( a , b )$ can be simultaneously diagonalized by Lemma E.51. Note that $G$ zeros out the eigenspace $\\mathbb { R } \\mathbf { 1 }$ , and is identity on its orthogonal complement. The result then follows from easy computations. □ ", + "bbox": [ + 173, + 412, + 825, + 457 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "The symmetry of BSB1 covariance matrices, especially the fact that they are isotropic in a codimension 1 subspace, is crucial to much of our analysis. ", + "bbox": [ + 173, + 470, + 823, + 501 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Ultrasymmetry. Indeed, as mentioned before, when investigating the asymptotics of the forward or backward dynamics, we encounter 4-tensor objects, such as $\\mathrm { \\frac { \\bar { d } V _ { } } { d \\Sigma } }$ or $\\mathrm { V } _ { B _ { \\phi } ^ { \\prime } }$ , that would seem daunting to analyze at first glance. However, under the BSB1 fixed point assumption, these 4- tensors contain considerable symmetry that somehow makes this possible. We formalize the notion of symmetry arising from such scenarios: ", + "bbox": [ + 173, + 513, + 825, + 592 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Definition E.53. Let $\\mathcal { T } : \\mathbb { R } ^ { B \\times B } \\mathbb { R } ^ { B \\times B }$ be a linear operator. Let $\\delta _ { i } \\in \\mathbb { R } ^ { B }$ be the vector with 0 everywhere except 1 in coordinate $i$ ; then $\\delta _ { i } \\delta _ { j } ^ { T }$ is the matrix with 0 everywhere except 1 in position $( i , j )$ . Write $[ k l | i j ] : = \\mathcal { T } ( \\delta _ { i } \\delta _ { j } ^ { T } ) _ { k l }$ . Suppose $\\tau$ has the property that for all $i , j , k , l \\in [ B ] =$ $\\{ 1 , \\ldots , B \\}$ ", + "bbox": [ + 173, + 594, + 825, + 659 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "• $[ k l | i j ] = [ \\pi ( k ) \\pi ( l ) | \\pi ( i ) \\pi ( j ) ]$ for all permutation $\\pi$ on $[ B ]$ , and • $[ i j | k l ] = [ j i | l k ]$ . ", + "bbox": [ + 214, + 666, + 647, + 708 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Then we say $\\tau$ is ultrasymmetric. ", + "bbox": [ + 174, + 718, + 397, + 733 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Remark E.54. In what follows, we will often “normalize” the representation “ $[ i j | k l ]$ ” to the unique “ $[ i ^ { \\prime } j ^ { \\prime } | k ^ { \\prime } l ^ { \\prime } ] ^ { \\cdot }$ ” that is in the same equivalence class according to Defn E.53 and such that $i ^ { \\prime } , j ^ { \\prime } , k ^ { \\prime } , l ^ { \\bar { \\prime } } \\in$ [4] and $i ^ { \\prime } \\overset { \\cdot } { \\leq } j ^ { \\prime } , k ^ { \\prime } \\overset { } { \\leq } l ^ { \\prime }$ unless $i ^ { \\prime } = l ^ { \\prime } , j ^ { \\prime } = k ^ { \\prime }$ , in which case the normalization is [12|21]. Explicitly, we have the following equivalence classes and their normalized representations ", + "bbox": [ + 174, + 737, + 825, + 792 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "We will study the eigendecomposition of an ultrasymmetric $\\tau$ as well as the projection $G ^ { \\otimes 2 } \\circ \\mathcal { T } \\circ$ $G ^ { \\otimes 2 } : \\Sigma \\mapsto { \\bar { G } } ( { \\mathcal { T } } \\{ { \\bar { G } } \\Sigma G \\} ) G$ , respectively with the following domains ", + "bbox": [ + 173, + 801, + 823, + 830 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "Definition E.55. Denote by $\\mathcal { H } _ { B }$ the space of symmetric matrices of dimension $B$ . Also write $\\mathcal { H } _ { B } ^ { G }$ for the space of symmetric matrices $\\Sigma$ of dimension $B$ such that $G \\Sigma G = \\Sigma$ (which is equivalent to saying rows of $\\Sigma$ sum up to 0). ", + "bbox": [ + 174, + 834, + 825, + 877 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "As in the case of $s$ , we omit subscript $B$ when it’s clear from context. ", + "bbox": [ + 174, + 887, + 632, + 902 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "The following subspaces of $\\mathcal { H } _ { B }$ and $\\mathcal { H } _ { B } ^ { G }$ will turn out to comprise the eigenspaces of $G ^ { \\otimes 2 } \\circ T \\circ G ^ { \\otimes 2 }$ . ", + "bbox": [ + 174, + 907, + 823, + 925 + ], + "page_idx": 39 + }, + { + "type": "table", + "img_path": "images/ce6fcb1bbec1530493d13de3f56facac8318b769165d956dd43918bb74dc8b62.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
class repr.
i=j=k=li=j=kori=j=li=jand k=l1111]
[1112
[11]22
i=j[11|23/
i=k=lorj=k=l1211j
i=k and j=l[12]|12]
i=kori=l[12|[13]
i=land j= k12|21j
k=l1233
all different1234
", + "bbox": [ + 361, + 101, + 617, + 256 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Definition E.56. Let $\\mathbb { L } _ { B } : = \\{ G D G : D$ diagonal, $\\mathrm { t r } D = 0 \\} \\subseteq \\mathcal { H } _ { B } ^ { G }$ and $\\mathbb { M } _ { B } : = \\{ \\Sigma \\in \\mathcal { H } _ { B } ^ { G } :$ $\\mathrm { D i a g } \\Sigma = 0 \\}$ . Note that $\\begin{array} { r } { \\dim \\mathbb { L } _ { B } = B - 1 , \\dim \\mathbb { M } _ { B } = \\frac { B ( B - 3 ) } { 2 } } \\end{array}$ and $\\mathbb { R } G ^ { B } \\oplus \\mathbb { L } _ { B } \\oplus \\mathbb { M } _ { B } = \\mathcal { H } _ { B } ^ { G }$ is an orthogonal decomposition w.r.t Frobenius inner product. ", + "bbox": [ + 173, + 279, + 826, + 327 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "For the eigenspaces of $\\tau$ , we also need to define ", + "bbox": [ + 173, + 338, + 493, + 353 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Definition E.57. For any nonzero $a , b \\in \\mathbb { R }$ , set ", + "bbox": [ + 173, + 356, + 485, + 371 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/574339128735ef069360bbab3a672394ce7f2698775a5d3b32e125814131bb0a.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { L } _ { \\boldsymbol { B } } ( \\boldsymbol { a } , \\boldsymbol { b } ) : = \\left( \\begin{array} { l l l l l } { \\boldsymbol { a } } & { \\boldsymbol { 0 } } & { - b } & { - b } & { \\cdots } \\\\ { \\boldsymbol { 0 } } & { - \\boldsymbol { a } } & { \\boldsymbol { b } } & { \\boldsymbol { b } } & { \\cdots } \\\\ { - \\boldsymbol { b } } & { \\boldsymbol { b } } & { \\boldsymbol { 0 } } & { \\boldsymbol { 0 } } & { \\cdots } \\\\ { - \\boldsymbol { b } } & { \\boldsymbol { b } } & { \\boldsymbol { 0 } } & { \\boldsymbol { 0 } } & { \\cdots } \\\\ { \\vdots } & { \\vdots } & { \\vdots } & { \\vdots } & { \\ddots } \\end{array} \\right) \\in \\mathcal { H } _ { \\boldsymbol { B } } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 333, + 377, + 665, + 459 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "In general, we say a matrix $M$ is $\\operatorname { L } _ { B } ( a , b )$ -shaped if $M = { \\cal P } \\mathrm { L } _ { B } ( a , b ) { \\cal P } ^ { T }$ for some permutation matrix $P$ . ", + "bbox": [ + 171, + 465, + 825, + 496 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Note that ", + "bbox": [ + 173, + 506, + 238, + 520 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Proposition E.58. $\\operatorname { L } ( B - 2 , 1 )$ -shaped matrices span $\\mathbb { L } _ { B }$ ", + "bbox": [ + 171, + 525, + 560, + 540 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/6273ee2ed5801506956596201ad420f0da94ae9a7d2d7d637a9b9e8aafb3a5b9.jpg", + "text": "$$\n\\begin{array} { r l } { { \\mathrm { L } ( B - 2 , 1 ) = \\frac { 1 } { B } [ ( - B + 1 , 1 , 1 , \\ldots , 1 ) ^ { \\otimes 2 } - ( 1 , - B + 1 , 1 , \\ldots , 1 ) ^ { \\otimes 2 } ] = } } & { { } } \\\\ { { \\otimes ^ { 2 } - ( 0 , 1 , 0 , \\ldots , 0 ) ^ { \\otimes 2 } ] G } } & { { } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 300, + 541, + 825, + 575 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Proof. Straightforward computation. ", + "bbox": [ + 174, + 588, + 416, + 604 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Lemma E.60. Let ${ \\cal L } = \\mathrm { L } _ { B } ( a , b )$ . Then $\\begin{array} { r } { G ^ { \\otimes 2 } \\{ L \\} = G L G = \\frac { a + 2 b } { B } \\mathrm { L } _ { B } ( B - 2 , 1 ) . } \\end{array}$ ", + "bbox": [ + 173, + 609, + 704, + 628 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Proof. $G L G$ can be written as the sum of outer products ", + "bbox": [ + 173, + 641, + 547, + 656 + ], + "page_idx": 40 + }, + { + "type": "equation", + "img_path": "images/32fd87c5009907fb0388bd49fa3d0eb1ec46c1818d332789cb1d43a50f2c93e1.jpg", + "text": "$$\n\\begin{array} { r l } & { G L G = a G _ { 1 } \\otimes G _ { 1 } - a G _ { 2 } \\otimes G _ { 2 } + \\underset { i = 3 } { \\overset { B } { \\sum } } - b G _ { 1 } \\otimes G _ { i } + b G _ { 2 } \\otimes G _ { i } - b G _ { 4 } \\otimes G _ { 1 } + b G _ { 4 } \\otimes G _ { 2 } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 194, + 661, + 805, + 878 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "We are now ready to discuss the eigendecomposition of ultrasymmetric operators. ", + "bbox": [ + 171, + 909, + 709, + 924 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "Theorem E.61 (Eigendecomposition of an ultrasymmetric operator). Let $\\mathcal { T } : \\mathbb { R } ^ { B \\times B } \\mathbb { R } ^ { B \\times B }$ be an ultrasymmetric linear operator. Then $\\tau$ has the following eigendecomposition. ", + "bbox": [ + 171, + 102, + 825, + 133 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "1. Two $^ { l }$ -dimensional eigenspace $\\mathbb { R } \\cdot \\mathrm { B S B } 1 ( \\lambda _ { \\mathrm { B S B } 1 , i } ^ { \\mathcal { T } } - \\alpha _ { 2 2 } , \\alpha _ { 2 1 } )$ with eigenvalue $\\lambda _ { \\mathrm { B S B } 1 , i } ^ { \\mathcal { T } }$ for $i = 1 , 2$ , where ", + "bbox": [ + 209, + 142, + 825, + 172 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/f0c97f6a27568b3e102719c30dcafb6187b0cb048e36aef1e98d26fa28053433.jpg", + "text": "$$\n\\begin{array} { l } { \\alpha _ { 1 1 } = [ 1 1 | 1 1 ] + [ 1 1 | 2 2 ] ( B - 1 ) } \\\\ { \\alpha _ { 1 2 } = 2 ( B - 1 ) [ 1 1 | 1 2 ] + ( B - 2 ) ( B - 1 ) [ 1 1 | 2 3 ] } \\\\ { \\alpha _ { 2 1 } = 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] } \\\\ { \\alpha _ { 2 2 } = [ 1 2 | 1 2 ] + 4 ( B - 2 ) [ 1 2 | 1 3 ] + [ 1 2 | 2 1 ] + ( B - 2 ) ( B - 3 ) [ 1 2 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 179, + 733, + 251 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "and $\\lambda _ { \\mathrm { B S B 1 , 1 } } ^ { \\mathcal { T } }$ and $\\lambda _ { \\mathrm { B S B 1 , 2 } } ^ { \\mathcal { T } }$ are the roots to the quadratic ", + "bbox": [ + 230, + 257, + 596, + 275 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/c424185da669f0bc4c2294b8bed1d5667fd782ec0210d81d895176044a9b1314.jpg", + "text": "$$\nx ^ { 2 } - ( \\alpha _ { 1 1 } + \\alpha _ { 2 2 } ) x + \\alpha _ { 1 1 } \\alpha _ { 2 2 } - \\alpha _ { 1 2 } \\alpha _ { 2 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 392, + 282, + 663, + 301 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "2. Two $( B - 1 )$ -dimensional eigenspaces $\\mathfrak { S } _ { B } \\cdot \\mathrm { L } ( \\lambda _ { \\mathbb { L } , i } ^ { \\mathcal { T } } - \\beta _ { 2 2 } , \\beta _ { 2 1 } )$ with eigenvalue $\\lambda _ { \\mathbb { L } , i } ^ { \\mathcal { T } }$ for $i = 1 , 2$ . Here $\\mathfrak { S } _ { B } \\cdot W$ denotes the linear span of the orbit of matrix $W$ under simultaneous permutation of its column and rows (by the same permutation), and ", + "bbox": [ + 214, + 330, + 823, + 377 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/45c8f510104ee9179c09fd070d8ccc0ad536678ea41f540d4fca2c843872f98c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\beta _ { 1 1 } = [ 1 1 | 1 1 ] - [ 1 1 | 2 2 ] } \\\\ & { \\beta _ { 1 2 } = 2 ( B - 2 ) ( [ 1 1 | 2 3 ] - [ 1 1 | 1 2 ] ) } \\\\ & { \\beta _ { 2 1 } = - [ 1 2 | 1 1 ] + [ 1 2 | 3 3 ] } \\\\ & { \\beta _ { 2 2 } = [ 1 2 | 2 1 ] + [ 1 2 | 1 2 ] + 2 ( B - 4 ) [ 1 2 | 1 3 ] - 2 ( B - 3 ) [ 1 2 | 3 4 ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 383, + 740, + 457 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "and λT and $\\lambda _ { \\mathbb { L } , 2 } ^ { \\mathcal { T } }$ are the roots to the quadratic ", + "bbox": [ + 230, + 462, + 547, + 482 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/56c32fede0d5b0c7684e4747e70dc06f2cc6392b66ded5a9a7e36d84304cb62d.jpg", + "text": "$$\nx ^ { 2 } - ( \\beta _ { 1 1 } + \\beta _ { 2 2 } ) x + \\beta _ { 1 1 } \\beta _ { 2 2 } - \\beta _ { 1 2 } \\beta _ { 2 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 395, + 489, + 660, + 508 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "3. Eigenspace M (dimension $B ( B \\mathrm { ~ - ~ } 3 ) / 2 )$ with eigenvalue $\\lambda _ { \\mathbb { M } } ^ { \\mathcal { T } } : = [ 1 2 | 1 2 ] + [ 1 2 | 2 1 ] -$ $4 [ 1 2 | 1 3 ] + 2 [ 1 2 | 3 4 ]$ . ", + "bbox": [ + 210, + 537, + 823, + 569 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "The proof is by careful, but ultimately straightforward, computation. ", + "bbox": [ + 174, + 579, + 622, + 594 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Proof. We will use the bracket notation of Defn E.53 to denote entries of $\\tau$ , and implicitly simplify it according to Remark E.54. ", + "bbox": [ + 176, + 609, + 823, + 638 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Item 1. Let $U \\in \\mathbb { R } ^ { B \\times B }$ be the BSB1 matrix. By ultrasymmetry of $\\tau$ and BSB1 symmetry of $A$ $\\tau \\{ U \\}$ is also BSB1. So we proceed to calculate the diagonal and off-diagonal entries of ${ \\mathcal { T } } \\{ G \\}$ . ", + "bbox": [ + 174, + 645, + 821, + 674 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "We have ", + "bbox": [ + 174, + 680, + 233, + 694 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/b362b834c60b1c42600a9b350f36e2bc7c3c6551801b9e9fc740688ea5941690.jpg", + "text": "$$\n\\begin{array} { r l } & { \\Gamma \\{ \\mathrm { B S B 1 } ( a , b ) \\} _ { 1 1 } = [ 1 1 | 1 1 ] a + 2 ( B - 1 ) [ 1 1 | 2 ] b + [ 1 1 | 2 2 ] ( B - 1 ) a + [ 1 1 | 2 3 ] ( B - 2 ) ( B - 1 ) b } \\\\ & { \\qquad = ( [ 1 1 | 1 1 ] + [ 1 1 | 2 ] ( B - 1 ) ) a + ( 2 ( B - 1 ) [ 1 1 | 1 2 ] + ( B - 2 ) ( B - 1 ) [ 1 1 | 2 3 ] ) b } \\\\ & { \\Gamma \\{ \\mathrm { B S B 1 } ( a , b ) \\} _ { 1 2 } = [ 1 2 | 1 2 ] b + 2 [ 1 2 | 1 1 ] a + ( B - 2 ) [ 1 2 | 3 3 ] a + 2 ( B - 2 ) [ 1 2 | 1 3 ] b } \\\\ & { \\qquad + [ 1 2 | 2 1 ] b + 2 ( B - 2 ) [ 1 2 | 2 3 ] b + ( B - 2 ) ( B - 3 ) [ 1 2 | 3 4 ] b } \\\\ & { \\qquad = ( 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] ) a } \\\\ & { \\qquad + ( [ [ 1 2 | 1 2 ] + 2 ( B - 2 ) [ 1 2 | 1 2 ] + [ 1 2 | 2 1 ] + 2 ( B - 2 ) [ 1 2 | 1 3 ] + ( B - 2 ) ( B - 3 ) } \\\\ & { \\qquad = ( 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] ) a } \\\\ & { \\qquad + ( [ [ 1 2 | 1 2 ] + 4 ( B - 2 ) [ 1 2 | 1 2 ] + [ 1 2 | 2 1 ] + ( B - 2 ) ( B - 3 ) [ 1 2 | 3 4 ] ) b } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 699, + 839, + 844 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Thus $\\mathrm { B S B 1 } ( \\omega _ { 1 } , \\gamma _ { 1 } )$ and $\\mathrm { B S B 1 } ( \\omega _ { 2 } , \\gamma _ { 2 } )$ are the eigenmatrices of $\\tau$ , where $( \\omega _ { 1 } , \\gamma _ { 1 } )$ and $( \\omega _ { 2 } , \\gamma _ { 2 } )$ are the eigenvectors of the matrix ", + "bbox": [ + 173, + 866, + 823, + 895 + ], + "page_idx": 41 + }, + { + "type": "equation", + "img_path": "images/62ca80e1e7c174d19d057d7267c1fea52fc63af5c4358ef8d7e691265adbca90.jpg", + "text": "$$\n\\left( \\begin{array} { c c } { { \\alpha _ { 1 1 } } } & { { \\alpha _ { 1 2 } } } \\\\ { { \\alpha _ { 2 1 } } } & { { \\alpha _ { 2 2 } } } \\end{array} \\right)\n$$", + "text_format": "latex", + "bbox": [ + 452, + 893, + 544, + 929 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "with ", + "bbox": [ + 173, + 103, + 207, + 117 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/2fe10eff0a584e2303e08e4003b7f56f852fde416c7c0e4c180007af4712af1d.jpg", + "text": "$$\n\\begin{array} { l } { \\alpha _ { 1 1 } = [ 1 1 | 1 1 ] + [ 1 1 | 2 2 ] ( B - 1 ) } \\\\ { \\alpha _ { 1 2 } = 2 ( B - 1 ) [ 1 1 | 1 2 ] + ( B - 2 ) ( B - 1 ) [ 1 1 | 2 3 ] } \\\\ { \\alpha _ { 2 1 } = 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] } \\\\ { \\alpha _ { 2 2 } = [ 1 2 | 1 2 ] + 4 ( B - 2 ) [ 1 2 | 1 3 ] + [ 1 2 | 2 1 ] + ( B - 2 ) ( B - 3 ) [ 1 2 | } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 122, + 699, + 195 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "The eigenvalues are the two roots $\\lambda _ { \\mathrm { B S B 1 , 1 } } ^ { \\mathcal { T } } , \\lambda _ { \\mathrm { B S B 1 , 2 } } ^ { \\mathcal { T } }$ to the quadratic ", + "bbox": [ + 173, + 199, + 617, + 217 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/f21f382454633487ef1a5b3156c4b69fd682217bb0587abb28aa692e14d1c545.jpg", + "text": "$$\nx ^ { 2 } - ( \\alpha _ { 1 1 } + \\alpha _ { 2 2 } ) x + \\alpha _ { 1 1 } \\alpha _ { 2 2 } - \\alpha _ { 1 2 } \\alpha _ { 2 1 }\n$$", + "text_format": "latex", + "bbox": [ + 364, + 222, + 632, + 241 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "and the corresponding eigenvectors are ", + "bbox": [ + 173, + 246, + 431, + 261 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/c12f9db6fafd4575e4fc28af8981dd832345b739a148c582c282fd30cbcd703d.jpg", + "text": "$$\n\\begin{array} { r } { ( \\omega _ { 1 } , \\gamma _ { 1 } ) = ( \\lambda _ { 1 } - \\alpha _ { 2 2 } , \\alpha _ { 2 1 } ) } \\\\ { ( \\omega _ { 2 } , \\gamma _ { 2 } ) = ( \\lambda _ { 2 } - \\alpha _ { 2 2 } , \\alpha _ { 2 1 } ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 403, + 263, + 593, + 301 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Item 2. We will study the image of $\\operatorname { L } _ { B } ( a , b )$ (Defn E.57) under $\\tau$ . We have ", + "bbox": [ + 171, + 311, + 671, + 328 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/58a4aaf67b67b6a35c06ed27c7a5abe8b5b358162a3071cf8f08466b8b812d55.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { T } [ \\{ \\Delta \\phi _ { i } \\} ] _ { n } = - \\mathcal { T } [ \\{ \\Delta \\phi _ { i } \\} ] _ { n } } \\\\ & { \\qquad - \\{ [ 1 1 1 \\{ 1 1 \\{ 1 9 \\} - 2 \\{ 1 2 \\} \\} \\Delta \\phi - 2 2 \\Lambda [ 1 1 2 \\phi _ { i } ] \\{ 1 2 \\{ 1 2 \\phi _ { i } - 8 \\} + 1 1 2 4 \\{ 2 \\Lambda ^ { 2 } - 2 \\} \\Lambda \\{ 1 1 2 \\phi _ { i } \\} \\} } \\\\ & { \\qquad - \\{ [ 1 1 1 \\{ 1 1 \\{ 1 9 \\} - 2 \\{ 1 5 \\} \\} - 6 \\Lambda ^ { 2 } - 2 \\Lambda ^ { 2 } \\{ 1 1 \\} \\{ 1 9 \\} \\Phi - 2 \\Lambda ^ { 2 } \\{ 1 1 \\} \\{ 1 1 2 \\} \\Phi - \\delta ] } \\\\ & { \\qquad - \\mathcal { T } [ \\{ \\Delta \\phi _ { i } \\} ] _ { n } } \\\\ & { \\mathcal { T } [ \\{ \\Delta \\phi _ { i } , \\phi _ { i } \\} ] _ { n } = - \\mathcal { T } [ \\{ \\Delta \\phi _ { i } \\} ] _ { n } } \\\\ & \\qquad - \\{ [ 2 1 1 \\{ 0 \\} - 1 2 \\{ 1 2 \\} \\{ 1 9 \\} + 1 2 1 \\} \\\\ & { \\qquad \\mathcal { T } [ \\{ \\Delta \\phi _ { i } \\} ] _ { n } = \\delta } \\\\ & { \\qquad - \\{ [ 1 1 1 \\{ 1 1 \\{ 1 \\} \\{ 1 1 ^ { 2 } \\{ \\Delta \\phi _ { i } - 3 \\phi _ { i } \\} \\} ] \\qquad \\mathrm { i . e . } \\Lambda ^ { \\prime } [ \\{ \\Delta \\phi _ { i } \\} ] _ { n } + \\delta ^ { \\prime } [ \\{ \\Delta \\phi _ { i } \\} ] _ { n } \\{ 1 2 \\{ 2 \\Lambda ^ { 2 } - 1 6 \\} \\Lambda ^ { 2 } \\{ 1 2 \\phi _ { i } - 3 \\phi _ { i } \\} } \\\\ & \\qquad - \\{ [ 1 1 1 \\{ 1 1 \\{ 1 \\} \\{ 2 \\Lambda ^ { 2 } \\} \\} ] \\sin [ 2 \\Lambda ^ { 2 } \\phi _ { i } \\} \\} \\\\ & { \\qquad - \\mathcal { T } [ \\{ \\Delta \\phi _ { i } \\} ] _ { n } } \\\\ & \\mathcal { T } [ \\{ \\Delta \\phi _ { i } \\} ] _ { n } = - \\mathcal { T } [ \\{ \\Delta \\phi _ { i } \\} ] _ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 169, + 329, + 816, + 657 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Thus $\\operatorname { L } ( a , b )$ transforms under $\\tau$ by the matrix ", + "bbox": [ + 173, + 656, + 482, + 671 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/25368836f0e5df2b5c304fc7fab24be2ad632b5112dc5a535f77c0f141b3f28c.jpg", + "text": "$$\n\\operatorname { L } ( a , b ) \\mapsto \\operatorname { L } ( { \\binom { \\beta _ { 1 1 } } { \\beta _ { 2 1 } } } \\quad \\beta _ { 1 2 } ) \\left( { a \\atop b } \\right) )\n$$", + "text_format": "latex", + "bbox": [ + 387, + 676, + 607, + 712 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "with ", + "bbox": [ + 173, + 715, + 207, + 729 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/411f0b5e0d9f7eb2485ce2a918e29e9890d3bfeea9c9e9448651acaf845cb17f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\beta _ { 1 1 } = [ 1 1 | 1 1 ] - [ 1 1 | 2 2 ] } \\\\ & { \\beta _ { 1 2 } = 2 ( B - 2 ) ( [ 1 1 | 2 3 ] - [ 1 1 | 1 2 ] ) } \\\\ & { \\beta _ { 2 1 } = - [ 1 2 | 1 1 ] + [ 1 2 | 3 3 ] } \\\\ & { \\beta _ { 2 2 } = [ 1 2 | 2 1 ] + [ 1 2 | 1 2 ] + 2 ( B - 4 ) [ 1 2 | 1 3 ] - 2 ( B - 3 ) [ 1 2 | 3 4 ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 733, + 710, + 806 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "So if $\\lambda _ { \\mathbb { L } , 1 } ^ { \\mathcal { T } }$ and $\\lambda _ { \\mathbb { L } , 2 } ^ { T }$ are the roots of the equation ", + "bbox": [ + 173, + 809, + 488, + 830 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/fcb66ca56679cc427b8bf23b20429d0e9fe241c0f2d293f41234e70953c8d18c.jpg", + "text": "$$\nx ^ { 2 } - ( \\beta _ { 1 1 } + \\beta _ { 2 2 } ) x + \\beta _ { 1 1 } \\beta _ { 2 2 } - \\beta _ { 1 2 } \\beta _ { 2 1 }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 835, + 629, + 854 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "then ", + "bbox": [ + 173, + 859, + 205, + 872 + ], + "page_idx": 42 + }, + { + "type": "equation", + "img_path": "images/bb1ca5abc3a165a83b6ec7d3f20ec1bdc2527445340d33d3ed52a77d805146fd.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { T } \\{ \\mathrm { L } ( \\lambda _ { \\mathbb { L } , 1 } ^ { \\mathcal { T } } - \\beta _ { 2 2 } , \\beta _ { 2 1 } ) \\} = \\lambda _ { \\mathbb { L } , 1 } ^ { \\mathcal { T } } \\mathrm { L } ( \\lambda _ { \\mathbb { L } , 1 } ^ { \\mathcal { T } } - \\beta _ { 2 2 } , \\beta _ { 2 1 } ) } \\\\ { \\mathcal { T } \\{ \\mathrm { L } ( \\lambda _ { \\mathbb { L } , 2 } ^ { \\mathcal { T } } - \\beta _ { 2 2 } , \\beta _ { 2 1 } ) \\} = \\lambda _ { \\mathbb { L } , 2 } ^ { \\mathcal { T } } \\mathrm { L } ( \\lambda _ { \\mathbb { L } , 2 } ^ { \\mathcal { T } } - \\beta _ { 2 2 } , \\beta _ { 2 1 } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 876, + 665, + 924 + ], + "page_idx": 42 + }, + { + "type": "text", + "text": "Similarly, any image of these eigenvectors under simultaneous permutation of rows and columns remains eigenvectors with the same eigenvalue. This derives Item 2. ", + "bbox": [ + 171, + 103, + 825, + 132 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Item 3. Let $M \\in \\mathbb { M }$ . We first show that $\\tau \\{ M \\}$ has zero diagonal. We have ", + "bbox": [ + 173, + 137, + 669, + 155 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/3ba179055a850de260ce7c84be85faef012bb4b1666d263da9cb28fbe2be4a82.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\mathcal T } \\{ M \\} _ { 1 1 } = [ 1 1 | 1 2 ] ( \\sum _ { i = 2 } ^ { B } M _ { 1 i } + M _ { i 1 } ) + [ 1 1 | 2 3 ] ( \\sum _ { i , j = 1 } ^ { B } M _ { i j } - \\left( \\sum _ { i = 2 } ^ { B } M _ { 1 i } + M _ { i 1 } \\right) ) } } \\\\ { { \\displaystyle \\quad \\quad = 0 + 0 = 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 160, + 766, + 224 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "which follows from $M { \\bf 1 } = 0$ by definition of $\\mathbb { M }$ . Similarly $\\mathcal { T } \\{ M \\} _ { i i } = 0$ for all $i$ . ", + "bbox": [ + 173, + 231, + 707, + 247 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Now we show that $M$ is an eigenmatrix. ", + "bbox": [ + 174, + 252, + 439, + 267 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/76a79731c1dfbe5c1d3bdb96a88c991579261180de44a1b16b5b1a4140cd1140.jpg", + "text": "$$\n\\begin{array} { l } { { r \\{ M \\} _ { 1 2 } = [ 1 2 | 1 2 ] M _ { 1 2 } + [ 1 2 | 1 1 ] 0 + [ 1 2 | 3 3 ] 0 + [ 1 2 | 1 3 ] \\left( \\displaystyle \\sum _ { i = 3 } ^ { B } M _ { 1 i } + \\displaystyle \\sum _ { i = 1 } ^ { B } M _ { i 2 } \\right) } } \\\\ { { \\qquad + [ 1 2 | 2 1 ] M _ { 2 1 } + [ 1 2 | 3 1 ] \\left( \\displaystyle \\sum _ { i = 3 } ^ { B } M _ { i \\mathrm { i } } + \\displaystyle \\sum _ { i = 1 } ^ { B } M _ { 2 i } \\right) + [ 1 2 | 3 4 ] \\displaystyle \\sum _ { \\qquad i \\ge 3 , j \\ge 3 } M _ { i j } } } \\\\ { { \\qquad = [ 1 2 | 1 2 ] M _ { 1 2 } - 2 M _ { 1 2 } [ 1 2 | 1 3 ] + M _ { 2 1 } [ 1 2 | 2 1 ] + [ 1 2 | 3 1 ] ( - 2 M _ { 1 2 } ) + [ 1 2 | 3 4 ] ( \\displaystyle \\sum _ { i = 3 } ^ { B } M _ { \\mathrm { i } 1 } - M _ { \\mathrm { i } } } } \\\\ { { \\qquad = M _ { 1 2 } ( [ 1 2 | 1 2 ] - 2 | 1 2 | 3 ] + [ 1 2 | 2 1 ] - 2 [ 1 2 | 3 1 ] + 2 [ 1 2 | 3 4 ] ) } } \\\\ { { \\qquad = M _ { 1 2 } ( [ 1 2 | 1 2 ] + [ 1 2 | 2 1 ] - 4 | 1 2 | 3 | + 2 [ 1 2 | 3 4 ] ) } } \\\\ { { \\qquad = M _ { 1 2 } M _ { 1 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 275, + 839, + 467 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Similarly $\\mathcal { T } \\{ M \\} _ { i j } = \\lambda _ { \\mathbb { M } } ^ { \\mathcal { T } } M _ { i j }$ for all $i \\neq j$ . ", + "bbox": [ + 174, + 472, + 455, + 489 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Note that the eigenspaces described above are in general not orthogonal under trace inner product, so $\\tau$ is not self-adjoint (relative to the trace inner product) typically. However, as we see next, after projection by $G ^ { \\otimes 2 }$ , it is self-adjoint. ", + "bbox": [ + 173, + 526, + 826, + 569 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Theorem E.62 (Eigendecomposition of a projected ultrasymmetric operator). Let $\\mathcal { T } : \\mathbb { R } ^ { B \\times B } $ $\\mathbb { R } ^ { B \\times B }$ be an ultrasymmetric linear operator. We write $\\dot { G } ^ { \\otimes 2 } \\circ \\mathcal { T } \\ \\uparrow \\ \\dot { \\mathcal { H } } _ { B } ^ { G }$ for the operator $\\mathcal { H } _ { B } ^ { G } $ $\\mathcal { H } _ { B } ^ { G } , \\Sigma \\mapsto \\mathcal { T } \\{ \\Sigma \\} \\mapsto G ( \\mathcal { T } \\{ \\Sigma \\} ) G$ , restricted to $\\Sigma \\in \\mathcal { H } _ { B } ^ { G }$ . Then $G ^ { \\otimes 2 } \\circ \\mathcal { T } \\ \\ u { \\mid } \\ \\mathcal { H } _ { B } ^ { G }$ has the following eigendecomposition. ", + "bbox": [ + 173, + 573, + 825, + 632 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "1. Eigenspace $\\mathbb { R } G$ with eigenvalue $\\lambda _ { G } ^ { G , 7 } \\ : = \\ B ^ { - 1 } ( ( B - 1 ) ( \\alpha _ { 1 1 } - \\alpha _ { 2 1 } ) - ( \\alpha _ { 1 2 } - \\alpha _ { 2 2 } ) ) ,$ where as in Thm E.61, ", + "bbox": [ + 210, + 641, + 823, + 674 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/ea80db799f9c0ef92700a73b1ee6043f1541d72d144b3416a8a37eb75c605567.jpg", + "text": "$$\n\\begin{array} { l } { \\alpha _ { 1 1 } = [ 1 1 | 1 1 ] + [ 1 1 | 2 2 ] ( B - 1 ) } \\\\ { \\alpha _ { 1 2 } = 2 ( B - 1 ) [ 1 1 | 1 2 ] + ( B - 2 ) ( B - 1 ) [ 1 1 | 2 3 ] } \\\\ { \\alpha _ { 2 1 } = 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] } \\\\ { \\alpha _ { 2 2 } = [ 1 2 | 1 2 ] + 4 ( B - 2 ) [ 1 2 | 1 3 ] + [ 1 2 | 2 1 ] + ( B - 2 ) ( B - 3 ) [ 1 2 | 3 4 ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 679, + 792, + 752 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "2. Eigenspace L with eigenvalue $\\lambda _ { \\mathbb { L } } ^ { G , \\mathcal { T } } : = B ^ { - 1 } ( ( B - 2 ) \\beta _ { 1 1 } + \\beta _ { 1 2 } + 2 ( B - 2 ) \\beta _ { 2 1 } + 2 \\beta _ { 2 2 } ) ,$ where as in Thm E.61, ", + "bbox": [ + 209, + 768, + 823, + 801 + ], + "page_idx": 43 + }, + { + "type": "equation", + "img_path": "images/c85d7b5cd45a61caa98e473f6ed1c933f17c68bcfef355cc640fb8d800df4001.jpg", + "text": "$$\n\\begin{array} { r l } & { \\beta _ { 1 1 } = [ 1 1 | 1 1 ] - [ 1 1 | 2 2 ] } \\\\ & { \\beta _ { 1 2 } = 2 ( B - 2 ) ( [ 1 1 | 2 3 ] - [ 1 1 | 1 2 ] ) } \\\\ & { \\beta _ { 2 1 } = - [ 1 2 | 1 1 ] + [ 1 2 | 3 3 ] } \\\\ & { \\beta _ { 2 2 } = [ 1 2 | 2 1 ] + [ 1 2 | 1 2 ] + 2 ( B - 4 ) [ 1 2 | 1 3 ] - 2 ( B - 3 ) [ 1 2 | 3 4 ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 808, + 745, + 880 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "3. Eigenspace M with eigenvalue $\\lambda _ { \\mathbb { M } } ^ { G , \\mathcal { T } } : = \\lambda _ { \\mathbb { M } } ^ { \\mathcal { T } } = [ 1 2 | 1 2 ] + [ 1 2 | 2 1 ] - 4 [ 1 2 | 1 3 ] + 2 [ 1 2 | 3 4 ] .$ ", + "bbox": [ + 205, + 905, + 807, + 926 + ], + "page_idx": 43 + }, + { + "type": "text", + "text": "Proof. Item 1 As in the proof of $\\mathrm { T h m } \\mathrm { E } . 6 1 $ , we find ", + "bbox": [ + 173, + 103, + 514, + 118 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/94638a1ea74ee94b90d7da04ea9d54c8a38d5c3df99ebdc95ef10cdf547573b5.jpg", + "text": "$$\n\\mathcal { T } \\{ \\mathrm { B S B 1 } ( a , b ) \\} = \\mathrm { B S B 1 } \\left( \\left( \\begin{array} { l l } { \\alpha _ { 1 1 } } & { \\alpha _ { 1 2 } } \\\\ { \\alpha _ { 2 1 } } & { \\alpha _ { 2 2 } } \\end{array} \\right) \\left( \\begin{array} { l } { a } \\\\ { b } \\end{array} \\right) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 336, + 125, + 658, + 160 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 166, + 217, + 180 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/d23b5000cda70c914c5e45d93f824bf41dbc1c195ba504af4f09170f12151032.jpg", + "text": "$$\n\\begin{array} { l } { \\alpha _ { 1 1 } = [ 1 1 | 1 1 ] + [ 1 1 | 2 2 ] ( B - 1 ) } \\\\ { \\alpha _ { 1 2 } = 2 ( B - 1 ) [ 1 1 | 1 2 ] + ( B - 2 ) ( B - 1 ) [ 1 1 | 2 3 ] } \\\\ { \\alpha _ { 2 1 } = 2 [ 1 2 | 1 1 ] + ( B - 2 ) [ 1 2 | 3 3 ] } \\\\ { \\alpha _ { 2 2 } = [ 1 2 | 1 2 ] + 4 ( B - 2 ) [ 1 2 | 1 3 ] + [ 1 2 | 2 1 ] + ( B - 2 ) ( B - 3 ) [ 1 2 | 3 4 ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 186, + 736, + 257 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "For $a = B - 1 , b = - 1$ so that $\\mathrm { B S B } 1 ( B - 1 , - 1 ) = B G$ , we get ", + "bbox": [ + 173, + 262, + 604, + 279 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/ec90681ac80dc3d310abf3968a83db70e3efa38d99e94e5bef506d320cf1c4a5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\qquad \\mathcal { T } \\{ \\mathrm { B S B } 1 ( B - 1 , - 1 ) \\} = \\mathrm { B S B } 1 ( ( B - 1 ) \\alpha _ { 1 1 } - \\alpha _ { 1 2 } , ( B - 1 ) \\alpha _ { 2 1 } - \\alpha _ { 2 2 } ) } \\\\ & { \\qquad \\boldsymbol { G } ^ { \\otimes 2 } \\circ \\mathcal { T } \\{ \\mathrm { B S B } 1 ( B - 1 , - 1 ) \\} = G \\mathrm { B S B } 1 \\left( ( B - 1 ) \\alpha _ { 1 1 } - \\alpha _ { 1 2 } , ( B - 1 ) \\alpha _ { 2 1 } - \\alpha _ { 2 2 } \\right) \\ G } \\\\ & { \\qquad = \\left( ( B - 1 ) ( \\alpha _ { 1 1 } - \\alpha _ { 2 1 } ) - ( \\alpha _ { 1 2 } - \\alpha _ { 2 2 } ) \\right) G } \\\\ & { \\qquad = B ^ { - 1 } \\left( ( B - 1 ) ( \\alpha _ { 1 1 } - \\alpha _ { 2 1 } ) - ( \\alpha _ { 1 2 } - \\alpha _ { 2 2 } ) \\right) \\mathrm { B S B } 1 ( B - 1 , - 1 ) } \\\\ & { \\qquad = \\lambda _ { G } ^ { G } \\mathcal { T } \\mathrm { B S B } 1 ( B - 1 , - 1 ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 284, + 813, + 383 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "by Lemma E.52. ", + "bbox": [ + 173, + 387, + 284, + 402 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Item 2. It suffices to show that $\\mathrm { L } ( B - 2 , 1 )$ is an eigenmatrix with the eigenvalue $\\lambda _ { \\mathbb { L } } ^ { G , T }$ ", + "bbox": [ + 174, + 410, + 746, + 428 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "As in the proof of Thm E.61, we find ", + "bbox": [ + 173, + 433, + 419, + 446 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/8bd80a3f945cff56f1bfd929e51c054c741cc6ccc8b9cade6766183b8bdc9f34.jpg", + "text": "$$\n{ \\mathcal { T } } \\{ \\mathrm { L } ( a , b ) \\} = \\mathrm { L } \\left( \\left( { \\begin{array} { c c } { \\beta _ { 1 1 } } & { \\beta _ { 1 2 } } \\\\ { \\beta _ { 2 1 } } & { \\beta _ { 2 2 } } \\end{array} } \\right) { \\binom { a } { b } } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 369, + 453, + 629, + 489 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 174, + 494, + 217, + 508 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/b8e9ff9794e936913846e0268cdaaf885e7c5f8d4f6fff6314d66c63f099cc70.jpg", + "text": "$$\n\\begin{array} { r l } & { \\beta _ { 1 1 } = [ 1 1 | 1 1 ] - [ 1 1 | 2 2 ] } \\\\ & { \\beta _ { 1 2 } = 2 ( B - 2 ) ( [ 1 1 | 2 3 ] - [ 1 1 | 1 2 ] ) } \\\\ & { \\beta _ { 2 1 } = - [ 1 2 | 1 1 ] + [ 1 2 | 3 3 ] } \\\\ & { \\beta _ { 2 2 } = [ 1 2 | 2 1 ] + [ 1 2 | 1 2 ] + 2 ( B - 4 ) [ 1 2 | 1 3 ] - 2 ( B - 3 ) [ 1 2 | 3 4 ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 515, + 710, + 588 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "So with $a = B - 2 , b = 1$ , we have ", + "bbox": [ + 173, + 592, + 408, + 608 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/f351b79fb0db3ba0a5ed5966c400206d1b1bc9cf3692add6523586f801b40f7b.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathcal { T } \\{ \\mathrm { L } ( B - 2 , 1 ) \\} = \\mathrm { L } ( ( B - 2 ) \\beta _ { 1 1 } + \\beta _ { 1 2 } , ( B - 2 ) \\beta _ { 2 1 } + \\beta _ { 2 2 } ) } \\\\ & { \\quad G ^ { \\otimes 2 } \\circ \\mathcal { T } \\{ \\mathrm { L } ( B - 2 , 1 ) \\} = G \\mathrm { L } ( ( B - 2 ) \\beta _ { 1 1 } + \\beta _ { 1 2 } , ( B - 2 ) \\beta _ { 2 1 } + \\beta _ { 2 2 } ) G } \\\\ & { \\quad \\quad \\quad \\quad \\quad = B ^ { - 1 } ( ( B - 2 ) \\beta _ { 1 1 } + \\beta _ { 1 2 } + 2 ( ( B - 2 ) \\beta _ { 2 1 } + \\beta _ { 2 2 } ) ) \\mathrm { L } ( B - 2 , 1 ) } \\\\ & { \\quad \\quad \\quad \\quad = \\lambda _ { \\mathrm { L } } ^ { G , \\top } \\mathrm { L } ( B - 2 , 1 ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 612, + 790, + 695 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "by Lemma E.60. ", + "bbox": [ + 173, + 699, + 284, + 713 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Item 3. The proof is exactly the same as in that of Thm E.61. ", + "bbox": [ + 173, + 720, + 573, + 736 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Noting that the eigenspaces of $\\cdot$ are orthogonal, we have ", + "bbox": [ + 173, + 751, + 604, + 766 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Proposition E.63. $G ^ { \\otimes 2 } \\circ \\mathcal { T } \\uparrow \\mathcal { H } ^ { G }$ for any ultrasymmetric operator $\\tau$ is self-adjoint. ", + "bbox": [ + 173, + 768, + 730, + 785 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "DOS Operators. In some cases, such as the original study of vanilla tanh networks by Schoenholz et al. (2016), the ultrasymmetric operator involved has a much simpler form: ", + "bbox": [ + 173, + 799, + 825, + 829 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Definition E.64. Let $\\mathcal { T } : \\mathcal { H } _ { B } \\mathcal { H } _ { B }$ be such that for any $\\Sigma \\in \\mathcal { H } _ { B }$ and any $i \\neq j \\in [ B ]$ , ", + "bbox": [ + 169, + 832, + 754, + 848 + ], + "page_idx": 44 + }, + { + "type": "equation", + "img_path": "images/e5596c503905585691c72cf39c4f58840773428f8e04dc8ecbc953481f204808.jpg", + "text": "$$\n\\begin{array} { l } { { \\mathcal { T } \\{ \\Sigma \\} _ { i i } = u \\Sigma _ { i i } } } \\\\ { { \\mathcal { T } \\{ \\Sigma \\} _ { i j } = v \\Sigma _ { i i } + v \\Sigma _ { j j } + w \\Sigma _ { i j } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 390, + 853, + 606, + 891 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "Then we say that $\\tau$ is diagonal-off-diagonal semidirect, or $D O S$ for short. We write more specifically $\\mathcal { T } = \\dot { \\mathrm { D O S } } _ { B } ( u , v , w )$ . ", + "bbox": [ + 173, + 895, + 820, + 924 + ], + "page_idx": 44 + }, + { + "type": "text", + "text": "$\\cdot$ and $\\mathrm { T h m ~ E . 6 1 }$ still hold for DOS operators, but we can simplify the results and reason about them in a more direct way. ", + "bbox": [ + 169, + 102, + 825, + 132 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Lemma E.65. Let $\\mathcal { T } : = \\mathrm { D O S } _ { B } ( u , v , w )$ . Then ${ \\mathcal { T } } \\{ \\mathrm { L } _ { B } ( a , b ) \\} = \\mathrm { L } _ { B } ( u a , w b - v a ) .$ . ", + "bbox": [ + 174, + 135, + 718, + 152 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Proof. Let $L : = \\operatorname { L } _ { B } ( a , b )$ . It suffices to verify the following, each of which is a direct computation. ", + "bbox": [ + 165, + 165, + 823, + 183 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "1. $\\gamma \\{ L \\} _ { 3 : B , 3 : B } = 0 .$ . \n2. $\\mathcal { T } \\{ L \\} _ { 1 , 2 } = \\mathcal { T } \\{ L \\} _ { 2 , 1 } = 0$ \n3. $\\mathcal { T } \\{ L \\} _ { 1 , 1 } = - \\mathcal { T } \\{ L \\} _ { 2 , 2 } = u a$ \n4. $\\mathcal { T } \\{ L \\} _ { 1 , i } = - \\mathcal { T } \\{ L \\} _ { 2 , i } = \\mathcal { T } \\{ L \\} _ { i , 1 } = - \\mathcal { T } \\{ L \\} _ { i , 2 } = v a - w b .$ ", + "bbox": [ + 210, + 193, + 633, + 285 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Lemma E.66. Let $\\mathcal { T } : = \\mathrm { D O S } _ { B } ( u , v , w )$ . Then $\\mathrm { L } _ { B } ( w - u , v )$ and $\\mathrm { L } _ { B } ( 0 , 1 )$ are its eigenvectors: ", + "bbox": [ + 173, + 318, + 805, + 333 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/293e10ab9e8a4eb310a3f4ad032d2d0faa3ad20f33805233d56c29d04b98ca32.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { T } \\{ \\mathrm { L } _ { B } ( w - u , v ) \\} = u \\mathrm { L } _ { B } ( w - u , v ) } \\\\ { \\mathcal { T } \\{ \\mathrm { L } _ { B } ( 0 , 1 ) \\} = w \\mathrm { L } _ { B } ( 0 , 1 ) \\qquad } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 374, + 338, + 622, + 375 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Proof. The map $( a , b ) \\mapsto ( u a , w b - v a )$ has eigenvalues $u$ and $w$ with corresponding eigenvectors $( w - u , v )$ and $( 0 , 1 )$ . By Lemma E.65, this implies the desired results. ", + "bbox": [ + 173, + 387, + 820, + 417 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Lemma E.67. Let $\\mathcal { T } : = \\mathrm { D O S } _ { B } ( u , v , w )$ . Then for any $L \\in \\mathbb { L } _ { B }$ $\\begin{array} { r } { { _ 3 } , G ^ { \\otimes 2 } \\circ \\mathcal { T } \\{ L \\} = \\frac { ( u - 2 v ) ( B - 2 ) + 2 w } { B } L . } \\end{array}$ ", + "bbox": [ + 171, + 422, + 821, + 444 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Proof. By Lemma E.60 and Lemma E.65, ${ \\cal G } ^ { \\otimes 2 } \\circ { \\mathcal T } \\{ \\mathrm { L } ( B - 2 , 1 ) \\} = { \\cal G } ^ { \\otimes 2 } \\{ \\mathrm { L } ( u ( B - 1 ) , w - v ( B -$ $\\begin{array} { r } { 1 ) ) \\} \\} = { \\frac { ( u - 2 v ) ( B - 2 ) + 2 w } { B } } \\mathrm { L } _ { B } ( B - 2 , 1 ) } \\end{array}$ . By permutation. Since they span mmetry, we also have the general res, this gives the conclusion we want. $\\mathrm { L } _ { B } ( B - 2 , 1 )$ $L$ $\\mathbb { L } _ { B }$ $\\boxed { \\begin{array} { r l } \\end{array} }$ ", + "bbox": [ + 173, + 458, + 826, + 507 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Lemma E.68. Let $\\mathcal { T } : = \\mathrm { D O S } _ { B } ( u , v , w )$ . Then $\\mathcal { T } \\{ \\mathrm { B S B } 1 _ { B } ( a , b ) \\} = \\mathrm { B S B } 1 _ { B } ( u a , w b + 2 v a ) .$ ", + "bbox": [ + 169, + 511, + 784, + 530 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Proof. Direct computation. ", + "bbox": [ + 173, + 542, + 354, + 559 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Lemma E.69. Let $\\mathcal { T } : = \\mathrm { D O S } _ { B } ( u , v , w )$ . Then $\\mathrm { B S B } 1 _ { B } ( w - u , w v )$ and $\\mathrm { B S B 1 } _ { B } ( 0 , 1 )$ are eigenvectors of $\\tau$ . ", + "bbox": [ + 173, + 564, + 823, + 595 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/0975cc68f6545a3ccb56ba1a4390a2afc160fa486351bb36f41b82160cad59a0.jpg", + "text": "$$\n\\begin{array} { c } { { \\mathcal { T } \\{ \\mathrm { B S B } 1 _ { B } ( u - w , 2 v ) \\} = u \\mathrm { B S B } 1 _ { B } ( u - w , 2 v ) } } \\\\ { { \\mathcal { T } \\{ \\mathrm { B S B } 1 _ { B } ( 0 , 1 ) \\} = w \\mathrm { B S B } 1 _ { B } ( 0 , 1 ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 336, + 599, + 661, + 637 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Proof. The linear map $( a , b ) \\mapsto ( u a , w b + 2 v a )$ has eigenvalues $u$ and $w$ with corresponding eigenvectors $( u - w , 2 v )$ and $( 0 , 1 )$ . The result then immediately follows from Lemma E.68. □ ", + "bbox": [ + 171, + 666, + 825, + 696 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Lemma E.70. Let $\\mathcal { T } : = \\mathrm { D O S } _ { B } ( u , v , w )$ . Then $\\begin{array} { r } { G ^ { \\otimes 2 } \\circ \\mathcal { T } \\{ G \\} = \\frac { ( B - 1 ) ( u - 2 v ) + w } { B } G . } \\end{array}$ ", + "bbox": [ + 173, + 703, + 717, + 724 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Proof. Direct computation with Lemma E.68 and Lemma E.52 ", + "bbox": [ + 173, + 736, + 589, + 752 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Definition E.71. Define ${ \\overline { { \\mathbb { M } } } } _ { B } : = \\{ \\Sigma \\in { \\cal S } _ { B } : \\mathrm { D i a g } \\Sigma = 0 \\}$ (so compared to $\\mathbb { M }$ , matrices in $\\overline { { \\mathbb { M } } }$ do not need to have zero row sums). In addition, for any $a , b \\in \\mathbb { R }$ , set ", + "bbox": [ + 173, + 760, + 825, + 790 + ], + "page_idx": 45 + }, + { + "type": "equation", + "img_path": "images/9fff47d36dc302a6ebe4fe8c9f208af951d7dca7acfe15724bfe99346bbace31.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathrm { L } } } _ { B } ( a , b ) : = \\left( \\begin{array} { l l l l } { a } & { - b } & { - b } & { \\cdots } \\\\ { - b } & { 0 } & { 0 } & { \\cdots } \\\\ { - b } & { 0 } & { 0 } & { \\cdots } \\\\ { \\vdots } & { \\vdots } & { \\vdots } & { \\ddots } \\end{array} \\right) \\in \\mathcal { H } _ { B } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 795, + 645, + 863 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Define $\\overline { { \\mathbb { L } } } _ { B } ( a , b ) : = \\mathrm { s p a n } ( P _ { 1 i } ^ { T } \\overline { { \\mathrm { L } } } _ { B } ( a , b ) P _ { 1 i } : i \\in [ B ] )$ where $P _ { 1 i }$ is the permutation matrix that swap the first entry with the ith entry, i.e. $\\overline { { \\mathbb { L } } } _ { B } ( a , b )$ is the span of the orbit of $\\textstyle { \\overline { { \\operatorname { L } } } } _ { B } ( a , b )$ under permuting rows and columns simultaneously. ", + "bbox": [ + 173, + 877, + 825, + 924 + ], + "page_idx": 45 + }, + { + "type": "text", + "text": "Note that ", + "bbox": [ + 173, + 103, + 238, + 118 + ], + "page_idx": 46 + }, + { + "type": "equation", + "img_path": "images/0dad303fae6054a6e5327a671eb2bd94f4cd74d31973a6ecd1178304f1916b31.jpg", + "text": "$$\n\\begin{array} { c } { \\mathrm { L } _ { B } ( a , b ) = \\overline { { \\mathrm { L } } } _ { B } ( a , b ) - P _ { 1 2 } ^ { T } \\overline { { \\mathrm { L } } } _ { B } ( a , b ) P _ { 1 2 } } \\\\ { \\mathrm { B S B 1 } _ { B } ( a , - 2 b ) = \\displaystyle \\sum _ { i = 1 } ^ { B } P _ { 1 i } ^ { T } \\overline { { \\mathrm { L } } } _ { B } ( a , b ) P _ { 1 i } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 122, + 656, + 188 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "So $\\mathrm { L } _ { B } ( a , b ) , \\mathrm { B S B 1 } _ { B } ( a , - 2 b ) \\in \\overline { { \\mathbb { L } } } _ { B } ( a , b )$ ", + "bbox": [ + 176, + 212, + 450, + 228 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Theorem E.72. Let $\\mathcal { T } : = \\mathrm { D O S } _ { B } ( u , v , w )$ . Suppose $w \\ne u$ . Then $\\mathcal { T } \\ \\lceil \\ \\mathcal { H } _ { B }$ has the following eigendecomposition: ", + "bbox": [ + 173, + 231, + 823, + 261 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "• ${ \\overline { { \\mathbb { M } } } } _ { B }$ has eigenvalue $w$ $( \\dim \\overline { { \\mathbb { M } } } _ { B } = B ( B - 1 ) / 2 )$ • $\\overline { { \\mathbb { L } } } _ { B } ( w - u , v )$ has eigenvalue of $u$ $\\iota ( \\dim \\mathbb { L } _ { B } ( w - u , v ) = B )$ ", + "bbox": [ + 215, + 270, + 633, + 314 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "If $w = u$ , then $\\mathbb { L } _ { B } ( w - u , v ) \\subseteq \\overline { { \\mathbb { M } } } _ { B }$ ", + "bbox": [ + 173, + 323, + 416, + 340 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Proof. The case of ${ \\overline { { \\mathbb { M } } } } _ { B }$ is obvious. We will show that $\\overline { { \\mathbb { L } } } _ { B } ( w - u , v )$ is an eigenspace with eigenvalue $u$ . Then by dimensionality consideration they are all of the eigenspaces of $\\tau$ . ", + "bbox": [ + 174, + 356, + 823, + 386 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Let $L : = \\overline { { \\mathrm { L } } } _ { B } ( w - u , v )$ . Then it’s not hard to see ${ \\mathcal { T } } \\{ L \\} = { \\overline { { \\mathrm { L } } } } _ { B } ( a , b )$ for some $a , b \\in \\mathbb { R }$ . It follows that $a$ has to be $u ( w - u )$ and $b$ has to be $- ( v ( w - u ) - w v ) = u v$ , which yields what we want. ", + "bbox": [ + 174, + 391, + 823, + 421 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Theorem E.73. Let $\\mathcal { T } : = \\mathrm { D O S } _ { B } ( u , v , w )$ . Then $G ^ { \\otimes 2 } \\circ \\mathcal { T } \\mid \\mathcal { H } _ { B } ^ { G } : \\mathcal { H } _ { B } ^ { G } \\mathcal { H } _ { B } ^ { G }$ has the following eigendecomposition: ", + "bbox": [ + 169, + 426, + 825, + 458 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "RG has eigenvalu e (B−1)(u−2v)+w (dim RG = 1) • $\\mathbb { M } _ { B }$ has eigenvalue w $( \\dim \\mathbb { M } _ { B } = B ( B - 3 ) / 2 )$ • $\\mathbb { L } _ { B }$ has eigenvalue $\\frac { ( B - 2 ) ( u - 2 v ) + 2 w } { B } ( \\dim \\mathbb { L } _ { B } = B - 1 )$ ", + "bbox": [ + 215, + 467, + 602, + 540 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Proof. The case of ${ \\mathbb { M } } _ { B }$ is obvious. The case for $\\mathbb { R } G$ follows from Lemma E.70. The case for $\\mathbb { L } _ { B }$ follows from Lemma E.67. By dimensionality considerations these are all of the eigenspaces. □ ", + "bbox": [ + 171, + 551, + 823, + 582 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "F FORWARD DYNAMICS ", + "text_level": 1, + "bbox": [ + 174, + 601, + 390, + 617 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "In this section we will be interested in studying the dynamics on PSD matrices of the form ", + "bbox": [ + 169, + 631, + 766, + 647 + ], + "page_idx": 46 + }, + { + "type": "equation", + "img_path": "images/ba1c1c45af726dfe65fef3f5d3c7bd7abdcb3d59e165c6603f304fd2806f3672.jpg", + "text": "$$\n\\Sigma ^ { l } = { \\bf V } _ { B _ { \\phi } } ( \\Sigma ^ { l - 1 } ) = \\mathbb { E } [ \\phi ( \\sqrt { B } G h / \\| G h \\| ) ^ { \\otimes 2 } : h \\sim \\mathcal { N } ( 0 , \\Sigma ^ { l - 1 } ) ]\n$$", + "text_format": "latex", + "bbox": [ + 289, + 652, + 707, + 674 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "where $\\Sigma ^ { l } \\in S _ { B }$ and $\\phi : \\mathbb { R } \\mathbb { R }$ . ", + "bbox": [ + 174, + 680, + 387, + 695 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "F.1 GLOBAL CONVERGENCE ", + "text_level": 1, + "bbox": [ + 174, + 712, + 388, + 727 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Basic questions regarding the dynamics Eq. (43) are 1) does it converge? 2) What are the limit points? 3) How fast does it converge? Here we answer these questions definitively when $\\phi = \\mathrm { i d }$ . ", + "bbox": [ + 173, + 738, + 825, + 767 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "The following is the key lemma in our analysis. ", + "bbox": [ + 173, + 773, + 486, + 789 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Lemma F.1. Consider the dynamics $\\Sigma ^ { l } = \\mathbb { E } [ ( h / \\| h \\| ) ^ { \\otimes 2 } : h \\sim \\mathcal { N } ( 0 , \\Sigma ^ { l - 1 } ) ]$ on $\\Sigma ^ { l } \\in { \\mathcal { S } } _ { A }$ . Suppose $\\Sigma ^ { 0 }$ is full rank. Then ", + "bbox": [ + 173, + 790, + 825, + 821 + ], + "page_idx": 46 + }, + { + "type": "equation", + "img_path": "images/8a42f7cf364871e0c5b59ba369a80fd69ef7d677ee620be92635eeb4dcf43cd1.jpg", + "text": "$$\n\\begin{array} { r } { I . \\ \\operatorname* { l i m } _ { l \\infty } \\Sigma ^ { l } = \\frac { 1 } { A } I . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 212, + 829, + 356, + 849 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "2. This convergence is exponential in the sense that, for any full rank $\\Sigma ^ { 0 }$ , there is a constant $K < 1$ such that $\\lambda _ { 1 } ( \\bar { \\Sigma } ^ { l } ) - \\lambda _ { A } ( \\Sigma ^ { l } ) < K ( \\lambda _ { 1 } ( \\Sigma ^ { l - 1 } ) - \\bar { \\lambda _ { A } } ( \\Sigma ^ { l - 1 } ) )$ for all $l \\geq 2$ . Here $\\lambda _ { 1 }$ (resp. $\\lambda _ { A }$ ) denotes that largest (resp. smallest) eigenvalue. ", + "bbox": [ + 212, + 857, + 825, + 901 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "3. Asymptotically, $\\begin{array} { r } { \\lambda _ { 1 } ( \\Sigma ^ { l } ) - \\lambda _ { A } ( \\Sigma ^ { l } ) = O ( ( 1 - \\frac { 2 } { A + 2 } ) ^ { l } ) . } \\end{array}$ ", + "bbox": [ + 210, + 906, + 581, + 926 + ], + "page_idx": 46 + }, + { + "type": "text", + "text": "Proof. Let ${ \\lambda _ { 1 } } ^ { l } \\ge { \\lambda _ { 2 } } ^ { l } \\ge \\dots \\ge { \\lambda _ { A } } ^ { l }$ be the eigenvalues of $\\Sigma ^ { l }$ . It’s easy to see that $\\begin{array} { r } { \\sum _ { i } \\lambda _ { i } ^ { l } = 1 } \\end{array}$ for all $l \\geq 1$ . So WLOG we assume $l \\geq 1$ and this equality holds. ", + "bbox": [ + 171, + 102, + 823, + 133 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "We will show the “exponential convergence” statement; that $\\begin{array} { r } { \\operatorname* { l i m } _ { l \\to \\infty } \\Sigma ^ { l } = \\frac { 1 } { A } I } \\end{array}$ then follows from the trace condition above. ", + "bbox": [ + 173, + 138, + 821, + 167 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Proof of Item 2. For brevity, we write suppress the superscript index $l$ , and use · to denote ·l. We will now compute the eigenvalues $\\overline { { \\lambda } } _ { 1 } \\geq \\cdots \\geq \\overline { { \\lambda } } _ { A }$ of $\\overline { { \\Sigma } }$ . ", + "bbox": [ + 171, + 174, + 823, + 205 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "First, notice that $\\overline { { \\Sigma } }$ and $\\Sigma$ can be simultaneously diagonalized, and by induction all of $\\{ \\Sigma ^ { l } \\} _ { l \\geq 0 }$ can be simultaneous diagonalized. Thus we will WLOG assume that $\\Sigma$ is diagonal, $\\Sigma = { \\mathrm { D i a g } } ( \\lambda _ { 1 } , \\bar { . } . . , \\lambda _ { A } )$ , so that $\\overline { { \\Sigma } } = \\mathrm { D i a g } ( \\gamma _ { 1 } , \\dots , \\gamma _ { A } )$ for some $\\{ \\gamma _ { i } \\} _ { i }$ . These $\\{ \\gamma _ { i } \\} _ { i }$ form the eigenvalues of $\\overline { { \\Sigma } }$ but a priori we don’t know whether they fall in decreasing order; in fact we will soon see that they do, and $\\gamma _ { i } = \\overline { { \\lambda } } _ { i }$ . ", + "bbox": [ + 173, + 209, + 826, + 284 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "We have ", + "bbox": [ + 173, + 290, + 233, + 304 + ], + "page_idx": 47 + }, + { + "type": "equation", + "img_path": "images/b558c01a3cf3eaccad735021fae7c7f5f268cc0673f64ad5c5ec6ea1b9cd863e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\gamma _ { 3 } = \\binom { \\pi } { 2 } ^ { - \\frac { 1 } { 2 } - \\theta / 2 } \\int _ { 0 } ^ { \\infty } \\frac { \\lambda \\sin ^ { 2 } \\theta } { \\sum _ { j } \\lambda _ { j } ^ { 2 } + \\theta ^ { 2 } } \\mathrm { e } ^ { - \\lambda _ { j } \\theta / 2 } } \\\\ & { \\quad - ( \\frac { \\pi } { 2 } ) ^ { - \\theta / 2 } \\int _ { 0 } ^ { \\infty } \\frac { \\lambda \\sin ^ { 2 } \\theta } { \\lambda _ { j } \\sin ^ { 2 } \\theta } \\mathrm { e } ^ { - \\lambda _ { j } \\theta / 2 } d \\mathrm { e } ^ { \\theta } \\mathrm { e } ^ { - \\gamma _ { 3 } \\sum \\theta / 2 } d \\mathrm { e } } \\\\ & { \\quad - ( \\frac { \\pi } { 2 } ) ^ { - \\theta / 2 } \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\lambda _ { j } \\cos ^ { 2 } \\theta \\mathrm { e } ^ { - \\lambda _ { j } \\theta / 2 } d \\mathrm { e } ^ { \\theta } \\mathrm { e } ^ { \\theta } \\mathrm { e } ^ { \\theta } \\mathrm { e } ^ { \\theta } \\mathrm { e } ^ { \\lambda _ { j } \\theta / 2 } d \\mathrm { e } } \\\\ & { \\quad - ( \\frac { \\pi } { 2 } ) ^ { - \\theta / 2 } \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\lambda _ { j } \\sin ^ { 2 } \\theta \\mathrm { e } ^ { - \\lambda _ { j } \\theta / 2 } d \\mathrm { e } ^ { \\theta } \\mathrm { e } ^ { \\theta } \\mathrm { e } ^ { \\lambda _ { j } \\theta / 2 } \\mathrm { e } \\mathrm { d } \\mathrm { e } } \\\\ & { \\quad - ( \\frac { \\pi } { 2 } ) ^ { - \\theta / 2 } \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\infty } \\lambda _ { j } \\cos ^ { 2 } \\theta \\mathrm { e } ^ { - \\lambda _ { j } \\theta / 2 } d \\mathrm { e } ^ { \\theta } \\mathrm { e } ^ { \\theta } \\mathrm { e } ^ { \\lambda _ { j } \\theta / 2 } \\prod _ { j } ^ { ( 1 ) } [ \\frac { \\pi } { 2 } ( 1 + 2 \\alpha \\lambda _ { j } ) ^ { - \\theta / 2 } \\mathrm { e } ^ { \\lambda _ { j } \\theta / 2 } } \\\\ & \\quad - ( \\frac { \\pi } { 2 } ) ^ { - \\theta / 2 } \\int _ { 0 } ^ { \\infty } \\lambda _ { j } ( 1 + 2 \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 303, + 761, + 599 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Therefore, ", + "bbox": [ + 173, + 623, + 245, + 637 + ], + "page_idx": 47 + }, + { + "type": "equation", + "img_path": "images/2bd2bf58a518be549bc43bb094f2af30a2d72413b2c9fd9ae837fc755ca6844f.jpg", + "text": "$$\n\\begin{array} { r } { \\gamma _ { i } - \\gamma _ { k } = \\displaystyle \\int _ { 0 } ^ { \\infty } \\prod _ { j = 1 } ^ { A } ( 1 + 2 s \\lambda _ { j } ) ^ { - 1 / 2 } ( \\lambda _ { i } ( 1 + 2 s \\lambda _ { i } ) ^ { - 1 } - \\lambda _ { k } ( 1 + 2 s \\lambda _ { k } ) ^ { - 1 } ) \\mathrm { d } s } \\\\ { = ( \\lambda _ { i } - \\lambda _ { k } ) \\displaystyle \\int _ { 0 } ^ { \\infty } \\prod _ { j = 1 } ^ { A } ( 1 + 2 s \\lambda _ { j } ) ^ { - 1 / 2 } ( 1 + 2 s \\lambda _ { i } ) ^ { - 1 } ( 1 + 2 s \\lambda _ { k } ) ^ { - 1 } \\mathrm { d } s } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 638, + 751, + 729 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Since the RHS integral is always positive, $\\lambda _ { i } \\geq \\lambda _ { k } \\implies \\gamma _ { i } \\geq \\gamma _ { k }$ and thus $\\gamma _ { i } = \\overline { { \\lambda } } _ { i }$ for each $i$ . ", + "bbox": [ + 173, + 748, + 787, + 766 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Define $\\begin{array} { r } { T ( \\lambda ) : = \\int _ { 0 } ^ { \\infty } \\prod _ { j = 1 } ^ { A } ( 1 + 2 s \\lambda _ { 1 } ) ^ { - 1 / 2 } ( 1 + 2 s \\lambda _ { 1 } ) ^ { - 1 } ( 1 + 2 s \\lambda _ { A } ) ^ { - 1 } \\mathrm { d } s , } \\end{array}$ , so that $\\overline { { { \\lambda } } } _ { 1 } \\mathrm { ~ - ~ } \\overline { { { \\lambda } } } _ { A } \\mathrm { ~ = ~ }$ $( \\lambda _ { 1 } - \\lambda _ { A } ) T ( \\lambda )$ . ", + "bbox": [ + 176, + 770, + 826, + 806 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "Note first that $\\begin{array} { r } { \\prod _ { j = 1 } ^ { A } ( 1 + 2 s \\lambda _ { j } ) ^ { - 1 / 2 } ( 1 + 2 s \\lambda _ { i } ) ^ { - 1 } ( 1 + 2 s \\lambda _ { k } ) ^ { - 1 } } \\end{array}$ is (strictly) log-convex and hence (strictly) convex in $\\lambda$ . Furthermore, $T ( \\lambda )$ is (strictly) convex because it is an integral of (strictly) convex functions. Thus $T$ is maximized over any convex region by its extremal points, and only by its extremal points because of strict convexity. ", + "bbox": [ + 173, + 813, + 825, + 873 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "The convex region we are interested in is given by ", + "bbox": [ + 176, + 878, + 501, + 895 + ], + "page_idx": 47 + }, + { + "type": "equation", + "img_path": "images/cb633bf16d19dbe9cad18c3dd893ca5de8d634b20fe235f3d9b458437256d293.jpg", + "text": "$$\n{ \\mathcal { A } } : = \\{ ( \\lambda _ { i } ) _ { i } : \\lambda _ { 1 } \\geq \\lambda _ { 2 } \\geq \\cdots \\geq \\lambda _ { A } \\geq 0 \\ \\& \\ \\sum _ { i } \\lambda _ { i } = 1 \\} .\n$$", + "text_format": "latex", + "bbox": [ + 307, + 895, + 687, + 928 + ], + "page_idx": 47 + }, + { + "type": "text", + "text": "The unit sum condition follows from the normalization $h / \\| h \\|$ of the iteration map. The extremal $k$ times points of $\\mathcal { A }$ are $\\{ \\omega ^ { k } : = ( \\widehat { 1 / k , 1 / k , \\cdot \\cdot \\cdot , 1 / k } , 0 , 0 , \\cdot \\cdot \\cdot , 0 ) \\} _ { k = 1 } ^ { A }$ . For $k = 1 , \\ldots , A - 1$ , we have ", + "bbox": [ + 169, + 102, + 826, + 152 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/ca46e7e7ebedab9cba2280ae1bac11f626164d2066daa5548366393b0c2e00f5.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle T ( \\omega ^ { k } ) = \\int _ { 0 } ^ { \\infty } ( 1 + 2 s / k ) ^ { - k / 2 - 1 } \\mathrm { d } s } \\\\ { \\displaystyle \\qquad = \\frac { 2 / k } { - k / 2 } ( 1 + 2 s / k ) ^ { - k / 2 } \\bigg \\rvert _ { 0 } ^ { \\infty } } \\\\ { \\displaystyle \\qquad = 1 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 377, + 156, + 619, + 243 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "But for $k = A$ , ", + "bbox": [ + 173, + 246, + 274, + 262 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/4285f15679297021fc878f06e9d03ed7b1882e47955212894ba30e76c9f75196.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle T ( \\omega ^ { A } ) = \\int _ { 0 } ^ { \\infty } ( 1 + 2 s / A ) ^ { - A / 2 - 2 } \\mathrm { d } s } \\\\ { \\displaystyle \\quad = \\frac { 2 / A } { - A / 2 - 1 } ( 1 + 2 s / A ) ^ { - k / 2 } \\bigg \\vert _ { 0 } ^ { \\infty } } \\\\ { \\displaystyle \\quad = \\frac { A } { A + 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 362, + 265, + 633, + 369 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "This shows that $T ( \\lambda ) \\leq 1$ , with equality iff $\\lambda = \\omega ^ { k }$ for $k = 1 , \\ldots , A - 1$ . In fact, because every point $\\lambda \\in { \\mathcal { A } }$ is a convex combination of $\\omega ^ { k }$ for $k = 1 , \\dots , A$ , $\\textstyle \\lambda = \\sum _ { k = 1 } ^ { A } a _ { k } \\omega ^ { k }$ , by convexity of $T$ we must have ", + "bbox": [ + 178, + 372, + 821, + 417 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/a5d3442b035cc681b4699b7aec0a64e2e396337e70c625f73face2b700be6718.jpg", + "text": "$$\n\\begin{array} { r } { T ( \\lambda ) \\leq \\displaystyle \\sum _ { k = 1 } ^ { A } a _ { k } T ( \\omega ^ { k } ) } \\\\ { = 1 - a _ { A } \\displaystyle \\frac { 2 } { A + 2 } } \\\\ { = 1 - \\lambda _ { A } \\displaystyle \\frac { 2 } { A + 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 421, + 419, + 573, + 527 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "where the last line follows because $\\omega ^ { A }$ is the only point with last coordinate nonzero so that $a _ { A } =$ $\\lambda _ { A }$ . ", + "bbox": [ + 173, + 530, + 823, + 560 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "We now show that the gap ${ \\lambda _ { 1 } } ^ { l } - { \\lambda _ { A } } ^ { l } 0$ as $l \\infty$ . There are two cases: if ${ \\lambda _ { A } } ^ { l }$ is bounded away from 0 infinitely often, then $T ( \\lambda ) < 1 - \\epsilon$ infinitely often for a fixed $\\epsilon > 0$ so that the gap indeed vanishes with $l$ . Now suppose otherwise, that ${ \\lambda _ { A } } ^ { l }$ converges to 0; we will show this leads to a contradiction. Notice that ", + "bbox": [ + 174, + 564, + 825, + 625 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/73ce29fc094adbe6ef80a9d6fbfaef85385513fa21bf5a3b6ee7b77fc81526c7.jpg", + "text": "$$\n\\begin{array} { c } { { \\overline { { { \\lambda } } } _ { A } = \\displaystyle \\int _ { 0 } \\displaystyle \\prod _ { j = 1 } ^ { \\infty } ( 1 + 2 s \\lambda _ { j } ) ^ { - 1 / 2 } \\lambda _ { A } ( 1 + 2 s \\lambda _ { A } ) ^ { - 1 } \\mathrm { d } s } } \\\\ { { \\overline { { { \\lambda } } } _ { A } / \\lambda _ { A } \\geq \\displaystyle \\int _ { 0 } \\displaystyle \\prod _ { j = 1 } ^ { \\infty } ( 1 + 2 s ( 1 - \\lambda _ { A } ) / ( A - 1 ) ) ^ { - ( A - 1 ) / 2 } ( 1 + 2 s \\lambda _ { A } ) ^ { - 3 / 2 } \\mathrm { d } s } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 626, + 748, + 718 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "where the first lines is Eq. (44) and the 2nd line follows from the convexity of $\\prod _ { j = 1 } ^ { A - 1 } ( 1 + 2 s \\lambda _ { j } ) ^ { - 1 / 2 }$ as a function of $\\left( \\lambda _ { 1 } , \\ldots , \\lambda _ { A - 1 } \\right)$ . By a simple application of dominated convergence, as $\\lambda _ { A } \\to 0$ , this integral converges to a particular simple form, ", + "bbox": [ + 176, + 723, + 823, + 770 + ], + "page_idx": 48 + }, + { + "type": "equation", + "img_path": "images/3cb75f0c93826fb624d775eb0ebce049b7d6bb6697e336659c0ad798d3166aac.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\overline { { \\lambda } } _ { A } / \\lambda _ { A } \\geq \\int _ { 0 } ^ { \\infty } \\prod _ { j = 1 } ^ { A } ( 1 + 2 s / ( A - 1 ) ) ^ { - ( A - 1 ) / 2 } \\mathrm { d } s } \\\\ { \\displaystyle \\qquad = 1 + \\frac { 2 } { A - 3 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 334, + 773, + 663, + 852 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Thus for large enough $l$ , ${ \\lambda _ { A } } ^ { l + 1 } / { \\lambda _ { A } } ^ { l }$ is at least $1 + \\epsilon$ for some $\\epsilon > 0$ , but this contradicts the convergence of $\\lambda _ { A }$ to 0. ", + "bbox": [ + 171, + 854, + 823, + 886 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "Altogether, this proves that ${ \\lambda _ { 1 } } ^ { l } - { \\lambda _ { A } } ^ { l } 0$ and therefore $\\lambda ^ { l } \\to \\omega ^ { A }$ as $l \\infty$ . Consequently, ${ \\lambda _ { A } } ^ { l }$ is bounded from below, say by $K ^ { \\prime }$ (where $K ^ { \\prime } > 0$ because by Eq. (44), ${ \\lambda _ { A } } ^ { l }$ is never 0), for all $l$ , and by Eq. (45), we prove Item 2 by taking $\\begin{array} { r } { K = 1 - K ^ { \\prime } \\frac { 2 } { A + 2 } } \\end{array}$ . In addition, asymptotically, the gap decreases exponentially as $\\begin{array} { r } { T ( \\omega ^ { A } ) ^ { l } = \\left( \\frac { A } { A + 2 } \\right) ^ { l } } \\end{array}$ , proving Item 3. ", + "bbox": [ + 173, + 891, + 823, + 925 + ], + "page_idx": 48 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 102, + 825, + 148 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Theorem F.2. Consider the dynamics $\\Sigma ^ { l } = \\mathbb { E } [ ( h / \\| h \\| ) ^ { \\otimes 2 } : h \\sim \\mathcal { N } ( 0 , \\Sigma ^ { l - 1 } ) ]$ on $\\Sigma ^ { l } \\in { \\cal S } _ { A }$ . Suppose $\\Sigma ^ { 0 } = M ^ { T } D M$ where $M$ is orthogonal and $D$ is a diagonal matrix. If $D$ has rank $C$ and $D _ { i i } \\neq $ $0 , \\forall 1 \\leq i \\leq C$ , then ", + "bbox": [ + 174, + 155, + 825, + 199 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "1. $\\begin{array} { r } { \\operatorname* { l i m } _ { l \\to \\infty } \\Sigma ^ { l } = \\frac { 1 } { C } M ^ { T } D ^ { \\prime } M } \\end{array}$ where $D ^ { \\prime }$ is the diagonal matrix with $D _ { i i } ^ { \\prime } = \\mathbb { I } ( D _ { i i } \\neq 0 )$ . ", + "bbox": [ + 209, + 208, + 777, + 227 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "2. This convergence is exponential in the sense that, for any $\\Sigma ^ { 0 }$ of rank $C$ , there is a constant $K < 1$ such that $\\lambda _ { 1 } ( \\dot { \\Sigma } ^ { l } ) - \\lambda _ { C } ( \\Sigma ^ { l } ) < K ( \\lambda _ { 1 } ( \\Sigma ^ { l \\bar { - } 1 } ) - \\dot { \\lambda } _ { C } ( \\Sigma ^ { \\bar { l } - 1 } ) )$ for all $l \\geq 2$ . Here $\\lambda _ { i }$ denotes the ith largest eigenvalue. ", + "bbox": [ + 210, + 236, + 823, + 280 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "3. Asymptotically, $\\begin{array} { r } { \\lambda _ { 1 } ( \\Sigma ^ { l } ) - \\lambda _ { C } ( \\Sigma ^ { l } ) = O ( ( 1 - \\frac { 2 } { C + 2 } ) ^ { l } ) . } \\end{array}$ . ", + "bbox": [ + 210, + 286, + 583, + 306 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Proof. Note that $\\Sigma ^ { l }$ can always be simultaneously diagonalized with $\\Sigma ^ { 0 }$ , so that $\\Sigma ^ { l } = { M } ^ { T } D ^ { l } { M }$ for some diagonal $D ^ { l }$ which has $D _ { i i } ^ { l } = 0 , \\forall 0 \\leq \\overline { { i } } \\leq \\mathbf { \\bar { C } }$ . Then we have $( D ^ { l } ) ^ { \\prime } = \\mathbb { E } [ ( h / \\| h \\| ) ^ { \\otimes 2 } :$ $h \\sim \\mathcal { N } ( 0 , ( D ^ { l - 1 } ) ^ { \\prime } ) ]$ , where $( D ^ { l } ) ^ { \\prime }$ means the diagonal matrix obtained from $D ^ { l }$ by deleting the dimensions with zero eigenvalues. The proof then finishes by Lemma F.1. □ ", + "bbox": [ + 173, + 321, + 825, + 381 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "From this it easily follows the following characterization of the convergence behavior. ", + "bbox": [ + 174, + 395, + 736, + 410 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Corollary F.3. Consider the dynamics of Eq. (43) for $\\phi = \\mathrm { i d }$ : $\\Sigma ^ { l } = B \\mathbb { E } [ ( G h / \\| G h \\| ) ^ { \\otimes 2 } : h \\sim$ $\\mathcal { N } ( 0 , \\Sigma ^ { l - 1 } ) ]$ on $\\Sigma ^ { l } \\in S _ { B }$ . Suppose $G \\Sigma G$ has rank $C < B$ and factors as $\\hat { \\Phi } \\bar { D \\Phi } ^ { T }$ where $\\mathbf { \\varPhi } ^ { \\prime \\prime } \\mathbf { \\varTheta } ^ { \\prime \\prime } \\in \\mathbb { R } ^ { C \\times C }$ is a diagonal matrix with no zero diagonal entries and $\\hat { \\Phi }$ is an $B \\times C$ matrix whose columns form an orthonormal basis of a subspace of $\\operatorname { i m } G$ . Then ", + "bbox": [ + 174, + 414, + 825, + 470 + ], + "page_idx": 49 + }, + { + "type": "equation", + "img_path": "images/99f2dab8b7eeb094b3c0b9a0a78c0a4684864c2e90adfa90e5c1d665ed3cd788.jpg", + "text": "$$\n\\begin{array} { r } { I . \\ \\operatorname* { l i m } _ { l \\infty } \\Sigma ^ { l } = \\frac { B } { C } \\hat { \\mathbb { e } } I _ { C } \\hat { \\mathbb { e } } ^ { T } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 214, + 479, + 393, + 500 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "2. This convergence is exponential in the sense that, for any $G \\Sigma ^ { 0 } G$ of rank $C$ , there is $a$ constant $K < 1$ such that $\\lambda _ { 1 } ( \\Sigma ^ { l } ) - \\lambda _ { C } ( \\Sigma ^ { l } ) < K ( \\bar { \\lambda } _ { 1 } ( \\Sigma ^ { l \\bar { - } 1 } ) - \\lambda _ { C } ( \\bar { \\Sigma } ^ { l - 1 } ) )$ for all $l \\geq 2$ . Here $\\lambda _ { i }$ denotes the ith largest eigenvalue. ", + "bbox": [ + 212, + 507, + 825, + 551 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "3. Asymptotically, $\\begin{array} { r } { \\lambda _ { 1 } ( \\Sigma ^ { l } ) - \\lambda _ { C } ( \\Sigma ^ { l } ) = O ( ( 1 - \\frac { 2 } { C + 2 } ) ^ { l } ) . } \\end{array}$ ", + "bbox": [ + 212, + 558, + 583, + 577 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "General nonlinearity. We don’t (currently) have a proof of any characterization of the basin of attraction for Eq. (43) for general $\\phi$ . Thus we are forced to resort to finding its fixed points manually and characterize their local convergence properties. ", + "bbox": [ + 174, + 590, + 825, + 633 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "F.2 LIMIT POINTS ", + "text_level": 1, + "bbox": [ + 174, + 650, + 312, + 665 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Batchnorm $B _ { \\phi }$ is permutation-equivariant, in the sense that $\\boldsymbol { B } _ { \\phi } ( \\pi h ) = \\pi \\boldsymbol { B } _ { \\phi } ( h )$ for any permtuation matrix $\\pi$ . Along with the case of $\\phi = \\mathrm { i d }$ studied above, this suggests that we look into BSB1 fixed points $\\Sigma ^ { * }$ . What are the BSB1 fixed points of $E q$ . (43)? ", + "bbox": [ + 173, + 676, + 825, + 719 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "F.2.1 SPHERICAL INTEGRATION ", + "text_level": 1, + "bbox": [ + 174, + 733, + 410, + 748 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "The main result $\\begin{array} { r } { \\left( \\begin{array} { c c c c c c c } { \\rho } & { \\rho } & { \\rho } & { \\rho } & { \\rho } & { \\rho } \\\\ { \\vdots } & { \\vdots } & { \\vdots } & { \\vdots } & { \\vdots } & { \\vdots } \\\\ { \\rho } & { \\rho } & { \\rho } & { \\rho } & { \\rho } & { \\rho } \\end{array} \\right) } \\end{array}$ ) of this section is an expression of the BSB1 fixed point diagonal and offdiagonal entries in terms of 1- and 2-dimensional integrals. This allows one to numerically compute such fixed points. ", + "bbox": [ + 174, + 757, + 825, + 801 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "By a simple symmetry argument, we have the following ", + "bbox": [ + 174, + 806, + 544, + 821 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Lemma F.4. Suppose $X$ is a random vector in $\\mathbb { R } ^ { B }$ , symmetric in the sense that for any permutation matrix $\\pi$ and any subset $U$ of $\\mathbb { R } ^ { B }$ measurable with respect to the distribution of $X$ , $P ( X \\in U ) =$ $P ( X \\in \\pi ( U ) )$ . Let $\\Phi : \\mathbb { R } ^ { B } \\mathbb { R } ^ { B }$ be a symmetric function in the sense that $\\Phi ( \\pi x ) = \\pi \\Phi ( x )$ for any permutation matrix $\\pi$ . Then $\\mathbb { E } \\Phi ( X ) \\dot { \\Phi ( X ) } ^ { T } = \\dot { \\mu } I + \\nu \\mathbf { 1 } \\mathbf { 1 } ^ { T }$ for some $\\mu$ and $\\nu$ . ", + "bbox": [ + 173, + 825, + 825, + 883 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "This lemma implies that $\\mathrm { V } _ { B _ { \\phi } } ( \\Sigma )$ is BSB1 whenever $\\Sigma$ is BSB1. In the following, We in fact show that for any BSB1 $\\Sigma$ , $\\mathrm { V } _ { B _ { \\phi } } ( \\Sigma )$ is the same, and this is the unique BSB1 fixed point of Eq. (43). ", + "bbox": [ + 173, + 893, + 823, + 924 + ], + "page_idx": 49 + }, + { + "type": "text", + "text": "Theorem F.5. The BSB1 fixed point $\\Sigma ^ { * } = \\mu ^ { * } I + \\nu ^ { * } \\mathbf { 1 1 } ^ { T }$ to Eq. (43) is unique and satisfies ", + "bbox": [ + 168, + 102, + 772, + 119 + ], + "page_idx": 50 + }, + { + "type": "equation", + "img_path": "images/27bb059bef075c5c7ac257a290594b3e3a647a0250fceece6911eb37114ba8c5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mu ^ { * } + \\nu ^ { * } = \\frac { \\Gamma \\left( \\frac { B - 1 } { 2 } \\right) } { \\Gamma \\left( \\frac { B - 2 } { 2 } \\right) \\sqrt { \\pi } } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } \\sin ^ { B - 3 } \\theta _ { 1 } \\phi ( \\sqrt { B } \\zeta _ { 1 } ( \\theta _ { 1 } ) ) ^ { 2 } } \\\\ & { \\qquad \\nu ^ { * } = \\left\\{ \\begin{array} { l l } { \\frac { B - 3 } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 2 } \\sin ^ { B - 3 } \\theta _ { 1 } \\sin ^ { B - 4 } \\theta _ { 2 } \\phi ( \\sqrt { B } \\zeta _ { 1 } ( \\theta _ { 1 } ) ) \\phi ( \\sqrt { B } \\zeta _ { 2 } ( \\theta _ { 1 } , \\theta _ { 2 } ) ) , } & { i f B \\geq 4 } \\\\ { \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\mathrm { d } \\theta \\phi ( - \\sqrt { 2 } \\sin \\theta ) \\phi ( \\sin \\theta / \\sqrt { 2 } - \\cos \\theta \\sqrt { 3 } / \\sqrt { 2 } ) , } & { i f B = 3 } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 121, + 828, + 202 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 202, + 217, + 215 + ], + "page_idx": 50 + }, + { + "type": "equation", + "img_path": "images/4667a7c7df8109ee5cada2fab1f6339b63ac8ec537bec16232dcd8bf01135ad3.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\zeta _ { 1 } ( \\theta ) = - \\sqrt { \\frac { B - 1 } { B } } \\cos \\theta } \\\\ { \\displaystyle \\zeta _ { 2 } ( \\theta _ { 1 } , \\theta _ { 2 } ) = \\frac { 1 } { \\sqrt { B ( B - 1 ) } } \\cos \\theta _ { 1 } - \\sqrt { \\frac { B - 2 } { B - 1 } } \\sin \\theta _ { 1 } \\cos \\theta _ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 307, + 214, + 691, + 291 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Proof. For any BSB1 $\\Sigma$ , $\\mathbb { e } ^ { T } \\Sigma \\mathbb { e } = \\mu I _ { B - 1 }$ for some $\\mu$ . Thus we can apply Proposition E.33 to $G \\Sigma G$ We observe that ${ \\mathcal { K } } ( v ; G \\Sigma G ) = 1$ , so that ", + "bbox": [ + 174, + 304, + 821, + 333 + ], + "page_idx": 50 + }, + { + "type": "equation", + "img_path": "images/b678164bde2db029ae91a52173081820771ca0da7bfa3b6a5093e8aa4115c60b.jpg", + "text": "$$\n\\begin{array} { r } { \\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( \\Sigma ) = \\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( G \\Sigma G ) = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathbb { e } v ) ^ { \\otimes 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 336, + 334, + 658, + 359 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Note that this is now independent of the specific values of $\\Sigma$ . Now, using the specific form of $\\mathbb { P }$ given by Eq. (33), we see that $\\mathrm { V } _ { B _ { \\phi } } ( \\Sigma ) _ { 1 1 }$ only depends on $\\theta _ { 1 }$ and $\\mathrm { V } _ { B _ { \\phi } } ( \\Sigma ) _ { 1 2 }$ only depends on $\\theta _ { 1 }$ and $\\theta _ { 2 }$ . Then applying Proposition E.33 followed by Lemma E.36 and Lemma E.37 yield the desired result. □ ", + "bbox": [ + 173, + 362, + 825, + 417 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "F.2.2 LAPLACE METHOD ", + "text_level": 1, + "bbox": [ + 174, + 433, + 362, + 446 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "In the case that $\\phi$ is positive-homogeneous, we can apply Laplace’s method Appendix E.1 and obtain the BSB1 fixed point in a much nicer form. ", + "bbox": [ + 173, + 457, + 823, + 486 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Definition F.6. For any $\\alpha \\geq 0$ , define $\\begin{array} { r } { K _ { \\alpha , B } : = \\mathsf { c } _ { \\alpha } \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\alpha \\right) ^ { - 1 } \\left( \\frac { B - 1 } { 2 } \\right) ^ { \\alpha } } \\end{array}$ where $\\mathrm { P } \\left( a , b \\right) : = \\Gamma ( a +$ $b ) / \\Gamma ( a )$ is the Pochhammer symbol. ", + "bbox": [ + 173, + 488, + 823, + 522 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Note that ", + "bbox": [ + 173, + 531, + 238, + 545 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Proposition F.7. $\\begin{array} { r } { \\operatorname* { l i m } _ { B \\to \\infty } K _ { \\alpha , B } = \\mathsf { c } _ { \\alpha } } \\end{array}$ . ", + "bbox": [ + 174, + 547, + 436, + 564 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "We give a closed form description of the BSB1 fixed point for a positive homogeneous $\\phi$ in terms of its J function. ", + "bbox": [ + 174, + 570, + 825, + 601 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Theorem F.8. Suppose $\\phi : \\mathbb { R } \\mathbb { R }$ is degree $\\alpha$ positive-homogeneous. For any BSB1 $\\Sigma \\in \\Sigma$ $\\boldsymbol { S } _ { B }$ , $\\mathrm { V } _ { B _ { \\phi } } ( \\Sigma )$ is BSB1. The diagonal entries are $K _ { \\alpha , B } \\mathbf { J } _ { \\phi } ( 1 )$ and the off-diagonal entries are $\\begin{array} { r } { K _ { \\alpha , B } \\mathbf { J } _ { \\phi } \\left( \\frac { - 1 } { B - 1 } \\right) } \\end{array}$ . Here $\\mathsf { C } _ { \\alpha }$ is as defined in Defn E.7 and $\\mathrm { J } _ { \\phi }$ is as defined in Defn E.20. Thus a BSB1 fixed point of Eq. (43) exists and is unique. ", + "bbox": [ + 173, + 602, + 826, + 670 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "Proof. Let $\\mathbb { P }$ be an $B \\times ( B - 1 )$ matrix whose columns form an orthonormal basis of $\\mathrm { i m } G : = \\{ G v :$ $v \\in \\mathbb { R } ^ { B } \\} = \\{ w \\in \\mathbb { R } ^ { B } : \\sum _ { i } w _ { i } = 0 \\}$ . Let $\\mathrm { B S B 1 } ( a , b )$ denote a matrix with $a$ on the diagonal and $b$ on the off-diagnals. Note that $\\mathrm { V } _ { \\phi }$ is positive homogeneous of degree $\\alpha$ , so for any $\\mu$ , ", + "bbox": [ + 173, + 684, + 825, + 727 + ], + "page_idx": 50 + }, + { + "type": "equation", + "img_path": "images/54710c2433eeeea667e8b47ef6d5887d87ab3b8f95e66d12c764b02826c34504.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad { \\nabla _ { \\phi } } ( \\phi \\mu I _ { B - 1 } ( I _ { B - 1 } + 2 s \\mu I _ { B - 1 } ) ^ { - 1 } \\mathfrak { g } ^ { T } ) } \\\\ & { = { \\nabla _ { \\phi } } ( \\frac { \\mu } { 1 + 2 s \\mu } \\frac { \\beta \\mathfrak { g } ^ { T } } { B } ) } \\\\ & { = \\left( \\frac { \\mu } { 1 + 2 s \\mu } \\right) ^ { \\alpha } \\nabla _ { \\phi } ( \\mathfrak { g } \\mathfrak { g } ^ { T } ) } \\\\ & { = \\left( \\frac { \\mu } { 1 + 2 s \\mu } \\right) ^ { \\alpha } \\nabla _ { \\phi } ( G ) } \\\\ & { = \\left( \\frac { \\mu } { 1 + 2 s \\mu } \\right) ^ { \\alpha } \\nabla _ { \\phi } ( G ) } \\\\ & { = \\left( \\frac { \\mu } { 1 + 2 s \\mu } \\frac { B - 1 } { B } \\right) ^ { \\alpha } \\nabla _ { \\phi } \\left( \\mathtt { B S B } \\left( 1 , \\frac { - 1 } { B - 1 } \\right) \\right) } \\\\ & { = \\left( \\frac { \\mu } { 1 + 2 s \\mu } \\frac { B - 1 } { B } \\right) ^ { \\alpha } \\mathrm { c } _ { \\phi } \\mathrm { J } _ { \\phi } \\left( \\mathtt { B S B } \\left( 1 , \\frac { - 1 } { B - 1 } \\right) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 729, + 581, + 925 + ], + "page_idx": 50 + }, + { + "type": "text", + "text": "So by Eq. (26), ", + "bbox": [ + 173, + 103, + 276, + 119 + ], + "page_idx": 51 + }, + { + "type": "equation", + "img_path": "images/38813a0add4302ce3297e224a625e6d51a5c04a46ef7496dbdc9d1aba655626c.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\zeta _ { B _ { \\phi } } ( \\Phi \\mu I \\mathbb { P } ^ { T } ) = B ^ { \\alpha } \\Gamma ( \\alpha ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { \\alpha - 1 } ( 1 + 2 s \\mu ) ^ { - ( B - 1 ) / 2 } \\left( \\frac { \\mu } { 1 + 2 s \\mu } \\frac { B - 1 } { B } \\right) ^ { \\alpha } \\mathrm { c } _ { \\alpha } \\mathrm { J } _ { \\phi } \\left( \\mathrm { B S B } 1 \\left( 1 , \\frac { \\sqrt { \\alpha - 1 } } { B } \\right) \\right) ^ { \\alpha - 1 } \\mathrm { d } s } } \\\\ { { \\displaystyle \\qquad = \\Gamma ( \\alpha ) ^ { - 1 } ( B - 1 ) ^ { \\alpha } \\mathrm { c } _ { \\alpha } \\mathrm { J } _ { \\phi } \\left( \\mathrm { B S B } 1 \\left( 1 , \\frac { - 1 } { B - 1 } \\right) \\right) \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { \\alpha - 1 } ( 1 + 2 s \\mu ) ^ { - ( B - 1 ) / 2 - \\alpha } \\mu ^ { \\alpha } } } \\\\ { { \\displaystyle \\qquad = \\Gamma ( \\alpha ) ^ { - 1 } ( B - 1 ) ^ { \\alpha } \\mathrm { c } _ { \\alpha } \\mathrm { J } _ { \\phi } \\left( \\mathrm { B S B } 1 \\left( 1 , \\frac { - 1 } { B - 1 } \\right) \\right) \\mathrm { B e t a } ( \\alpha , ( B - 1 ) / 2 ) 2 ^ { - \\alpha } } } \\\\ { { \\displaystyle \\qquad = \\mathrm { c } _ { \\alpha } \\frac { \\Gamma ( ( B - 1 ) / 2 ) } { \\Gamma ( \\alpha + ( B - 1 ) / 2 ) } \\left( \\frac { B - 1 } { 2 } \\right) ^ { \\alpha } \\mathrm { J } _ { \\phi } \\left( \\mathrm { B S B } 1 \\left( 1 , \\frac { - 1 } { B - 1 } \\right) \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 127, + 839, + 272 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Corollary F.9. Suppose $\\phi : \\mathbb { R } \\mathbb { R }$ is degree α positive-homogeneous. If $\\Sigma ^ { * }$ is the BSB1 fixed point of Eq. (43) as given by Thm F.8, then $G \\Sigma ^ { * } G = \\mu ^ { * } I _ { B - 1 }$ where $\\mu ^ { * } = \\bar { K _ { \\alpha , B } } \\big ( \\mathrm { J } _ { \\phi } ( 1 ) - \\mathrm { J } _ { \\phi } \\big ( - 1 / \\bar { ( } B -$ 1))\u0001 ", + "bbox": [ + 173, + 309, + 826, + 356 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "By setting $\\alpha = 1$ and $\\phi =$ relu, we get ", + "bbox": [ + 174, + 368, + 429, + 383 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Corollary F.10. For any BSB1 $\\Sigma \\in S _ { B }$ , $\\mathrm { V } _ { B _ { \\mathrm { r e l u } } } ( \\Sigma )$ is BSB1 with diagonal entries $\\frac { 1 } { 2 }$ and off-diagonal entries $\\textstyle { \\frac { 1 } { 2 } } \\mathrm { J _ { 1 } } \\left( { \\frac { - 1 } { B - 1 } } \\right)$ , so that $\\begin{array} { r } { G ^ { \\otimes 2 } \\{ \\mathsf { V } _ { \\mathcal { B } _ { \\mathrm { r e l u } } } ( \\Sigma ) \\} = G ( \\mathsf { V } _ { \\mathcal { B } _ { \\mathrm { r e l u } } } ( \\Sigma ) ) G = \\left( \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\mathsf { J } _ { 1 } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) G . } \\end{array}$ ", + "bbox": [ + 173, + 387, + 826, + 429 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "By setting $\\alpha = 1$ and $\\phi = \\operatorname { i d } = x \\mapsto \\operatorname { r e l u } ( x ) - \\operatorname { r e l u } ( - x )$ , we get ", + "bbox": [ + 174, + 440, + 604, + 457 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Corollary F.11. For any BSB1 $\\Sigma \\in S _ { B }$ , $\\mathrm { V } _ { B _ { \\mathrm { i d } } } ( \\Sigma )$ is BSB1 with diagonal entries 1 and off-diagonal entries $\\frac { - 1 } { B - 1 }$ , so that $G ^ { \\otimes 2 } \\{ \\mathrm { V } _ { { \\cal B } _ { \\mathrm { i d } } } ( \\Sigma ) \\} = B / ( \\stackrel { } { B } - 1 ) G$ . ", + "bbox": [ + 176, + 462, + 823, + 493 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Remark F.12. One might hope to tweak the Laplace method for computing the fixed point to work for the Fourier method, but because there is no nice relation between $\\mathrm { V } _ { \\phi } ( c \\Sigma )$ and $\\mathrm { V } _ { \\phi } ( \\bar { \\Sigma } )$ in general, we cannot simplify Eq. (32) as we can Eq. (26) and Eq. (27). ", + "bbox": [ + 173, + 498, + 826, + 542 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "F.2.3 GEGENBAUER EXPANSION ", + "text_level": 1, + "bbox": [ + 174, + 559, + 413, + 574 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "It turns out that the BSB1 fixed point can be described very cleanly using the Gegenbauer coefficients of $\\phi$ . ", + "bbox": [ + 173, + 584, + 825, + 614 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "When $\\Sigma$ is BSB1, $\\mathcal { K } ( v , \\Sigma ) = 1$ for all $v \\in S ^ { B - 2 }$ , so that by Proposition E.34, ", + "bbox": [ + 173, + 618, + 686, + 636 + ], + "page_idx": 51 + }, + { + "type": "equation", + "img_path": "images/2d384116786658b6a89016ad5c510d262a15ae33b673cefb1c07bb1d20bd6783.jpg", + "text": "$$\n\\begin{array} { r l } & { \\Sigma ^ { * } = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathbb { e } v ) ^ { \\otimes 2 } } \\\\ & { \\Sigma _ { a b } ^ { * } = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathbb { e } _ { a , : } v ) \\phi ( \\sqrt { B } \\mathbb { e } _ { b , : } v ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 366, + 643, + 633, + 700 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "for any $a , b \\in [ B ]$ , independent of the actual values of $\\Sigma$ . Here $\\Phi _ { a , \\mathbf { \\lambda } }$ : is the $a$ th row of $\\mathbb { P }$ . ", + "bbox": [ + 173, + 708, + 740, + 724 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Theorem F.13. If $\\phi ( { \\sqrt { B - 1 } } x ) \\in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \\frac { B - 3 } { 2 } } )$ has Gegenbauer expansion $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ , then $\\Sigma ^ { * } = \\mu ^ { * } I + \\nu ^ { * } \\mathbf { 1 1 } ^ { T }$ with ", + "bbox": [ + 173, + 729, + 825, + 771 + ], + "page_idx": 51 + }, + { + "type": "equation", + "img_path": "images/c890e5774ced7165c625b41ed79ec6d67048a55ee28369fe9cd367adfd374d9a.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\mu ^ { * } = \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } } \\\\ { { \\displaystyle \\quad = \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } \\left( \\binom { B - 4 + l } { l } - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } } \\\\ { { \\displaystyle \\nu ^ { * } = \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 302, + 781, + 696, + 909 + ], + "page_idx": 51 + }, + { + "type": "text", + "text": "Proof. By the reproducing property Fact E.46, we have ", + "bbox": [ + 173, + 103, + 539, + 119 + ], + "page_idx": 52 + }, + { + "type": "equation", + "img_path": "images/5158098bde92b1859896e440afaa05a5bca0adb7d89781cec2d7f658a109122b.jpg", + "text": "$$\n\\begin{array} { r l } { \\Phi _ { \\alpha \\beta } ^ { \\alpha } = } & { \\langle \\phi ( \\sqrt { D } w _ { \\alpha \\beta } ; 0 ) \\phi ( \\sqrt { D } w _ { \\alpha \\beta } ) \\rangle _ { \\beta \\alpha \\beta } , } \\\\ & { = \\langle \\phi ( \\sqrt { D } w - 1 ) \\phi _ { \\alpha \\beta } , 1 \\phi _ { \\alpha \\beta } \\rangle _ { \\beta \\alpha } , } \\\\ & { = \\langle \\phi ( \\sqrt { D } w _ { \\alpha \\beta } - 1 ) c _ { \\alpha \\beta } , 1 \\phi _ { \\alpha \\beta } \\rangle _ { \\beta \\alpha } , } \\\\ & { = \\langle \\displaystyle \\sum _ { j = 0 } ^ { \\infty } a _ { j } \\frac { 1 } { c _ { \\alpha \\beta - 1 , j } } c _ { j } ^ { ( \\alpha \\beta \\alpha ) } ( \\phi _ { \\alpha \\beta } ) , \\displaystyle \\sum _ { j = 0 } ^ { \\infty } a _ { j } \\frac { 1 } { c _ { \\alpha \\beta - 1 , j } } c _ { j } ^ { ( \\alpha \\beta \\alpha , 3 ) } ( \\phi _ { \\beta \\beta } , + ) \\rangle _ { \\alpha \\beta } , } \\\\ & { = \\langle \\displaystyle \\sum _ { j = 0 } ^ { \\infty } a _ { j } Z _ { \\alpha \\alpha \\beta - 1 , j } ^ { \\beta \\alpha } , \\displaystyle \\sum _ { j = 0 } ^ { \\infty } a _ { j } Z _ { \\alpha \\beta - 2 , j } ^ { \\beta \\alpha } \\rangle _ { j \\alpha \\beta } \\rangle _ { j \\alpha \\beta - 1 } \\rangle _ { j \\alpha \\beta - 2 } \\rangle } \\\\ & { = \\langle \\displaystyle \\sum _ { j = 0 } ^ { \\infty } a _ { j } Z _ { \\alpha \\beta - 2 , j } ^ { \\beta \\alpha \\beta - 2 \\beta } , \\qquad \\mathrm { f o r ~ a l l ~ } \\qquad \\mathrm { ~ a l l ~ } \\qquad \\mathrm { ~ f o r ~ a l l ~ } \\qquad \\mathrm { ~ a l l ~ } \\qquad \\mathrm { ~ f o r ~ a ~ \\beta ~ } } \\\\ & { = \\displaystyle \\sum _ { j = 0 } ^ { \\infty } a _ { j } ^ { 2 } Z _ { \\alpha \\beta - 2 } ^ { \\beta \\alpha - 2 \\beta } ( \\phi _ { \\alpha \\beta } ) , } \\\\ & { = \\displaystyle \\sum _ { j = 0 } ^ { \\infty } a _ { j } ^ { 2 } \\frac { 1 } { c _ { \\alpha \\beta - 1 , j } } c _ { j } ^ { ( \\alpha \\beta \\alpha ) } ( \\phi _ { \\alpha \\beta } , \\phi _ { \\beta \\beta } ) , } \\\\ & = \\displaystyle \\sum _ { j = 0 } ^ { \\infty } a _ { j } ^ { 2 } \\frac { 1 } { c _ { \\alpha \\beta - 1 , j } } c _ { j } ^ { ( \\alpha \\beta \\alpha ) } ( \\ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 261, + 125, + 735, + 381 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "This gives the third equation. Since $\\mu ^ { * } = \\Sigma _ { 1 1 } ^ { * } - \\Sigma _ { 1 2 } ^ { * }$ , straightforward arithmetic yields the first equation. The second follows from Fact E.41. ", + "bbox": [ + 173, + 383, + 823, + 412 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "We will see later that the rate of gradient explosion also decomposes nicely in terms of Gegenbauer coefficients $\\mathrm { \\ T h m ~ G } . 5 )$ . However, this is not quite true for the eigenvalues of the forward convergence (Thms F.22 and F.24). ", + "bbox": [ + 174, + 428, + 823, + 472 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "F.3 LOCAL CONVERGENCE ", + "text_level": 1, + "bbox": [ + 174, + 487, + 377, + 502 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "In this section we consider linearization of the dynamics given in Eq. (43). Thus we must consider linear operators on the space of PSD linear operators $\\boldsymbol { S } _ { B }$ . To avoid confusion, we use the following notation: If $\\tau : S _ { B } S _ { B }$ (for example the Jacobian of $\\mathrm { V } _ { B _ { \\phi } }$ ) and $\\Sigma \\in S _ { B }$ , then write $\\mathcal { T } \\{ \\Sigma \\}$ for the image of $\\Sigma$ under $\\tau$ . ", + "bbox": [ + 174, + 512, + 825, + 569 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "A priori, the Jacobian of $\\mathrm { V } _ { B _ { \\phi } }$ at its BSB1 fixed point may seem like a very daunting object. But a moment of thought shows that ", + "bbox": [ + 173, + 575, + 823, + 604 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Proposition F.14. The Jacobian $\\frac { \\mathrm { d } \\mathrm { V } _ { B _ { \\phi } } } { \\mathrm { d } \\Sigma } \\big | _ { \\Sigma } : \\mathcal { H } _ { B } \\mathcal { H } _ { B }$ is ultrasymmetric for any BSB1 $\\Sigma$ ", + "bbox": [ + 171, + 608, + 761, + 632 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Now we prepare several helper reults in order to make progress understanding batchnorm. ", + "bbox": [ + 171, + 638, + 763, + 655 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Definition F.15. Define $\\mathsf { n } ( x ) = \\sqrt { B } x / \\| x \\|$ , i.e. division by sample standard deviation. ", + "bbox": [ + 173, + 659, + 743, + 676 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Batchnorm $B _ { \\phi }$ can be decomposed as the composition of three steps, $\\phi \\circ { \\mathsf { n } } \\circ G$ , where $G$ is meancentering, n is division by standard deviation, and $\\phi$ is coordinate-wise application of nonlinearity. We have, as operators $\\mathcal { H } \\to \\mathcal { H }$ , ", + "bbox": [ + 174, + 685, + 823, + 728 + ], + "page_idx": 52 + }, + { + "type": "equation", + "img_path": "images/8feafcf751e9fcceda24fc7316756581a99891dcdaec23065e5775f83e6c08ef.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { \\mathrm { d } \\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( \\Sigma ) } { \\mathrm { d } \\Sigma } = \\frac { \\mathrm { d } \\mathbb { E } [ \\mathcal { B } _ { \\phi } ( z ) ^ { \\otimes 2 } : z \\in \\mathcal { N } ( 0 , \\Sigma ) ] } { \\mathrm { d } \\Sigma } } \\\\ & { \\quad \\quad \\quad = \\frac { \\mathrm { d } \\mathbb { E } [ ( \\phi \\circ \\mathfrak { n } ) ( x ) ^ { \\otimes 2 } : x \\in \\mathcal { N } ( 0 , G \\Sigma G ) ] } { \\mathrm { d } \\Sigma } } \\\\ & { \\quad \\quad \\quad = \\frac { \\mathrm { d } \\mathrm { V } _ { ( \\phi \\circ \\mathfrak { n } ) } ( \\Sigma ^ { G } ) } { \\mathrm { d } \\Sigma ^ { G } } \\circ \\frac { \\mathrm { d } \\Sigma ^ { G } } { \\mathrm { d } \\Sigma } } \\\\ & { \\quad \\quad \\quad = \\frac { \\mathrm { d } \\mathrm { V } _ { ( \\phi \\circ \\mathfrak { n } ) } ( \\Sigma ^ { G } ) } { \\mathrm { d } \\Sigma ^ { G } } \\circ G ^ { \\otimes 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 330, + 732, + 666, + 868 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "Definition F.16. With $\\Sigma ^ { * }$ being the unique BSB1 fixed point, write $\\begin{array} { r l r } { \\mathcal { U } } & { { } : = } & { G ^ { \\otimes 2 } } \\end{array}$ ◦ $\\frac { \\mathrm { d } \\mathrm { V } _ { ( \\phi \\circ \\mathfrak { n } ) } ( \\Sigma ^ { G } ) } { \\mathrm { d } \\Sigma ^ { G } } \\Big | _ { \\Sigma ^ { G } = G \\Sigma ^ { * } G } : \\mathcal { H } ^ { G } \\to \\mathcal { H } ^ { G }$ ", + "bbox": [ + 173, + 888, + 823, + 928 + ], + "page_idx": 52 + }, + { + "type": "text", + "text": "It turns out to be advantageous to study $\\mathcal { U }$ first and relate its eigendecomposition back to that of $\\frac { \\mathrm { d } \\mathrm { V } _ { B _ { \\phi } } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\big | _ { \\Sigma = \\Sigma ^ { * } }$ ) \f\fΣ=Σ∗ , where Σ∗ is the BSB1 fixed point, by applying Lemma F.17. ", + "bbox": [ + 174, + 102, + 825, + 140 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "Lemma F.17. Let $X$ and $Y$ be two vector spaces. Consider linear operators $A : X \\to Y , B : Y \\to$ $X$ . Then ", + "bbox": [ + 173, + 147, + 823, + 178 + ], + "page_idx": 53 + }, + { + "type": "equation", + "img_path": "images/59bbbebb4d219a1d1a9df3190ec29d8d195379cfe413d9ef54892885a27e1b6f.jpg", + "text": "$$\nI . { \\mathrm { ~ \\ r a n k { \\ } } } A B = { \\mathrm { r a n k } } B A\n$$", + "text_format": "latex", + "bbox": [ + 212, + 195, + 379, + 210 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "2. If $v \\in Y$ is an eigenvector of $A B$ with nonzero eigenvalue, then $X \\ni B v \\neq 0$ and $B v$ is an eigenvector of $B A$ of the same eigenvalue ", + "bbox": [ + 205, + 226, + 825, + 256 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "3. Suppose $A B$ has $k = { \\mathrm { r a n k } } A B$ linearly independent eigenvectors $\\{ v _ { i } \\} _ { i = 1 } ^ { k }$ of nonzero eigenvalues $\\{ \\lambda _ { i } \\} _ { i = 1 } ^ { k }$ . Then $B A$ has $k$ linearly independent eigenvectors $\\{ B v _ { i } \\} _ { i = 1 } ^ { k }$ with the same eigenvalues $\\{ \\lambda _ { i } \\} _ { i = 1 } ^ { k }$ which are all eigenvectors of $B A$ with nonzero eigenvalues, up to linear combinations within eigenvectors with the same eigenvalue. ", + "bbox": [ + 210, + 270, + 825, + 332 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "With $A = G ^ { \\otimes 2 }$ and $\\begin{array} { r } { B = \\frac { \\mathrm { d } \\mathrm { V } _ { ( \\phi \\circ \\mathsf { n } ) } \\left( \\Sigma ^ { G } \\right) } { \\mathrm { d } \\Sigma ^ { G } } } \\end{array}$ , $\\mathrm { T h m } \\mathrm { E } . 6 1 $ implies that $A B$ and $B A$ can both be diagonalized, and this lemma implies that all nonzero eigenvalues of $\\frac { \\mathrm { d } \\mathrm { V } _ { B _ { \\phi } } ( \\Sigma ) } { \\mathrm { d } \\Sigma }$ can be recovered from those of $\\mathcal { U }$ . ", + "bbox": [ + 176, + 348, + 826, + 392 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "Proof. (Item 1) Observe rank $A B = \\operatorname { r a n k } A B A B \\leq \\operatorname { r a n k } B A$ . By symmetry the two sides are in fact equal. ", + "bbox": [ + 174, + 449, + 825, + 479 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "(Item 2) $B v$ cannot be zero or otherwise $A B v \\ = \\ A 0 \\ = \\ 0$ , contradicting the fact that $v$ is an eigenvector with nonzero eigenvalue. Suppose $\\lambda$ is the eigenvalue associated to $v$ . Then $B A ( B v ) =$ $\\bar { B ( A B v ) } = B ( \\lambda v ) = \\lambda B v$ , so $B v$ is an eigenvector of $B A$ with the same eigenvalue. ", + "bbox": [ + 173, + 484, + 825, + 529 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "(Item 3) Item 2 shows that $\\{ B v _ { i } \\} _ { i = 1 } ^ { k }$ are eigenvectors $B A$ with the same eigenvalues $\\{ \\lambda _ { i } \\} _ { i = 1 } ^ { k }$ . The eigenspaces with different eigenvalues are linearly independent, so it suffices to show that if $\\{ B v _ { i _ { j } } \\} _ { j }$ are eigenvectors of the same eigenvalue $\\lambda _ { s }$ , then they are linearly independent. But $\\textstyle \\sum _ { j } a _ { j } B v _ { i _ { j } } =$ $\\begin{array} { r } { \\mathrm { ~ 0 ~ \\implies ~ } \\sum _ { j } a _ { j } v _ { i _ { j } } = 0 } \\end{array}$ because $B$ is injective on eigenvectors by Item 2, so that $a _ { j } = 0$ identically. Hence $\\{ B v _ { j } \\} _ { j }$ is linearly independent. ", + "bbox": [ + 173, + 534, + 825, + 611 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "Since rank $B A = k$ , these are all of the eigenvectors with nonzero eigenvalues of $B A$ up to linearly combinations. ", + "bbox": [ + 173, + 616, + 823, + 646 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "Lemma F.18. Let $f : \\mathbb { R } ^ { B } \\to \\mathbb { R } ^ { A }$ be measurable, and $\\Sigma \\in { \\cal S } _ { B }$ be invertible. Then for any $\\Lambda \\ \\in$ $\\mathbb { R } ^ { B \\times B }$ , with $\\langle \\cdot , \\cdot \\rangle$ denoting trace inner product, ", + "bbox": [ + 171, + 718, + 821, + 750 + ], + "page_idx": 53 + }, + { + "type": "equation", + "img_path": "images/136abb01353151bf4d59c02a1df160dacf194c588bc92a1eceba50fbf3b098c6.jpg", + "text": "$$\n{ \\frac { \\mathrm { d } } { \\mathrm { d } \\Sigma } } \\mathbb { E } [ f ( z ) : z \\sim { \\mathcal { N } } ( 0 , \\Sigma ) ] \\{ \\Lambda \\} = { \\frac { 1 } { 2 } } \\mathbb { E } [ f ( z ) \\left. \\Sigma ^ { - 1 } z z ^ { T } \\Sigma ^ { - 1 } - \\Sigma ^ { - 1 } , \\Lambda \\right. : z \\sim { \\mathcal { N } } ( 0 , \\Sigma ) ]\n$$", + "text_format": "latex", + "bbox": [ + 215, + 770, + 782, + 799 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "If $f$ is in addition twice-differentiable, then ", + "bbox": [ + 173, + 819, + 459, + 834 + ], + "page_idx": 53 + }, + { + "type": "equation", + "img_path": "images/38f09e548a2625c67083bb2b0596b572d187d728e66e7911fcc29a50a3b4e759.jpg", + "text": "$$\n{ \\frac { \\mathrm { d } } { \\mathrm { d } \\Sigma } } \\mathbb { E } [ f ( z ) : z \\sim { \\mathcal { N } } ( 0 , \\Sigma ) ] = { \\frac { 1 } { 2 } } \\mathbb { E } \\left[ { \\frac { \\mathrm { d } ^ { 2 } f ( z ) } { \\mathrm { d } z ^ { 2 } } } : z \\sim { \\mathcal { N } } ( 0 , \\Sigma ) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 299, + 854, + 696, + 888 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "whenever both sides exist. ", + "bbox": [ + 173, + 909, + 346, + 924 + ], + "page_idx": 53 + }, + { + "type": "text", + "text": "Proof. Let $\\Sigma _ { t } , t \\in ( - \\epsilon , \\epsilon )$ be a smooth path in $\\boldsymbol { S } _ { B }$ , with $\\Sigma _ { 0 } = \\Sigma$ . Write $\\begin{array} { r } { \\frac { d } { d t } \\Sigma _ { t } = \\dot { \\Sigma } _ { t } } \\end{array}$ . Then ", + "bbox": [ + 169, + 101, + 764, + 121 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/893c7b5c78406bf9ce8b6f2b08a24bd2ca4c9c23c951b49650671236fc99ccb5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { d } { d t } \\mathbb { E } [ f ( z ) : z \\sim \\mathcal { N } ( 0 , \\Sigma _ { \\varepsilon } ) ] } \\\\ & { = \\frac { d } { d t } ( 2 \\pi ) ^ { - \\delta / 2 } \\mathrm { d } \\mathrm { e t } \\Sigma _ { \\varepsilon } ^ { - 1 / 2 } \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma _ { \\varepsilon } \\cdot 1 } { _ 1 } ^ { t } f ( z ) \\Big | _ { \\mathrm { L } = 0 } } \\\\ & { = ( 2 \\pi ) ^ { - \\delta / 2 } \\frac { 1 } { 2 } \\mathrm { d } \\mathrm { e t } \\Sigma _ { \\varepsilon } ^ { - 1 / 2 } \\mathrm { e t } \\left( \\sum _ { 0 } ^ { - 1 } \\bar { \\Sigma } _ { 0 } ^ { T } \\Sigma _ { 0 } ^ { - 1 } \\right) \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma _ { 0 } ^ { - 1 } z ^ { T } \\int \\bar { f } ( z ) } } \\\\ & { \\qquad + ( 2 \\pi ) ^ { - \\delta / 2 } \\mathrm { d } \\mathrm { e t } \\Sigma _ { 0 } ^ { - 1 / 2 } \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma _ { 0 } ^ { T } \\Sigma _ { 0 } ^ { - 1 } \\cdot 2 } \\frac { 1 } { 2 } z ^ { T } ( - \\Sigma _ { 0 } ^ { - 1 } \\Sigma _ { 0 } ^ { - 1 } ) z ( z ) ^ { - 1 } \\mathrm { e / \\Sigma } } \\\\ & { = \\frac { - 1 } { 2 } ( 2 \\pi ) ^ { - \\delta / 2 } \\mathrm { d } \\mathrm { e t } \\Sigma _ { 0 } ^ { - 1 / 2 } \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma _ { 0 } ^ { - 1 } z } f ( z ) ( ( \\mathrm { r } \\Sigma _ { 0 } ^ { - 1 } \\Sigma _ { 0 } - z ^ { T } \\Sigma _ { 0 } ^ { - 1 } \\Sigma _ { 0 } ^ { - 1 } \\Sigma _ { 0 } ^ { - 1 } \\Sigma _ { 0 } ^ { - 1 } \\Sigma _ { 0 } ^ { - 1 } ) } \\\\ & = \\frac { 1 } { 2 } ( 2 \\pi ) ^ { - \\delta / 2 } \\mathrm { d } \\mathrm { e t } \\Sigma _ { 0 } ^ { - 1 / 2 } \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma _ { 0 } ^ { - 1 } z } f ( z ) ( \\Sigma ^ { - 1 } z ^ { T } \\Sigma ^ { - 1 } - \\Sigma ^ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 136, + 764, + 368 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "Note that ", + "bbox": [ + 173, + 381, + 238, + 395 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/6bd855d0a34ae8c926865f13638e705ed5708abe1f6127755e30afd95f128e73.jpg", + "text": "$$\n\\begin{array} { c } { { v ^ { T } \\displaystyle \\frac { \\mathrm { d } } { \\mathrm { d } z } e ^ { \\frac { - 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } = - v ^ { T } \\Sigma ^ { - 1 } z e ^ { \\frac { - 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } } } \\\\ { { w ^ { T } \\left( \\displaystyle \\frac { \\mathrm { d } ^ { 2 } } { \\mathrm { d } z ^ { 2 } } e ^ { \\frac { - 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\right) v = \\left( w ^ { T } \\Sigma ^ { - 1 } z z ^ { T } \\Sigma ^ { - 1 } v - w ^ { T } \\Sigma ^ { - 1 } v \\right) e ^ { \\frac { - 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } } } \\\\ { { { } } } \\\\ { { { } = \\left. \\Sigma ^ { - 1 } z z ^ { T } \\Sigma ^ { - 1 } - \\Sigma ^ { - 1 } , v w ^ { T } \\right. e ^ { \\frac { - 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 410, + 736, + 501 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "so that as a cotensor, ", + "bbox": [ + 173, + 515, + 310, + 529 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/fa5bbe731fdff6c3c49f90c3ff8222c46da4294bc39cc2e711d566e1cd74b0c9.jpg", + "text": "$$\n{ \\frac { \\mathrm { d } ^ { 2 } } { \\mathrm { d } z ^ { 2 } } } e ^ { { \\frac { - 1 } { 2 } } z ^ { T } \\Sigma ^ { - 1 } z } \\{ \\Lambda \\} = \\left. \\Sigma ^ { - 1 } z z ^ { T } \\Sigma ^ { - 1 } - \\Sigma ^ { - 1 } , \\Lambda \\right. e ^ { { \\frac { - 1 } { 2 } } z ^ { T } \\Sigma ^ { - 1 } z }\n$$", + "text_format": "latex", + "bbox": [ + 299, + 545, + 699, + 575 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "for any Λ RB×B . ", + "bbox": [ + 173, + 592, + 305, + 608 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "Therefore, ", + "bbox": [ + 173, + 616, + 245, + 631 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/07fbfddc9076bd9c9f6e25bb0f7833818d170e52906d17d3f8b9e7bd08ed58a9.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { \\frac { \\displaystyle \\mathrm { d } \\mathbb { E } [ f ( z ) : z \\sim { \\mathcal { N } ( 0 , \\Sigma ) } ] } { \\mathrm { d } \\Sigma } \\{ \\Lambda \\} = \\frac { 1 } { 2 } ( 2 \\pi ) ^ { - B / 2 } \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } \\int \\mathrm { d } z f ( z ) \\frac { \\displaystyle \\mathrm { d } ^ { 2 } e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } } { \\mathrm { d } ^ { 2 } z } \\{ \\Lambda \\} } \\\\ & { } & { \\quad \\quad \\quad \\quad = \\frac { 1 } { 2 } ( 2 \\pi ) ^ { - B / 2 } \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } \\int \\mathrm { d } z e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\frac { \\displaystyle \\mathrm { d } ^ { 2 } f ( z ) } { \\displaystyle \\mathrm { d } ^ { 2 } z } \\{ \\Lambda \\} } \\\\ & { } & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad ( \\mathrm { b y ~ i n t e g r a t i o n ~ b y ~ p a r t s } ) } \\\\ & { } & { \\quad \\quad \\quad \\quad = \\frac { 1 } { 2 } \\mathbb { E } \\left[ \\frac { \\mathrm { d } ^ { 2 } f ( z ) } { \\displaystyle \\mathrm { d } ^ { 2 } z } : z \\sim \\mathcal { N } ( 0 , \\Sigma ) \\right] \\{ \\Lambda \\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 646, + 784, + 773 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "Note that for any $\\Lambda \\in \\mathcal { H } _ { B }$ with $\\| \\boldsymbol { \\Lambda } \\| _ { o p } < \\mu ^ { * }$ , ", + "bbox": [ + 174, + 849, + 470, + 867 + ], + "page_idx": 54 + }, + { + "type": "equation", + "img_path": "images/74d2f72e1596a85466307a68f58a70cbdf8b5b19f734ced60695a2cc31b624b1.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } [ \\phi \\circ \\mathsf { n } ( z ) ^ { \\otimes 2 } : z \\in \\mathcal { N } ( 0 , \\mu ^ { * } G + \\Lambda ) ] = \\mathbb { E } [ \\phi \\circ \\mathsf { n } ( G z ) ^ { \\otimes 2 } : z \\in \\mathcal { N } ( 0 , \\mu ^ { * } I + \\Lambda ) ] } \\\\ & { \\qquad = \\mathbb { E } [ \\mathcal { B } _ { \\phi } ( z ) ^ { \\otimes 2 } : z \\in \\mathcal { N } ( 0 , \\mu ^ { * } I + \\Lambda ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 241, + 882, + 754, + 924 + ], + "page_idx": 54 + }, + { + "type": "text", + "text": "so that we have, for any $\\Lambda \\in \\mathcal { H } _ { B }$ , ", + "bbox": [ + 173, + 103, + 397, + 118 + ], + "page_idx": 55 + }, + { + "type": "equation", + "img_path": "images/16b6ff7e832db8ed00d518c81a984a474f912c58ed4326b6c28b3b96a1c69c29.jpg", + "text": "$$\n\\begin{array} { r l } { u \\{ \\Delta \\} } & { = \\frac { \\mathrm { d } \\mathrm { V } _ { \\mathrm { S } } \\alpha ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\Big | _ { \\Sigma = \\mu ^ { * } } \\{ \\Delta \\} } \\\\ & { \\phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\\\ & { = \\frac { 1 } { 2 } \\mathbb { E } [ B _ { \\delta } ( z ) ^ { \\otimes 2 } \\langle a ^ { \\mu - 1 } z z ^ { T } - \\mu ^ { \\mu - 1 } I , \\Delta \\rangle : z \\sim \\mathcal { N } ( 0 , \\mu ^ { * } I ) ] } \\\\ & { \\phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\\\ & { \\phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\\\ & { = \\frac { 1 } { 2 } \\mu ^ { \\mu _ { \\mathrm { S } } \\alpha - 2 } \\mathbb { E } [ B _ { \\delta } ( z ) ^ { \\otimes 2 } \\langle z z ^ { T } , \\Delta \\rangle : z \\sim \\mathcal { N } ( 0 , \\mu ^ { * } I ) ] - \\frac { 1 } { 2 } \\mu ^ { \\mu _ { \\mathrm { S } } - 1 } \\Sigma ^ { * } \\langle I , \\Delta \\rangle } \\\\ & { \\phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\\\ & { \\phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\\\ & { = ( 2 \\mu ^ { * } ) ^ { - 1 } \\left( \\mathbb { E } [ B _ { \\delta } ( z ) ^ { \\otimes 2 } \\langle z z ^ { T } G , \\Delta \\rangle : z \\sim \\mathcal { N } ( 0 , I ) ] - \\mathbb { E } ^ { * } \\langle I , \\Delta \\rangle \\right) } \\\\ & { \\phantom { a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 251, + 123, + 743, + 332 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Let’s extend to all matrices by this formula: ", + "bbox": [ + 173, + 352, + 460, + 367 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Definition F.19. Define ", + "bbox": [ + 174, + 371, + 334, + 386 + ], + "page_idx": 55 + }, + { + "type": "equation", + "img_path": "images/5781f56b647d564c5043682726133ac7fa6381cd995f6cc8a90aa3911b40e071.jpg", + "text": "$$\n\\begin{array} { r l } & { \\tilde { \\mathcal { U } } : \\mathbb { R } ^ { B \\times B } S _ { B } , } \\\\ & { \\qquad \\Lambda \\mapsto ( 2 \\mu ^ { * } ) ^ { - 1 } ( \\mathbb { E } [ \\mathcal { B } _ { \\phi } ( z ) ^ { \\otimes 2 } \\langle G z z ^ { T } G , \\Lambda \\rangle : z \\sim \\mathcal { N } ( 0 , I ) ] - \\Sigma ^ { * } \\langle I , \\Lambda \\rangle ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 240, + 388, + 751, + 431 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "F.3.1 SPHERICAL INTEGRATION ", + "text_level": 1, + "bbox": [ + 174, + 462, + 410, + 477 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "$\\mathbf { S o } \\tilde { \\mathcal { U } } \\mid \\mathcal { H } _ { B } = \\mathcal { U } \\mid \\mathcal { H } _ { B }$ . Ultimately we will apply $\\cdot$ to $G ^ { \\otimes 2 } \\circ \\tilde { \\mathcal { U } } \\mid \\mathcal { H } _ { B } = G ^ { \\otimes 2 } \\circ \\mathcal { U }$ Definition F.20. Write $\\begin{array} { r } { \\tau _ { i j } = \\frac { 1 } { 2 } \\left( \\delta _ { i } \\delta _ { j } ^ { T } + \\delta _ { j } \\delta _ { i } ^ { T } \\right) } \\end{array}$ . ", + "bbox": [ + 174, + 484, + 764, + 523 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Then ", + "bbox": [ + 173, + 532, + 210, + 546 + ], + "page_idx": 55 + }, + { + "type": "equation", + "img_path": "images/71087193cd1b7365905d08dc6364b432e518779e34985f0d441494fec18e3f7d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\tilde { \\mathcal { U } } \\{ \\tau _ { i j } \\} _ { k l } = ( 2 \\mu ^ { * } ) ^ { - 1 } \\left( \\mathbb { E } [ \\mathcal { B } _ { \\phi } ( z ) _ { k } \\mathcal { B } _ { \\phi } ( z ) _ { l } ( G z ) _ { i } ( G z ) _ { j } : z \\sim \\mathcal { N } ( 0 , I ) ] - \\Sigma _ { k l } ^ { * } \\mathbb { I } ( i = j ) \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\mathrm { f o r d e d } \\ M [ i = j ] , } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 192, + 550, + 779, + 651 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Here we will realize $\\mathbb { P }$ as the matrix in Eq. (33). ", + "bbox": [ + 176, + 664, + 486, + 679 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Definition F.21. Define $W _ { i j | k l } : = \\mathbb { E } [ \\phi ( \\mathsf { n } ( \\mathtt { e } x ) ) _ { k } \\phi ( \\mathsf { n } ( \\mathtt { e } x ) ) _ { l } ( \\mathtt { e } x ) _ { i } ( \\mathtt { e } x ) _ { j } : x \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] .$ ", + "bbox": [ + 169, + 681, + 764, + 700 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "Then $\\tilde { \\mathcal { U } } \\{ \\tau _ { i j } \\} _ { k l } = ( 2 \\mu ^ { * } ) ^ { - 1 } ( W _ { i j | k l } - \\Sigma _ { k l } ^ { * } \\mathbb { I } ( i = j ) )$ . If we can evaluate $W _ { i j | k l }$ then we can use Thm E.61 to compute the eigenvalues of $G ^ { \\otimes 2 } \\circ u$ . It’s easy to see that $W _ { i j | k l }$ is ultrasymmetric Thus WLOG we can take $i , j , k , l$ from $\\{ 1 , 2 , 3 , 4 \\}$ . ", + "bbox": [ + 173, + 710, + 825, + 760 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "By Lemma E.35, and the fact that $x \\mapsto \\Phi x$ is an isometry, ", + "bbox": [ + 173, + 765, + 552, + 780 + ], + "page_idx": 55 + }, + { + "type": "equation", + "img_path": "images/9326e02668a25e27973dc5f15a9cfdf59e00d57428e97fa472cd91c2d2f17723.jpg", + "text": "$$\n\\begin{array} { l } { { W _ { i j | k l } = \\mathbb E [ r ^ { 2 } \\phi ( \\sqrt { B } \\mathfrak e v ) _ { k } \\phi ( \\sqrt { B } \\mathfrak e v ) _ { l } ( \\mathfrak e v ) _ { i } ( \\mathfrak e v ) _ { j } : x \\sim \\mathcal N ( 0 , I _ { B - 1 } ) ] } } \\\\ { { \\ = ( B - 5 ) ( B - 3 ) ( B - 1 ) ( 2 \\pi ) ^ { - 2 } \\times } } \\\\ { { \\displaystyle \\qquad \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } \\cdot \\cdot \\cdot \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 4 } \\phi ( \\sqrt { B } \\mathfrak e v ) _ { k } \\phi ( \\sqrt { B } \\mathfrak e v ) _ { l } ( \\mathfrak e v ) _ { i } ( \\mathfrak e v ) _ { j } \\sin ^ { B - 3 } \\theta _ { 1 } \\cdot \\cdot \\cdot \\sin ^ { B - 6 } \\theta _ { 4 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 199, + 785, + 800, + 863 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "If WLOG we further assume that $k , l \\in \\{ 1 , 2 \\}$ (by ultrasymmetry), then there is no dependence on $\\theta _ { 3 }$ and $\\theta _ { 4 }$ inside $\\phi$ . So we can expand $( \\mathbb { e } v ) _ { i }$ and $( \\mathbb { e } v ) _ { j }$ in trigonometric expressions as in Eq. (34) and integrate out $\\theta _ { 3 }$ and $\\theta _ { 4 }$ via Lemma E.29. We will not write out this integral explicitly but instead focus on other techniques for evaluating the eigenvalues. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 55 + }, + { + "type": "text", + "text": "F.3.2 GEGENBAUER EXPANSION ", + "text_level": 1, + "bbox": [ + 174, + 103, + 413, + 118 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "Now let’s compute the local convergence rate via Gegenbauer expansion. By differentiating Proposition E.34 through a path $\\Sigma _ { t } \\in { \\cal S } _ { B } , t \\in ( - \\epsilon , \\epsilon )$ , we get ", + "bbox": [ + 173, + 128, + 823, + 160 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/d1ec10d504a6aaa1771b932778af9178ed351029ee904f02b277b7a953798c60.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { { ^ \\prime } } _ { B _ { \\phi } } ( { \\Sigma } _ { t } ) = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathfrak { e } v ) ^ { \\otimes 2 } \\frac { d } { d t } K ( v ; { \\Sigma } _ { t } ) } \\\\ { = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathfrak { e } v ) ^ { \\otimes 2 } K ( v ; { \\Sigma } _ { t } ) \\left( \\frac { B - 1 } { 2 } ( v ^ { T } { \\Sigma } _ { \\boxdot } { v } ^ { - 1 } v ) ^ { - 1 } v ^ { T } { \\Sigma } _ { \\boxdot } { } ^ { - 1 } \\frac { d { \\Sigma } _ { \\boxdot } { \\Sigma } } { d t } { \\Sigma } _ { \\boxdot } { } ^ { - 1 } v - \\frac { 1 } { 2 } \\operatorname { t r } ( { \\Sigma } _ { \\boxdot } { \\dot } ^ { - 1 } \\frac { d { \\Sigma } _ { \\boxdot } { \\Sigma } } { d t } ) ^ { \\otimes 2 } \\right) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 169, + 861, + 238 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "where $\\Sigma _ { \\Pi } = \\Sigma _ { \\Pi t } = \\Phi ^ { T } \\Sigma _ { t ^ { \\mathbb { C } } } \\in S _ { B - 1 }$ (as introduced below Defn E.3). At $\\Sigma _ { 0 } = \\Sigma ^ { * }$ , the BSB1 fixed point, we have $\\Sigma _ { \\perp 0 } = \\mu ^ { * } I$ , and ", + "bbox": [ + 173, + 247, + 825, + 279 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/df2e7266e1bfbd1c0c1df3fbe9962c188f7ba01f8c128042c38371e94ed34c10.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { d } { d t } \\nabla _ { B _ { \\phi } } ( \\Sigma _ { t } ) \\bigg | _ { t = 0 } = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathfrak { e } v ) ^ { \\otimes 2 } K ( v ; \\Sigma ^ { * } ) \\mu ^ { * - 1 } \\left( \\frac { B - 1 } { 2 } v ^ { T } \\frac { d \\Sigma _ { \\boxplus } } { d t } v - \\frac { 1 } { 2 } \\operatorname { t r } ( \\frac { d \\Sigma _ { \\boxplus } } { d t } ) \\right) } \\\\ & { \\qquad = \\mu ^ { * - 1 } \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathfrak { e } v ) ^ { \\otimes 2 } \\left( \\frac { B - 1 } { 2 } v ^ { T } \\frac { d \\Sigma _ { \\boxplus } } { d t } v - \\frac { 1 } { 2 } \\operatorname { t r } ( \\frac { d \\Sigma _ { \\boxplus } } { d t } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 286, + 795, + 359 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "If $\\textstyle { \\frac { d } { d t } } \\Sigma | _ { t = 0 } = G$ , then the term in the parenthesis vanishes. This shows that $\\left. \\frac { \\mathrm { d } \\mathrm { V } \\mathfrak { s } _ { \\phi } } { \\mathrm { d } \\Sigma } \\right| _ { \\Sigma = \\Sigma ^ { * } } \\left\\{ G \\right\\} = 0 .$ ", + "bbox": [ + 173, + 368, + 810, + 397 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "Eigenvalue for L. If $\\begin{array} { r } { \\frac { d } { d t } \\Sigma | _ { t = 0 } = \\operatorname { L } ( B - 2 , 1 ) = B G ( \\delta _ { 1 } ^ { \\otimes 2 } - \\delta _ { 2 } ^ { \\otimes 2 } ) G } \\end{array}$ where $\\delta _ { 1 } = ( 1 , 0 , \\ldots , 0 ) , \\delta _ { 2 } =$ $( 0 , 1 , 0 , \\ldots , 0 )$ , both in $\\mathbb { R } ^ { B }$ (Proposition E.59), then we know by Thm E.62 that $G ^ { \\otimes 2 } \\circ$ $\\begin{array} { r } { \\frac { d } { d t } \\mathrm { V } _ { B _ { \\phi } } ( \\Sigma _ { t } ) \\big | _ { t = 0 } } \\end{array}$ is a multiple of $\\mathrm { L } ( B - 2 , 1 )$ . We compute ", + "bbox": [ + 173, + 414, + 825, + 464 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/223c4d8ac7f53663f782499f0cb0db04905921e602d0efbcd585a3e6322b8032.jpg", + "text": "$$\n\\left( \\frac { d } { d t } \\nabla _ { B _ { \\phi } } ( \\Sigma _ { t } ) \\bigg | _ { t = 0 } \\right) _ { a b } = B \\frac { B - 1 } { 2 } \\mu ^ { * - 1 } \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathbb { e } _ { a , : } v ) \\phi ( \\sqrt { B } \\mathbb { e } _ { b , : } v ) ( ( \\mathbb { e } _ { 1 , : } v ) ^ { 2 } - ( \\mathbb { e } _ { 2 , : } v ) ^ { 2 } ) .\n$$", + "text_format": "latex", + "bbox": [ + 187, + 474, + 808, + 512 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "Clearly, $\\begin{array} { r } { \\frac { d } { d t } \\mathrm { V } _ { B _ { \\phi } } ( \\Sigma _ { t } ) \\big | _ { t = 0 } } \\end{array}$ is L-shaped. ", + "bbox": [ + 173, + 522, + 418, + 542 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "Now define the quantity $\\begin{array} { r l r } { A ( { a } , { b } ; { c } ) } & { : = } & { \\mathbb { E } _ { v \\sim S ^ { B - 2 } } \\phi ( \\sqrt { B } \\mathbb { e } _ { a , : } v ) \\phi ( \\sqrt { B } \\mathbb { e } _ { b , : } v ) ( \\mathbb { e } _ { c , : } v ) ^ { 2 } } \\end{array}$ , so that $\\begin{array} { r l r } { \\left( \\frac { d } { d t } \\nabla _ { \\mathcal { B } _ { \\phi } } ( \\Sigma _ { t } ) \\big | _ { t = 0 } \\right) _ { a b } } & { { } = } & { \\frac { B ( B - 1 ) } { 2 } \\mu ^ { * - 1 } ( \\mathcal { A } ( a , b ; 1 ) - \\mathcal { A } ( a , b ; 2 ) ) } \\end{array}$ B(B−1)2 µ∗−1(A(a, b; 1) − A(a, b; 2)). Because e is an isometry, $\\begin{array} { r } { \\sum _ { b } ( \\Phi _ { b , : } v ) ^ { 2 } = \\| v \\| ^ { 2 } = 1 } \\end{array}$ for all $v \\in S ^ { B - 2 }$ . Thus ", + "bbox": [ + 173, + 547, + 825, + 602 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/f21ca23acdcd8e631f22824dc1c2398643d631f801818c400946867b556159d8.jpg", + "text": "$$\n\\sum _ { c = 1 } ^ { B } A ( a , b ; c ) = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathbb { e } _ { a , : } v ) \\phi ( \\sqrt { B } \\mathbb { e } _ { b , : } v ) = \\Sigma _ { a b } ^ { * } .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 611, + 691, + 655 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "By symmetry, $\\mathcal { A } ( a , b ; a ) = \\mathcal { A } ( a , b ; b )$ and for any $c , c ^ { \\prime } \\notin \\{ a , b \\} , A ( a , b ; c ) = A ( a , b ; c ^ { \\prime } )$ . So we have, for any $a , b , c$ not equal, ", + "bbox": [ + 171, + 665, + 825, + 695 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/c22ea6adefc817930b1c0bc868f9dd96304e19a46b32feb0bd6dd9fcc32cf651.jpg", + "text": "$$\n\\begin{array} { r } { A ( a , a ; a ) + ( B - 1 ) A ( a , a ; c ) = \\Sigma _ { a a } ^ { * } } \\\\ { 2 A ( a , b ; b ) + ( B - 2 ) A ( a , b ; c ) = \\Sigma _ { a b } ^ { * } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 367, + 704, + 629, + 743 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "So the eigenvalue associated to $\\mathrm { L } ( B - 2 , 1 )$ is, So by Lemma E.60, ${ \\cal G } ^ { \\otimes 2 } \\circ \\mathcal { U } \\{ \\mathrm { L } ( B - 2 , 1 ) \\} =$ $\\lambda _ { \\mathbb { L } } ^ { \\uparrow } \\mathrm { L } ( B - 2 , 1 )$ , where ", + "bbox": [ + 171, + 752, + 823, + 785 + ], + "page_idx": 56 + }, + { + "type": "equation", + "img_path": "images/8acc160824880f90d4df9cbb2dc15288100af15803ab797e6484fe196e6e36cd.jpg", + "text": "$$\n\\begin{array} { r l } & { \\overset { \\uparrow } { \\underset { \\mathrm { L } } { \\mathrm { T } } } = B \\frac { B - 1 } { 2 } \\mu ^ { * - 1 } \\frac { ( \\frac { d } { d t } \\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( \\Sigma _ { t } ) | _ { t = 0 } ) _ { 1 1 } - 2 ( \\frac { d } { d t } \\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( \\Sigma _ { t } ) \\big | _ { t = 0 } ) _ { 1 3 } } { B } } \\\\ & { \\quad = B \\frac { B - 1 } { 2 } \\mu ^ { * - 1 } \\frac { A ( 1 , 1 ; 1 ) - A ( 1 , 1 ; 2 ) - 2 ( A ( 1 , 3 ; 1 ) - A ( 1 , 3 ; 2 ) ) } { B } } \\\\ & { \\quad = \\frac { B - 1 } { 2 } \\mu ^ { * - 1 } ( A ( 1 , 1 ; 1 ) - \\frac { 1 } { B - 1 } ( \\Sigma _ { a a } ^ { * } - A ( 1 , 1 ; 1 ) ) - 2 ( A ( 1 , 3 ; 1 ) - \\frac { 1 } { B - 2 } ( \\Sigma _ { a b } ^ { * } - 2 A ( 1 , 3 ; 1 ) - \\frac { 1 } { B - 2 } ( \\Sigma _ { a b } ^ { * } - 2 A ( 1 , 3 ; 1 ) - \\frac { 1 } { B - 2 } \\Sigma _ { a b } ^ { * } ) ) } \\\\ & { \\quad = \\frac { B - 1 } { 2 } \\mu ^ { * - 1 } ( \\frac { B } { B - 1 } A ( 1 , 1 ; 1 ) - \\frac { 1 } { B - 1 } \\Sigma _ { a a } ^ { * } - 2 \\frac { B } { B - 2 } A ( 1 , 3 ; 1 ) + \\frac { 2 } { B - 2 } \\Sigma _ { a b } ^ { * } ) \\qquad ( 4 8 ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 794, + 839, + 931 + ], + "page_idx": 56 + }, + { + "type": "text", + "text": "Thus, if $\\phi ( \\sqrt { B - 1 } x ) \\in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \\frac { B - 3 } { 2 } } )$ has Gegenbauer expansion $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ then by Proposition E.43, ", + "bbox": [ + 173, + 101, + 826, + 137 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/7396d7e8ca759f3b0deb4f4a7f383797f947a6cfe1d4379565262fe9bce796ec.jpg", + "text": "$$\n\\begin{array} { r l } { { } } & { { \\ x ^ { 2 } \\phi ( \\sqrt { B - 1 } x ) } } \\\\ { { } } & { { = \\displaystyle \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } x ^ { 2 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } } \\\\ { { } } & { { = \\displaystyle \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } \\biggl ( \\kappa _ { 2 } ( l , \\frac { B - 3 } { 2 } ) C _ { l + 2 } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } } \\\\ { { } } & { { \\qquad + \\kappa _ { 0 } ( l , \\frac { B - 3 } { 2 } ) C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) + \\kappa _ { - 2 } ( l , \\frac { B - 3 } { 2 } ) C _ { l - 2 } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) \\biggr ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 186, + 145, + 725, + 285 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "where $C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x )$ is understood to be 0 for $l < 0$ ", + "bbox": [ + 488, + 287, + 812, + 309 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/d29d6b5bd114f96edd49775411137242f935aef9733580511df84f71c1192d24.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle = \\sum _ { l = 0 } ^ { \\infty } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\\\ { \\displaystyle \\quad \\quad \\quad \\left( \\frac { a _ { l - 2 } } { c _ { B - 1 , l - 2 } } \\kappa _ { 2 } ( l - 2 , \\frac { B - 3 } { 2 } ) + \\frac { a _ { l } } { c _ { B - 1 , l } } \\kappa _ { 0 } ( l , \\frac { B - 3 } { 2 } ) + \\frac { a _ { l + 2 } } { c _ { B - 1 , l + 2 } } \\kappa _ { - 2 } ( l + 2 , \\frac { B - 3 } { 2 } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 311, + 812, + 390 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/69fd10c25cc422631c47a9b5a7cfc0e967bbd5144118d682d961bf1d907b0266.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle = \\sum _ { l = 0 } ^ { \\infty } c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\\\ { \\displaystyle \\quad \\quad \\left( \\frac { a _ { l - 2 } c _ { B - 1 , l } } { c _ { B - 1 , l - 2 } } \\kappa _ { 2 } ( l - 2 , \\frac { B - 3 } { 2 } ) + a _ { l } \\kappa _ { 0 } ( l , \\frac { B - 3 } { 2 } ) + \\frac { a _ { l + 2 } c _ { B - 1 , l } } { c _ { B - 1 , l + 2 } } \\kappa _ { - 2 } ( l + 2 , \\frac { B - 3 } { 2 } ) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 420, + 810, + 497 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "We can evaluate, for any $a , b$ (possibly equal), ", + "bbox": [ + 174, + 522, + 475, + 537 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/53a2cb0cc4ee834f64a3794f4afa8700f15479a06ab13bce7260f25b46815f35.jpg", + "text": "$$\n\\begin{array} { l } { { 4 ( a , b ; b ) = \\displaystyle \\operatorname * { \\mathbb { E } } _ { v \\sim S ^ { B - 2 } } \\phi ( \\sqrt { B } \\mathfrak { S } a , v ) \\phi ( \\sqrt { B } \\mathfrak { e } _ { b , : } v ) ( \\mathfrak { e } _ { b , : } v ) ^ { 2 } } } \\\\ { { \\displaystyle \\quad = \\operatorname * { \\mathbb { E } } _ { v \\sim S ^ { B - 2 } } \\phi ( \\sqrt { B - 1 } \\hat { \\mathfrak { e } } a , v ) \\phi ( \\sqrt { B - 1 } \\hat { \\mathfrak { e } } _ { b , : } v ) ( \\hat { \\mathfrak { e } } _ { b , : } v ) ^ { 2 } \\frac { B - 1 } { B } } } \\\\ { { \\displaystyle \\quad = \\frac { B - 1 } { B } \\sum _ { l = 0 } ^ { \\infty } c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( \\langle \\hat { \\mathfrak { e } } _ { a , : } , \\hat { \\mathfrak { e } } _ { b , : } \\rangle ) } } \\\\ { { \\displaystyle a _ { l } \\left( \\frac { a _ { l - 2 } c _ { B - 1 , l } } { c _ { B - 1 , l - 2 } } \\kappa _ { 2 } ( l - 2 , \\frac { B - 3 } { 2 } ) + a _ { l } \\kappa _ { 0 } ( l , \\frac { B - 3 } { 2 } ) + \\frac { a _ { l + 2 } c _ { B - 1 , l } } { c _ { B - 1 , l + 2 } } \\kappa _ { - 2 } ( l + 2 , \\frac { B - 3 } { 2 } ) \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 542, + 839, + 684 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "By Eq. (48), the eigenvalue associated with $\\mathrm { L } ( B - 2 , 1 )$ is ", + "bbox": [ + 171, + 703, + 558, + 719 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/a19e635857c616ec6398669395b222d09ff246ded56f3a7b40c78e81be336fe5.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\frac { B - 1 } { 2 } \\mu ^ { * - 1 } ( \\frac { B } { B - 1 } A ( 1 , 1 ; 1 ) - \\frac { 1 } { B - 1 } \\Sigma _ { a a } ^ { * } - 2 \\frac { B } { B - 2 } A ( 1 , 3 ; 1 ) + \\frac { 2 } { B - 2 } \\Sigma _ { a b } ^ { * } ) } } \\\\ { { = \\displaystyle \\frac { B - 1 } { 2 } \\mu ^ { * - 1 } \\sum _ { l = 0 } ^ { \\infty } c _ { B - 1 , l } ^ { - 1 } ( \\gamma _ { l } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) - \\tau _ { l } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( \\frac { - 1 } { B - 1 } ) ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 228, + 727, + 772, + 801 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 808, + 217, + 821 + ], + "page_idx": 57 + }, + { + "type": "equation", + "img_path": "images/fc8e9d3cc57aeff2571f58392b7ba57bf6f2430be954e72fad734cf0328c7851.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { \\ u _ { \\iota } : = a _ { l } \\left( \\frac { a _ { l - 2 } c _ { B - 1 , l } } { c _ { B - 1 , l - 2 } } \\kappa _ { 2 } ( l - 2 , \\frac { B - 3 } { 2 } ) + a _ { l } \\kappa _ { 0 } ( l , \\frac { B - 3 } { 2 } ) + \\frac { a _ { l + 2 } c _ { B - 1 , l } } { c _ { B - 1 , l + 2 } } \\kappa _ { - 2 } ( l + 2 , \\frac { B - 3 } { 2 } ) \\right) - \\frac { 1 } { B - 1 } \\mathrm { ~ c ~ o ~ n ~ s ~ t ~ } , } } \\\\ { \\displaystyle { \\ u _ { \\tilde { \\iota } } : = \\frac { 2 ( B - 1 ) } { B - 2 } \\gamma \\ u _ { l } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 827, + 839, + 898 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "Thus, ", + "bbox": [ + 173, + 909, + 214, + 924 + ], + "page_idx": 57 + }, + { + "type": "text", + "text": "Theorem F.22. Suppose φ(√B − 1x) has Gegenbauer expansion P∞l=0 al 1cB−1,l $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ . Then the eigenvalue of $\\mathcal { U }$ with respect to the eigenspace $\\mathbb { L }$ is a ratio of quadratic forms ", + "bbox": [ + 173, + 102, + 826, + 138 + ], + "page_idx": 58 + }, + { + "type": "equation", + "img_path": "images/efd0a3c9aef199d3017e8e72f9d70ab6a08070e997fc46a9c99505071c34fb97.jpg", + "text": "$$\n\\lambda _ { \\mathbb { L } } ^ { \\uparrow } = \\frac { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } w _ { B - 1 , l } + a _ { l } a _ { l + 2 } u _ { B - 1 , l } } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } v _ { B - 1 , l } }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 140, + 629, + 176 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 179, + 217, + 193 + ], + "page_idx": 58 + }, + { + "type": "equation", + "img_path": "images/48954d12412e86937a47d0d86323cbe386ec8ca9ef72081b57167b1e9e1ca103.jpg", + "text": "$$\n\\begin{array} { r l } & { r _ { ( k - 1 ) , ( k ) } = \\varepsilon _ { ( k - 1 ) , ( k ) } ^ { - 1 } \\left( C _ { i } ^ { ( k ^ { 2 } - 2 ) } ( 1 ) - C _ { i } ^ { ( k ^ { 2 } - 2 ) } ( \\frac { 1 } { B } ) \\right) } \\\\ & { r _ { ( k - 1 ) , ( k ^ { 2 } - 1 ) } ^ { - 1 } \\left( r _ { ( k ) } \\left( \\frac { 1 } { B } - \\frac { 1 } { 2 } \\right) - \\frac { 1 } { 1 - 1 } \\right) e _ { ( k - 1 ) } ^ { r _ { ( k ) } } \\left( C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( 1 ) - \\frac { 2 ( B - 1 ) } { B - 2 } C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( \\frac { 1 } { B - 1 } ) \\right) } \\\\ & { \\quad \\quad \\quad - \\frac { 1 } { ( B } - 3 + 1 ) ( B - 3 ) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & & { = \\frac { 1 } { ( B - 5 ) + ( B ) ( B - 1 + 3 ) ( B - 3 + 2 ) ) } e _ { ( k - 1 ) } ^ { r _ { ( k ) } } \\left( C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( 1 ) - \\frac { 2 ( B - 1 ) } { B - B } C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( \\frac { 1 } { B - 1 } ) \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & & { = \\frac { 1 } { ( B - 1 ) ( B - 3 + 2 ) ( B - 3 + 2 ) ) } \\left( C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( 1 ) - \\frac { 2 ( B - 1 ) } { B } C _ { i } ^ { ( k ^ { 2 } - 3 ) } ( \\frac { 1 } { B - 1 } ) \\right) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 193, + 833, + 478 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "Note that $v _ { B - 1 , 0 } = w _ { B - 1 , 0 } = u _ { B - 1 , 0 } = 0$ , so that there is in fact no dependence on $a _ { 0 }$ , as expected since batchnorm is invariant under additive shifts of $\\phi$ . ", + "bbox": [ + 173, + 483, + 823, + 512 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "We see that $\\lambda _ { \\mathbb { L } } ^ { \\uparrow } \\geq 1$ iff ", + "bbox": [ + 174, + 518, + 325, + 536 + ], + "page_idx": 58 + }, + { + "type": "equation", + "img_path": "images/4824d8bac021b4d6ee7a142ec03914082297b1b5a61b53a1ad6bc13ab37aa0f5.jpg", + "text": "$$\n0 \\leq \\sum _ { l = 1 } ^ { \\infty } a _ { l } ^ { 2 } ( w _ { B - 1 , l } - v _ { B - 1 , l } ) + a _ { l } a _ { l + 2 } u _ { B - 1 , l } .\n$$", + "text_format": "latex", + "bbox": [ + 341, + 539, + 656, + 580 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "This is a quadratic form on the coefficients $\\{ a _ { l } \\} _ { l }$ . We now analyze it heuristically and argue that the eigenvalue is $\\geq 1$ typically when $\\phi ( \\sqrt { B - 1 } x )$ explodes sufficiently as $x \\to 1$ or $x \\to - 1$ ; in other words, the more explosive $\\phi$ is, the less likely it is to induce Eq. (43) to converge to a BSB1 fixed point. ", + "bbox": [ + 173, + 590, + 825, + 648 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "Heuristically, $C _ { l + 2 } ^ { ( \\frac { B - 3 } { 2 } ) } \\big ( \\frac { - 1 } { B - 1 } \\big )$ is negligible compared to $C _ { l + 2 } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 )$ for sufficiently large $B$ and $l$ , so that $\\begin{array} { r l r } { w _ { B - 1 , l } ~ - ~ v _ { B - 1 , l } } & { { } \\approx } & { \\left( \\frac { l ( B - 3 + l ) ( B - 3 ) } { ( B - 5 + 2 l ) ( B - 1 + 2 l ) } - 1 \\right) c _ { B - 1 , l } ^ { - 1 } C _ { l + 2 } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) \\quad = } \\end{array}$ $\\begin{array} { r } { \\left( \\frac { B - 7 } { 4 } + O ( B ^ { - 2 } ) + O ( l ^ { - 2 } ) \\right) c _ { B - 1 , l } ^ { - 1 } C _ { l + 2 } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) } \\end{array}$ and is positive. For small $l$ , we can calculate ", + "bbox": [ + 173, + 655, + 826, + 727 + ], + "page_idx": 58 + }, + { + "type": "equation", + "img_path": "images/bace1a357d0deebbb803d5150b4228b1fcb811c6268b73f18d0e80ae6583d4cc.jpg", + "text": "$$\n\\begin{array} { r l } & { w _ { B - 1 , 1 } - v _ { B - 1 , 1 } = \\displaystyle \\frac { - 2 B } { 1 + B } < 0 } \\\\ & { w _ { B - 1 , 2 } - v _ { B - 1 , 2 } = \\displaystyle \\frac { B ( B + 1 ) ( B ^ { 2 } - 9 B + 1 2 ) } { 2 ( B - 1 ) ( B + 3 ) } \\geq 0 , \\forall B \\geq 1 0 } \\\\ & { w _ { B - 1 , 3 } - v _ { B - 1 , 3 } = \\displaystyle \\frac { ( B - 3 ) ( B + 3 ) B ^ { 2 } ( 2 B ^ { 3 } - 1 9 B ^ { 2 } + 1 6 B + 1 3 ) } { 6 ( B - 1 ) ^ { 2 } ( B + 1 ) ( B + 5 ) } \\geq 0 , \\forall B \\geq 1 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 217, + 731, + 779, + 832 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "In fact, plotting $w _ { B - 1 , l } - v _ { B - 1 , l }$ for various values of $l$ suggests that for $B \\geq 1 0$ , $w _ { B - 1 , l } - v _ { B - 1 , l } \\geq$ 0 for all $l \\geq 1$ (Fig. 10). ", + "bbox": [ + 173, + 832, + 823, + 862 + ], + "page_idx": 58 + }, + { + "type": "text", + "text": "Thus, the more “linear” $\\phi$ is, the larger $a _ { 1 } ^ { 2 }$ is compared to the rest of $\\{ a _ { l } \\} _ { l }$ , and the more likely that the eigenvalue is $< ~ 1$ . In the case that $a _ { l } a _ { l + 2 } = 0 \\forall l$ , then indeed the eigenvalue is $< ~ 1$ precisely when $a _ { 1 } ^ { 2 }$ is sufficiently large. Because higher degree Gegenbauer polynomials explodes more violently as $x \\pm 1$ , this is consistent with our claim. ", + "bbox": [ + 173, + 867, + 825, + 924 + ], + "page_idx": 58 + }, + { + "type": "image", + "img_path": "images/ea53df2404b9d1ec200fce8d9dce20ea8f850633079e937330d90fb992d6c9ad.jpg", + "image_caption": [ + "Figure 10: Plots of $w _ { B - 1 , l } - v _ { B - 1 , l }$ over $B$ for various $l$ " + ], + "image_footnote": [], + "bbox": [ + 302, + 101, + 689, + 256 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "Eigenvalue for M We use a similar technique shows the Gegenbauer expansion of the eigenvalue for $\\mathbb { M }$ . ", + "bbox": [ + 169, + 305, + 825, + 334 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "Define $\\tilde { \\mathcal { A } } _ { a b } ( c , d ) : = \\mathbb { E } _ { v \\sim S ^ { B - 2 } } \\phi ( \\sqrt { B } \\mathbb { e } _ { a , : } v ) \\phi ( \\sqrt { B } \\mathbb { e } _ { b , : } v ) ( \\mathbb { e } _ { c , : } v ) ( \\mathbb { e } _ { d , : } v )$ . Then $\\tilde { \\mathcal { A } } _ { a b } ( c , c ) = \\mathcal { A } ( a , b ; c )$ . Note the ultrasymmetries $\\begin{array} { l l l } { \\tilde { \\mathcal { A } } _ { a b } ( c , d ) } & { = } & { \\tilde { \\mathcal { A } } _ { \\pi ( a ) \\pi ( b ) } ( \\pi ( c ) , \\pi ( d ) ) } \\end{array}$ for any permutation $\\pi$ , and $\\tilde { \\mathcal { A } } _ { a b } ( c , d ) = \\tilde { \\mathcal { A } } _ { a b } ( d , c ) = \\tilde { \\mathcal { A } } _ { b a } ( c , d ) = \\tilde { \\mathcal { A } } _ { b a } ( d , c )$ . ", + "bbox": [ + 173, + 339, + 826, + 391 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "By Eq. (46), we have ", + "bbox": [ + 173, + 396, + 316, + 411 + ], + "page_idx": 59 + }, + { + "type": "equation", + "img_path": "images/247a94293ba2d6b8d94bbcc237dc8544b19aa0b3e4e115f77b48ad9ce2492f3e.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\tilde { \\mathcal { U } } \\{ \\delta _ { c } \\delta _ { d } ^ { T } \\} _ { a b } } \\\\ & { = \\mu ^ { * - 1 } \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathfrak { e } _ { a , : } v ) \\phi ( \\sqrt { B } \\mathfrak { e } _ { b , : } v ) \\left( \\frac { B - 1 } { 2 } ( \\mathfrak { e } _ { c , : } v ) ( \\mathfrak { e } _ { d , : } v ) - \\frac { 1 } { 2 } \\left( \\mathbb { I } ( c = d ) - \\frac { 1 } { B } \\right) \\right) } \\\\ & { = \\mu ^ { * - 1 } \\frac { B - 1 } { 2 } \\tilde { A } ( a , b ; c , d ) - \\mu ^ { * - 1 } \\frac { 1 } { 2 } \\Sigma _ { a b } ^ { * } \\left( \\mathbb { I } ( c = d ) - \\frac { 1 } { B } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 412, + 794, + 505 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "By $\\cdot$ , $G ^ { \\otimes 2 } \\circ u$ has eigenspace $\\mathbb { M }$ with eigenvalue ", + "bbox": [ + 174, + 505, + 560, + 520 + ], + "page_idx": 59 + }, + { + "type": "equation", + "img_path": "images/4aa799a6cc16006ee9e683f781e3a36b3e7a3939da992ac96626b990d7a015ed.jpg", + "text": "$$\n\\begin{array} { r l } & { \\lambda _ { \\mathbb { M } } ^ { \\uparrow } = \\tilde { \\mathcal { U } } \\{ \\delta _ { 1 } \\delta _ { 2 } ^ { T } \\} _ { 1 2 } + \\tilde { \\mathcal { U } } \\{ \\delta _ { 2 } \\delta _ { 1 } ^ { T } \\} _ { 1 2 } - 4 \\tilde { \\mathcal { U } } \\{ \\delta _ { 1 } \\delta _ { 3 } ^ { T } \\} _ { 1 2 } + 2 \\tilde { \\mathcal { U } } \\{ \\delta _ { 3 } \\delta _ { 4 } ^ { T } \\} _ { 1 2 } } \\\\ & { \\quad = 2 ( \\tilde { \\mathcal { U } } \\{ \\delta _ { 1 } \\delta _ { 2 } ^ { T } \\} _ { 1 2 } - 2 \\tilde { \\mathcal { U } } \\{ \\delta _ { 1 } \\delta _ { 3 } ^ { T } \\} _ { 1 2 } + \\tilde { \\mathcal { U } } \\{ \\delta _ { 3 } \\delta _ { 4 } ^ { T } \\} _ { 1 2 } ) } \\\\ & { \\quad = \\frac { B - 1 } { \\mu ^ { * } } ( \\tilde { A } _ { 1 2 } ( 1 2 ) - \\tilde { A } _ { 1 2 } ( 1 3 ) + \\tilde { A } _ { 1 2 } ( 3 4 ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 287, + 521, + 710, + 594 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "Thus we need to evaluate $\\tilde { A } _ { 1 2 } ( 1 2 ) , \\tilde { A } _ { 1 2 } ( 1 3 ) , \\tilde { A } _ { 1 2 } ( 3 4 )$ . We will do so by exploiting linear dependences between different values of $\\tilde { A } _ { a b } ( c , d )$ , the computed values of $\\tilde { \\mathcal { A } } _ { a b } ( c , c ) = \\mathcal { A } ( a , b ; c )$ , and the value of $\\tilde { A } _ { a b } ( a , b )$ computed below in Lemma F.23. Indeed, we have ", + "bbox": [ + 171, + 598, + 826, + 647 + ], + "page_idx": 59 + }, + { + "type": "equation", + "img_path": "images/1291c18441dbd39dc38b2d0a4444be55bd9025ef873202f1a6e9f8d682e05496.jpg", + "text": "$$\n\\sum _ { d = 1 } ^ { B } \\tilde { \\mathcal { A } } _ { a b } ( c , d ) = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathbb { e } _ { a , : } v ) \\phi ( \\sqrt { B } \\mathbb { e } _ { b , : } v ) ( \\mathbb { e } _ { c , : } v ) \\sum _ { d = 1 } ^ { B } ( \\mathbb { e } _ { d , : } v ) = 0 .\n$$", + "text_format": "latex", + "bbox": [ + 256, + 647, + 738, + 691 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "Leveraging the symmmetries of $\\tilde { \\mathcal { A } }$ , we get ", + "bbox": [ + 176, + 694, + 452, + 710 + ], + "page_idx": 59 + }, + { + "type": "equation", + "img_path": "images/672a4c16f2bc4078690fd20568ebfa8c7b4afb9d7a173653d8c7fbf91557d463.jpg", + "text": "$$\n\\begin{array} { r } { ( B - 3 ) \\tilde { A } _ { 1 2 } ( 3 4 ) + 2 \\tilde { A } _ { 1 2 } ( 1 3 ) + \\tilde { A } _ { 1 2 } ( 3 3 ) = 0 } \\\\ { ( B - 2 ) \\tilde { A } _ { 1 2 } ( 1 3 ) + \\tilde { A } _ { 1 2 } ( 1 1 ) + \\tilde { A } _ { 1 2 } ( 1 2 ) = 0 . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 346, + 710, + 651, + 752 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "Expressing in terms of $\\tilde { \\mathcal { A } } _ { 1 2 } ( 1 1 )$ and $\\tilde { A } _ { 1 2 } ( 1 2 )$ , we get ", + "bbox": [ + 173, + 755, + 524, + 772 + ], + "page_idx": 59 + }, + { + "type": "equation", + "img_path": "images/86cae5e42d56ad315b0233cfdc7903ba40eb3842481043022e3694e143434923.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 3 ) = \\frac { - 1 } { B - 2 } ( \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 1 ) + \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 2 ) ) } } \\\\ & { } & { \\tilde { \\mathcal { A } } _ { 1 2 } ( 3 4 ) = \\frac { - 1 } { B - 3 } ( 2 \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 3 ) + \\tilde { \\mathcal { A } } _ { 1 2 } ( 3 3 ) ) } \\\\ & { } & { = \\frac { - 1 } { B - 3 } ( \\frac { - 2 } { B - 2 } ( \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 1 ) + \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 2 ) ) + \\frac { 1 } { B - 2 } ( \\Sigma _ { 1 2 } ^ { * } - 2 \\tilde { \\mathcal { A } } _ { 1 2 } ( 2 2 ) ) ) } \\\\ & { } & { \\mathbf { B } \\mathbf { y } \\operatorname { E q . } ( 4 7 ) } \\\\ & { } & { = \\frac { - 1 } { ( B - 3 ) ( B - 2 ) } ( \\Sigma _ { 1 2 } ^ { * } - 4 \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 1 ) - 2 \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 2 ) ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 222, + 773, + 774, + 922 + ], + "page_idx": 59 + }, + { + "type": "text", + "text": "These relations allow us to simplify ", + "bbox": [ + 174, + 103, + 410, + 118 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/d59b1bdc2b07c2a9276c08cc7228e318fa6ce47af79cc4325ef1fc36929678f6.jpg", + "text": "$$\n\\lambda _ { \\mathtt { M } } ^ { \\uparrow } = \\frac { B - 1 } { ( B - 3 ) \\mu ^ { * } } \\left( \\frac { - 1 } { B - 2 } \\Sigma _ { 1 2 } ^ { * } + \\frac { 2 ( B - 1 ) } { B - 2 } \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 1 ) + ( B - 1 ) \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 2 ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 132, + 746, + 167 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "By Eq. (49), we know the Gegenbauer expansion of $\\tilde { \\mathcal { A } } _ { 1 2 } ( 1 1 )$ . The following shows the Gegenbauer expansion of $\\tilde { \\mathcal { A } } _ { 1 2 } ( 1 2 )$ . ", + "bbox": [ + 171, + 181, + 823, + 215 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Lemma F.23. Suppose $\\phi ( \\sqrt { B - 1 } x ) , x \\phi ( \\sqrt { B - 1 } x ) \\in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \\frac { B - 3 } { 2 } } )$ , and $\\phi ( \\sqrt { B - 1 } x )$ has Gegenbauer expansion P∞l=0 al 1cB−1,l $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ . Then ", + "bbox": [ + 173, + 222, + 826, + 263 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/2155f08cb0f3bcf8f8c0587ba2dfe8125e56dc1699e09aa4838ca84f92511b59.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\tilde { \\mathcal { A } } _ { 1 2 } ( 1 2 ) = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B } \\mathbb { e } _ { 1 , : } v ) \\phi ( \\sqrt { B } \\mathbb { e } _ { 2 , : } v ) ( \\mathbb { e } _ { 1 , : } v ) ( \\mathbb { e } _ { 2 , : } v ) } \\\\ { \\displaystyle = \\sum _ { l = 0 } ^ { \\infty } \\alpha _ { l } a _ { l } ^ { 2 } + \\beta _ { l } a _ { l } a _ { l + 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 276, + 702, + 348 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 361, + 217, + 375 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/7b70565702f61a2d303512c613da01794a2331767a2f1001f31914faa57c1986.jpg", + "text": "$$\n\\begin{array} { r l } & { v _ { l } = \\frac { B - 1 } { B } \\times \\left\\{ \\begin{array} { l l } { \\left( \\frac { l + 1 } { B - 1 + 2 l } \\right) ^ { 2 } c _ { B - 1 , l + 1 } ^ { - 1 } C _ { l + 1 } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) + \\left( \\frac { B + l - 4 } { B + 2 l - 5 } \\right) ^ { 2 } c _ { B - 1 , l - 1 } ^ { - 1 } C _ { l - 1 } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) } & { i f l \\sin ( j ) } \\\\ { \\left( \\frac { 1 } { B - 1 } \\right) ^ { 2 } c _ { B - 1 , 1 } ^ { - 1 } C _ { 1 } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) } & { o t h } \\end{array} \\right. } \\\\ & { 3 _ { l } = \\frac { B - 1 } { B } \\times \\frac { 2 ( l + 1 ) ( B + l - 2 ) } { ( B + 2 l - 1 ) ^ { 2 } } c _ { B - 1 , l + 1 } ^ { - 1 } C _ { l + 1 } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 386, + 839, + 481 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "erwise ", + "bbox": [ + 815, + 422, + 882, + 435 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Proof. Let $\\psi ( x ) = x \\phi ( { \\sqrt { B - 1 } } x )$ . Then by Proposition E.43, we have ", + "bbox": [ + 173, + 517, + 640, + 535 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/736ba87e5e95e4445754c87fffdd457cc19c7769aa4e9a9919bc7b90e3948e5c.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\psi ( x ) = \\sum _ { l = 0 } ^ { \\infty } a _ { l } c _ { B - 1 , l } ^ { - 1 } \\frac { 1 } { B - 3 + 2 l } \\left( ( l + 1 ) C _ { l + 1 } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) + ( B + l - 4 ) C _ { l - 1 } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) \\right) } } \\\\ { { \\displaystyle \\quad = \\sum _ { l = 0 } ^ { \\infty } \\left( a _ { l - 1 } \\frac { l } { B + 2 l - 3 } + a _ { l + 1 } \\frac { B + l - 3 } { B + 2 l - 3 } \\right) c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 232, + 547, + 766, + 633 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Then ", + "bbox": [ + 173, + 645, + 210, + 659 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/e420f4c4e24bd9ce2424375683e599d47cb00bd57d27fc7e4a0d6717acaa445b.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\tilde { A } _ { 1 2 } ( 1 2 ) = \\frac { B - 1 } { B } \\sum _ { v \\sim S ^ { B - 2 } } \\psi ( \\hat { \\bf e } _ { 1 , : } v ) \\psi ( \\hat { \\bf e } _ { 2 , : } v ) } \\\\ { \\displaystyle = \\frac { B - 1 } { B } \\sum _ { l = 0 } ^ { \\infty } \\left( a _ { l - 1 } \\frac { l } { B + 2 l - 3 } + a _ { l + 1 } \\frac { B + l - 3 } { B + 2 l - 3 } \\right) ^ { 2 } c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 197, + 671, + 802, + 746 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Rearranging the sum in terms of $a _ { l }$ gives the result. ", + "bbox": [ + 173, + 757, + 511, + 773 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Combining all of the above, we obtain the following ", + "bbox": [ + 173, + 811, + 517, + 828 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "Theorem F.24. Suppose φ(√B − 1x) has Gegenbauer expansion P∞l=0 al 1cB−1,l C $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ ( B−32 )l (x). Then the eigenvalue of $\\mathcal { U }$ with respect to the eigenspace M is a ratio of quadratic forms ", + "bbox": [ + 174, + 837, + 826, + 873 + ], + "page_idx": 60 + }, + { + "type": "equation", + "img_path": "images/703945d21dda693ede286a1a74268dd0a6ecb5b39f26657a111c9ef31fafabd5.jpg", + "text": "$$\n\\lambda _ { \\mathbb { M } } ^ { \\uparrow } = \\frac { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\tilde { w } _ { B - 1 , l } + a _ { l } a _ { l + 2 } \\tilde { u } _ { B - 1 , l } } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } v _ { B - 1 , l } }\n$$", + "text_format": "latex", + "bbox": [ + 367, + 886, + 630, + 922 + ], + "page_idx": 60 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 104, + 217, + 118 + ], + "page_idx": 61 + }, + { + "type": "equation", + "img_path": "images/a358dbc57c352e6b9e85e1f6009929bbbe5dcd899dca332f8d7d4dd64cac1e42.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\frac { 2 ( B - 1 ) ^ { 5 } } { ( B - 1 ) ^ { 5 / 2 } } \\textstyle ( \\frac { ( B - 3 - l ) ( B - 2 + l ) } { ( B - 1 ) ^ { 6 } } \\textstyle ( \\frac { ( B - 3 - l ) ( B - 2 + 1 ) } { B - 1 + 2 l } C _ { l } ^ { ( \\frac { 2 3 - 2 } { 2 } ) } \\textstyle ( \\frac { - 1 } { B - 1 } ) } \\\\ & { \\qquad + \\frac { ( B - 2 ) ( l + 1 ) ( B - 2 + 1 ) } { B - 1 + 2 l + 3 } C _ { l } ^ { ( \\frac { 2 4 - 3 } { 2 } ) } \\textstyle ( \\frac { - 1 } { B - 1 } ) } \\\\ & { \\qquad + \\frac { ( 4 - 3 ) ( B - 1 + 2 + 1 ) ( B - 3 + 2 ) } { ( B + 1 + 1 ) ( B - 3 + 2 l ) } C _ { l } ^ { ( \\frac { 2 3 - 2 } { 2 } ) } \\textstyle ( \\frac { - 1 } { B - 1 } ) ) } \\\\ & { \\qquad + \\frac { ( 4 - 3 ) ( B - 1 ) } { ( B + 2 l + 1 ) ( B - 3 + 2 l ) } \\textstyle ( \\frac { - 1 } { B + 2 l } ) ) } \\\\ & { \\qquad \\frac { B - 1 } { ( B - 3 ) ^ { 2 } ( B - 2 ) } \\textstyle \\bigg [ } \\\\ & { \\qquad ( \\frac { 2 ( B - 1 ) ^ { 2 } ( B - 2 ) ( B + 2 l - 3 ) ( 2 B + 3 + 2 l ^ { 2 } - 6 l - 5 ) } { B ( B + 2 l - 5 ) ( B + 2 l - 5 ) ( B + 2 l - 1 ) } - ( B - 3 + 2 l ) ) C _ { l } ^ { ( \\frac { 2 3 - 2 } { 2 } ) } ( \\frac { - 1 } { B } ) } \\\\ & { \\qquad + \\frac { ( B - 2 ) ( B - 1 ) ^ { 2 } ( l + 1 ) ^ { 2 } } { B ( B + 2 l - 1 ) ( B - 2 l + 1 ) ^ { 2 } } C _ { l } ^ { ( \\frac { 2 3 - 2 } { 2 } ) } ( \\frac { - 1 } { B - 1 } ) } \\\\ & \\qquad + \\frac { ( B - 2 ) ( B - 1 ) ^ { 2 } ( l + 1 ) ^ { 2 } } { B ( B + 2 l - 1 ) } C _ { l } ^ { ( \\frac { 2 3 - 2 } { 2 } ) } ( \\frac { - 1 } { B - 1 } ) \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 196, + 839, + 458 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "where $C _ { - 1 } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) = 0$ by convention. ", + "bbox": [ + 173, + 536, + 419, + 559 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "Note that $\\tilde { u } _ { B - 1 , l }$ and $\\tilde { w } _ { B - 1 , l }$ depends only on $C _ { \\bullet } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right)$ which is much smaller than $C _ { \\bullet } ^ { ( \\frac { B - 3 } { 2 } ) }$ (1) for degree larger than 0. Thus the eigenvalue for $\\mathbb { M }$ is typically much smaller than the eigenvalue for $\\mathbb { L }$ (though there are counterexamples, like sin). ", + "bbox": [ + 173, + 608, + 826, + 669 + ], + "page_idx": 61 + }, + { + "type": "text", + "text": "Differentiating Eq. (26) In what follows, let $\\phi$ be a positive-homogeneous function of degree $\\alpha$ . We begin by studying $\\frac { \\mathrm { d } \\mathrm { V } _ { ( \\phi \\circ \\mathsf { n } ) } ( \\Sigma ^ { G } ) } { \\mathrm { d } \\Sigma ^ { G } }$ . We will differentiate Eq. (26) directly at $G ^ { \\otimes 2 } \\{ \\Sigma ^ { * } \\} = G \\Sigma ^ { * } G$ where $\\Sigma ^ { * }$ is the BSB1 fixed point given by Thm F.8. To that end, consider a smooth path $\\Sigma _ { t } \\ \\in$ ${ \\cal S } _ { B } ^ { G } , t \\in ( - \\epsilon , \\epsilon )$ for some $\\epsilon > 0$ , with $\\Sigma _ { 0 } = G \\Sigma ^ { * } G$ . Set $\\Sigma _ { \\perp t } = \\Phi ^ { T } \\Sigma _ { t ^ { \\mathbb { F } } } \\in { \\mathcal S } _ { B - 1 }$ , so that $\\Sigma _ { t } =$ $\\mathbb { e } \\sum _ { \\mathbf { \\Pi } } { _ { t } } \\mathbb { e } ^ { T }$ and $\\Sigma _ { \\perp 0 } = \\mu ^ { * } I _ { B - 1 }$ where $\\begin{array} { r } { \\mu ^ { * } = K _ { \\alpha , B } ( \\mathrm { J } _ { \\phi } ( 1 ) - \\mathrm { J } _ { \\phi } ( \\frac { - 1 } { B - 1 } ) ) } \\end{array}$ as in Thm F.8. If we write $\\dot { \\Sigma } _ { \\perp }$ and $\\dot { \\Sigma }$ for the time derivatives, we have ", + "bbox": [ + 173, + 827, + 825, + 924 + ], + "page_idx": 61 + }, + { + "type": "equation", + "img_path": "images/5158844f36ece0d61b984d322032309d4dc0838fbe289512a20e2c6855a89892.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\left. B ^ { - \\alpha } \\Gamma ( \\alpha ) \\frac { 1 } { d t } { \\cal M } _ { \\mathrm { v } } ^ { \\mathrm { i } } V ( \\phi _ { \\mathrm { c o s s } } ^ { \\mathrm { o u p } } ( x ^ { \\mathrm { i } } \\mathrm { f r a l } ^ { \\mathrm { o } ^ { T } } ) \\biggr | _ { \\mathrm { l o s s } } ^ { 2 } \\right. } \\\\ & { = \\left. \\overline { { B ^ { - \\alpha } \\Gamma ( \\alpha ) } } \\frac { d } { d t } { \\cal V } _ { \\mathrm { a } } \\varepsilon ( \\mathrm { P } _ { \\mathrm { A L } } ^ { \\mathrm { i } } \\mathrm { P } _ { \\mathrm { E } } ^ { \\mathrm { o u p } } ) \\right| _ { \\mathrm { t e s o } } } \\\\ & { = \\int _ { 0 } ^ { \\infty } \\mathrm { d } s s ^ { - 1 } \\left[ \\frac { d } { d t } \\mathrm { d } { \\bf f } \\left( I + 2 s \\Sigma _ { 3 2 } \\right) ^ { - 1 / 2 } \\right] _ { \\mathrm { c o s s } } ^ { - 1 / 2 } \\biggr | _ { \\mathrm { v e l o } } \\times { \\cal V } _ { \\mathrm { s g } } \\left( \\beta \\Sigma _ { 2 0 } \\left( I + 2 s \\Sigma _ { 2 0 } \\right) ^ { - 1 } \\mathrm { e } ^ { T } \\right) } \\\\ & { \\qquad + \\operatorname { d } \\mathrm { e t } ( I + 2 s \\Sigma _ { 3 2 } ) ^ { - 1 / 2 } \\frac { d } { d t } { \\cal V } _ { \\mathrm { v e l o } } \\left( \\phi \\Sigma _ { 3 2 } \\left( I + 2 s \\Sigma _ { 3 2 } \\right) ^ { - 1 } \\mathrm { e } ^ { T } \\right) \\biggr | _ { \\mathrm { l o s s } } \\biggr ] } \\\\ & { = \\int _ { 0 } ^ { \\infty } \\mathrm { d } s s ^ { - 1 } \\left[ - \\frac { 3 } { 1 + 2 s \\beta \\beta \\sigma ^ { 4 } } \\operatorname { d i f } \\left( I + 2 s \\Sigma _ { 3 2 } \\right) ^ { - 1 / 2 } \\operatorname { d i f } \\left( \\tilde { \\Sigma } _ { \\mathrm { f o l } } \\right) \\times \\left( \\frac { \\beta ^ { \\prime } } { 1 + 2 s \\beta \\sigma ^ { 4 } } \\right) ^ { \\alpha } \\nabla _ { \\mathrm { s g } } \\left( G \\right) \\right. } \\\\ & { \\qquad \\left. + \\operatorname { d i f } \\left( I + 2 s \\Sigma _ { 1 0 } \\right) ^ { - 1 / 2 } \\frac { d } { \\mathrm { d } s \\sigma } \\mathrm { d } t \\beta \\left( \\Sigma \\right) \\right] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 117, + 769, + 369 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "(apply Lemma F.26 and Lemma F.27) ", + "bbox": [ + 562, + 372, + 813, + 387 + ], + "page_idx": 62 + }, + { + "type": "equation", + "img_path": "images/016c32edc10a9b2329af79ce8be7a42444fbe4d95a502a2632ccd79ee0848121.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle = \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { \\alpha - 1 } \\bigg [ - \\frac { s } { 1 + 2 s \\mu ^ { * } } ( 1 + 2 s \\mu ^ { * } ) ^ { - ( B - 1 ) / 2 } \\mathrm { t r } \\left( \\dot { \\Sigma } _ { \\bigstar \\mathrm { D 0 } } \\right) \\times \\left( \\frac { \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } \\right) ^ { \\alpha } \\mathrm { V } _ { \\phi } ( G ) } } \\\\ { { \\displaystyle \\qquad + \\left( 1 + 2 s \\mu ^ { * } \\right) ^ { - ( B - 1 ) / 2 } \\left( \\frac { \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } \\right) ^ { \\alpha - 1 } \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\bigg \\vert _ { \\Sigma = G } \\left\\{ ( 1 + 2 s \\mu ^ { * } ) ^ { - 2 } \\mathrm { e } \\dot { \\Sigma } _ { \\big \\mathrm { D 0 } } \\mathrm { e } ^ { T } \\right\\} \\bigg ] } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 187, + 390, + 813, + 465 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "(using fact that $\\mathrm { d } \\mathrm { V } _ { \\phi } / \\mathrm { d } \\Sigma$ is degree $( \\alpha - 1 )$ -positive homogeneous) ", + "bbox": [ + 375, + 465, + 813, + 481 + ], + "page_idx": 62 + }, + { + "type": "equation", + "img_path": "images/0181db7ab92eba941cee028540f125d54f2b2c6ce3a88e99cc0ea3d5f0505e87.jpg", + "text": "$$\n\\begin{array} { r l } & { = - \\left( \\displaystyle \\int _ { 0 } ^ { \\infty } ( 1 + 2 s \\mu ^ { * } ) ^ { - \\frac { B - 1 } { 2 } - 1 - \\alpha } \\mu ^ { * \\alpha } s ^ { \\alpha } \\right) \\mathrm { t r } \\left( \\dot { \\Sigma } _ { \\bigtriangledown \\mathrm { D } 0 } \\right) { \\bf V } _ { \\phi } ( G ) } \\\\ & { \\quad + \\left( \\displaystyle \\int _ { 0 } ^ { \\infty } ( 1 + 2 s \\mu ^ { * } ) ^ { - \\frac { B - 1 } { 2 } - 1 - \\alpha } \\mu ^ { * \\alpha - 1 } s ^ { \\alpha - 1 } \\right) \\frac { \\mathrm { d } { \\bf V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\biggr | _ { \\Sigma = G } \\left\\{ \\mathrm { e } \\dot { \\Sigma } _ { \\bigtriangledown \\mathrm { D } 0 } \\mathrm { e } ^ { T } \\right\\} } \\\\ & { = - \\mu ^ { * - 1 } 2 ^ { - 1 - \\alpha } \\mathrm { B e t a } \\left( \\frac { B - 1 } { 2 } , \\alpha + 1 \\right) \\mathrm { t r } \\left( \\dot { \\Sigma } _ { \\bigtriangledown 0 } \\right) { \\bf V } _ { \\phi } ( G ) } \\\\ & { \\quad + \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { B e t a } \\left( \\frac { B + 1 } { 2 } , \\alpha \\right) \\frac { \\mathrm { d } { \\bf V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\biggr | _ { \\Sigma = G } \\left\\{ \\dot { \\Sigma } _ { 0 } \\right\\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 186, + 484, + 689, + 626 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "(apply Lemma F.28) ", + "bbox": [ + 676, + 626, + 813, + 642 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "if $\\frac { B - 1 } { 2 } + 2 > \\alpha$ (precondition for Lemma F.28). ", + "bbox": [ + 173, + 647, + 493, + 664 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "With some trivial simplifications, we obtain the following ", + "bbox": [ + 173, + 669, + 553, + 684 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "Lemma F.25. Let $\\phi$ be positive-homogeneous of degree α. Consider a smooth path $\\Sigma _ { t } \\in S _ { B } ^ { G }$ with $\\Sigma _ { 0 } = \\mu ^ { * } G$ . If $\\frac { B - 1 } { 2 } + 2 > \\alpha$ , then ", + "bbox": [ + 176, + 686, + 823, + 718 + ], + "page_idx": 62 + }, + { + "type": "equation", + "img_path": "images/e90a87fe229ebe7925b63c6e709f63513e9006f0d055b0c08592bfb639325338.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\frac { d } { d t } \\nabla _ { ( \\phi \\circ \\mathbf { n } ) } ( \\Sigma _ { t } ) \\bigg | _ { t = 0 } } \\\\ & { = \\frac { \\mathrm { d } \\mathrm { V } _ { ( \\phi \\circ \\mathbf { n } ) } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\bigg | _ { \\Sigma = \\mu ^ { * } G } \\{ \\dot { \\Sigma } _ { 0 } \\} } \\\\ & { = - \\alpha B ^ { \\alpha } \\mu ^ { * - 1 } 2 ^ { - 1 - \\alpha } \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\alpha + 1 \\right) ^ { - 1 } \\mathrm { t r } \\left( \\dot { \\Sigma } _ { \\Sigma 0 } \\right) \\mathrm { V } _ { \\phi } ( G ) } \\\\ & { \\quad + B ^ { \\alpha } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\bigg | _ { \\Sigma = G } \\left\\{ \\dot { \\Sigma } _ { 0 } \\right\\} } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 292, + 722, + 705, + 873 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "where $\\mathrm { P } \\left( a , b \\right) = \\Gamma ( a + b ) / \\Gamma ( a )$ is the Pochhammer symbol. ", + "bbox": [ + 173, + 873, + 570, + 890 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "Lemma F.26. For any $\\begin{array} { r } { s \\in \\mathbb { R } , \\frac { d } { d t } \\operatorname* { d e t } ( I + 2 s \\Sigma _ { \\Sigma } ) ^ { - 1 / 2 } = - s \\operatorname* { d e t } ( I + 2 s \\Sigma _ { \\Sigma } ) ^ { - 1 / 2 } \\operatorname { t r } ( ( I + 2 s \\Sigma _ { \\Sigma } ) ^ { - 1 / 2 } = 0 , 1 ) } \\end{array}$ $2 s \\Sigma _ { \\perp } ) ^ { - 1 } d \\Sigma _ { \\perp } / d t )$ . ", + "bbox": [ + 173, + 891, + 825, + 925 + ], + "page_idx": 62 + }, + { + "type": "text", + "text": "For Σ\u00030 = µI, this is also equal to − s1+2sµ $\\begin{array} { r l r } { \\mathrm { ~ } } & { { } } & \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm { ~ } \\mathrm ~ \\end{array}$ at $t = 0$ . ", + "bbox": [ + 174, + 102, + 728, + 122 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Proof. Straightforward computation. ", + "bbox": [ + 174, + 135, + 416, + 151 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Lemma F.27. For any $s \\in \\mathbb { R }$ , $\\begin{array} { r } { \\frac { d } { d t } \\Sigma _ { \\bigtriangledown } ( I + 2 s \\Sigma _ { \\bigtriangledown } ) ^ { - 1 } = ( I + 2 s \\Sigma _ { \\bigtriangledown } ) \\frac { d \\Sigma _ { \\bigtriangledown } } { d t } ( I + 2 s \\Sigma _ { \\bigtriangledown } ) ^ { - 1 } . } \\end{array}$ ", + "bbox": [ + 169, + 157, + 745, + 178 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Proof. Straightforward computation. ", + "bbox": [ + 174, + 190, + 418, + 205 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Lemma F.28. For $a > b + 1$ , $\\begin{array} { r } { \\int _ { 0 } ^ { \\infty } ( 1 + 2 s \\mu ) ^ { - a } s ^ { b } \\mathrm { d } s = ( 2 \\mu ) ^ { - 1 - b } \\operatorname { B e t a } ( b + 1 , a - 1 - b ) . } \\end{array}$ ", + "bbox": [ + 173, + 210, + 761, + 231 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Proof. Apply change of variables $\\begin{array} { r } { x = \\frac { 2 \\mu s } { 1 + 2 \\mu s } } \\end{array}$ . ", + "bbox": [ + 174, + 244, + 475, + 263 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "This immediately gives the following consequence. ", + "bbox": [ + 176, + 279, + 509, + 294 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Theorem F.29. Let $\\phi : \\mathbb { R } \\mathbb { R }$ be any function with finite first and second Gaussian moments. Then φ \f\fΣ=BSB1(a,b) : HGB → HGB has the following eigendecomposition: ", + "bbox": [ + 173, + 296, + 825, + 333 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "· $\\mathbb { R } G$ has eigenvalu e (B−1)(u−2v)+w (dim RG = 1) • $\\mathbb { M } _ { B }$ has eigenvalue $w$ $( \\dim \\mathbb { M } _ { B } = B ( B - 3 ) / 2 )$ • $\\mathbb { L } _ { B }$ has eigenvalue $\\frac { ( B - 2 ) ( u - 2 v ) + 2 w } { B } ( \\dim \\mathbb { L } _ { B } = B - 1 )$ ", + "bbox": [ + 215, + 343, + 602, + 415 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 424, + 217, + 438 + ], + "page_idx": 63 + }, + { + "type": "equation", + "img_path": "images/0e120bec69fcc4f8fe0a7d939f3d8c65f033a221c4c5363a570609c82f4c07b1.jpg", + "text": "$$\nu = \\left. \\frac { \\partial \\mathrm { V } _ { \\phi } ( \\Sigma ) _ { 1 1 } } { \\partial \\Sigma _ { 1 1 } } \\right. _ { \\Sigma = \\mathrm { B S B } 1 ( a , b ) } , \\quad v = \\left. \\frac { \\partial \\mathrm { V } _ { \\phi } ( \\Sigma ) _ { 1 2 } } { \\partial \\Sigma _ { 1 1 } } \\right. _ { \\Sigma = \\mathrm { B S B } 1 ( a , b ) } , \\quad w = \\left. \\frac { \\partial \\mathrm { V } _ { \\phi } ( \\Sigma ) _ { 1 2 } } { \\partial \\Sigma _ { 1 2 } } \\right. _ { \\Sigma = \\mathrm { B S B } 1 ( a , b ) } .\n$$", + "text_format": "latex", + "bbox": [ + 189, + 441, + 805, + 479 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Theorem F.30. Let $\\phi : \\mathbb { R } \\mathbb { R }$ be positive-homogeneous of degree $\\alpha$ . Then for any $p \\neq 0 , c \\in \\mathbb { R }$ , Σ=BSB1(p,cp) : HGB → HGB has the following eigendecomposition: ", + "bbox": [ + 173, + 483, + 825, + 520 + ], + "page_idx": 63 + }, + { + "type": "equation", + "img_path": "images/d400e22da4a6a095cf92ae74952850349716e17cb7371c307b1b99f04049aa5e.jpg", + "text": "$$\n\\begin{array} { c } { { \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\frac 1 B \\left( ( B - 2 ) \\alpha \\left[ { \\bf J } _ { \\phi } ( 1 ) - { \\bf J } _ { \\phi } \\left( c \\right) \\right] + \\left( 2 + c ( B - 2 ) \\right) { \\bf J } _ { \\phi } ^ { \\prime } \\left( c \\right) \\right) } } \\\\ { { \\mathrm { , ~ } } } \\\\ { { \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\frac 1 B \\left( ( B - 1 ) \\alpha \\left[ { \\bf J } _ { \\phi } ( 1 ) - { \\bf J } _ { \\phi } \\left( c \\right) \\right] + \\left( 1 + c ( B - 1 ) \\right) { \\bf J } _ { \\phi } ^ { \\prime } \\left( c \\right) \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 357, + 556, + 787, + 618 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "Proof. By Proposition E.22, $\\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } } { \\mathrm { d } \\Sigma } \\Big | _ { \\Sigma = \\mathrm { B S B } 1 ( p , c p ) }$ is $\\mathrm { D O S } ( u , v , w )$ with ", + "bbox": [ + 173, + 631, + 629, + 654 + ], + "page_idx": 63 + }, + { + "type": "equation", + "img_path": "images/833e3a87bb00547e9d6636be777679b860592b62a3dcd40ddda141e741da4383.jpg", + "text": "$$\n\\begin{array} { r l } & { \\boldsymbol { u } = \\frac { \\partial N _ { \\phi } ( E ( \\Sigma ) ) _ { 1 1 } } { \\partial \\Sigma _ { 1 1 } } \\Bigg | _ { \\{ \\Sigma = \\mathrm { R } \\ S \\} \\cup \\{ \\rho _ { \\phi } \\} } } \\\\ & { \\quad = c _ { \\phi , \\Phi } p ^ { i - 1 } \\dot { \\Sigma } _ { \\phi , \\Phi } ( 1 ) } \\\\ & { \\quad = \\frac { \\partial N _ { \\phi } ( \\Sigma ) _ { 1 2 } } { \\partial \\Sigma _ { 1 1 } } \\Bigg | _ { \\{ \\Sigma = \\mathrm { R } \\ S \\mathrm { M } \\{ \\rho _ { \\phi } \\} } } \\\\ & { \\quad = \\frac { 1 } { 2 } c _ { \\phi , \\Phi } \\Sigma _ { i \\alpha } ^ { ( i \\alpha - 2 ) } \\Sigma _ { \\Delta \\phi } ^ { \\mathrm { t } _ { \\alpha } } ( \\Omega _ { \\phi } \\langle c _ { \\phi } \\rangle - c _ { \\phi , \\Phi } J _ { \\phi } ^ { \\prime } ( c _ { \\phi } ) \\Bigg ) } \\\\ & { \\quad = \\frac { 1 } { 2 } c _ { \\phi , \\Phi } p ^ { i - 1 } \\left( \\Omega _ { \\phi } c _ { \\phi } ( \\Sigma ) - c _ { \\phi } ^ { \\dagger } \\right) ^ { \\alpha } } \\\\ & { \\quad = \\frac { \\partial N _ { \\phi } ( \\Sigma ) _ { 1 2 } } { \\partial \\Sigma _ { 1 2 } } \\Bigg | _ { \\{ \\Sigma = \\mathrm { R } \\ S \\mathrm { M } \\{ \\rho _ { \\phi } \\} } } \\\\ & { \\quad = c _ { \\phi , \\Phi } \\Sigma _ { i \\alpha } ^ { ( i \\alpha - 1 ) / 2 } \\Sigma _ { \\mathrm { a r } } ^ { ( i \\alpha - 1 ) / 2 } \\Sigma _ { \\phi } ^ { \\mathrm { t } _ { \\alpha } } ( c _ { \\phi } ) } \\\\ & { \\quad = c _ { \\phi , \\Phi } \\pi ^ { ( i \\alpha - 1 ) / 2 } \\Sigma _ { 1 \\phi } ^ { ( i \\alpha - 1 ) / 2 } \\Sigma _ { \\phi } ^ { \\mathrm { t } _ { \\alpha } } ( c _ { \\phi } ) } \\\\ & { \\quad = c _ { \\phi , \\Phi } \\pi ^ { ( i \\alpha - 1 ) / 2 } \\Sigma _ { 1 \\phi } ^ { ( i \\alpha - 1 ) / 2 } \\Gamma _ { \\phi } ^ { ( i \\alpha ) } ( c _ { \\phi } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 660, + 655, + 901 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "With $\\cdot$ , we can do the computation: ", + "bbox": [ + 174, + 909, + 465, + 924 + ], + "page_idx": 63 + }, + { + "type": "text", + "text": "• $\\mathbb { M } _ { B }$ has eigenvalue $w = \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\mathsf { J } _ { \\phi } ^ { \\prime } \\left( c \\right)$ ", + "bbox": [ + 217, + 102, + 490, + 121 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "• $\\mathbb { L } _ { B }$ has eigenvalue ", + "bbox": [ + 215, + 128, + 357, + 143 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/2882c7cca66706c6b5e5e1bba18cd44be92f1083b9707eac8671ec9344530791.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\frac { ( u - 2 v ) ( B - 2 ) + 2 w } { B } } \\ ~ } \\\\ { { \\displaystyle = \\frac { 1 } { B } \\left( ( B - 2 ) \\left( \\mathsf { c } _ { \\alpha } \\alpha p ^ { \\alpha - 1 } \\mathbf { J } _ { \\phi } ( 1 ) - \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\left( \\alpha \\mathbf { J } _ { \\phi } \\left( c \\right) - c \\mathbf { J } _ { \\phi } ^ { \\prime } \\left( c \\right) \\right) \\right) + 2 \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\mathbf { J } _ { \\phi } ^ { \\prime } \\left( c \\right) \\right) } \\ ~ } \\\\ { { \\displaystyle = \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\frac { 1 } { B } \\left( ( B - 2 ) \\alpha \\left[ \\mathsf { J } _ { \\phi } ( 1 ) - \\mathsf { J } _ { \\phi } \\left( c \\right) \\right] + ( 2 + c ( B - 2 ) ) \\mathbf { J } _ { \\phi } ^ { \\prime } \\left( c \\right) \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 147, + 808, + 243 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "• $\\mathbb { R } G$ has eigenvalue ", + "bbox": [ + 215, + 272, + 361, + 287 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/4d2001216ed8cefb40efa6df839899dff4d07d8a0bd6f9714e01011c517eb4a3.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\frac { ( B - 1 ) ( u - 2 v ) + w } { B } } \\ ~ } \\\\ { { \\displaystyle = \\frac { 1 } { B } \\left( ( B - 1 ) \\left( \\mathsf { c } _ { \\alpha } \\alpha p ^ { \\alpha - 1 } \\mathbf { J } _ { \\phi } ( 1 ) - \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\left( \\alpha \\mathbf { J } _ { \\phi } \\left( c \\right) - c \\mathbf { J } _ { \\phi } ^ { \\prime } \\left( c \\right) \\right) \\right) + \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\mathbf { J } _ { \\phi } ^ { \\prime } \\left( c \\right) \\right) } \\ ~ } \\\\ { { \\displaystyle = \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\frac { 1 } { B } \\left( ( B - 1 ) \\alpha \\left[ \\mathbf { J } _ { \\phi } ( 1 ) - \\mathbf { J } _ { \\phi } \\left( c \\right) \\right] + \\left( 1 + c ( B - 1 ) \\right) \\mathbf { J } _ { \\phi } ^ { \\prime } \\left( c \\right) \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 258, + 291, + 802, + 387 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "We record the following consequence which will be used frequently in the sequel. ", + "bbox": [ + 173, + 428, + 710, + 444 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Theorem F.31. Let $\\phi : \\mathbb { R } \\mathbb { R }$ be positive-homogeneous of degree $\\alpha$ . Then $G ^ { \\otimes 2 } \\circ \\left. \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } } { \\mathrm { d } \\Sigma } \\right| _ { \\Sigma = G }$ $\\mathcal { H } _ { B } ^ { G } \\to \\mathcal { H } _ { B } ^ { G }$ has the following eigendecomposition: ", + "bbox": [ + 174, + 448, + 823, + 482 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/d73db303e52cd342a84d0b8c6ac3ffb3fc14e2cd16c030e80f74b9642ecb2ee5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\iota e \\lambda _ { \\mathbb { M } } ^ { G , \\phi } ( B , \\alpha ) : = \\mathsf { c } _ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha - 1 } \\operatorname { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) } \\\\ & { } \\\\ & { \\mathrm { \\ } ^ { \\beta \\lambda _ { \\mathbb { L } } ^ { G , \\phi } } ( B , \\alpha ) : = \\mathsf { c } _ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha - 1 } \\frac { 1 } { B } \\left( ( B - 2 ) \\alpha \\left[ \\operatorname { J } _ { \\phi } ( 1 ) - \\operatorname { J } _ { \\phi } \\left( \\frac { - 1 } { B - 1 } \\right) \\right] + \\frac { B } { B - 1 } \\operatorname { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } \\\\ & { } \\\\ & { e \\lambda _ { G } ^ { G , \\phi } ( B , \\alpha ) : = \\mathsf { c } _ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha } \\alpha \\left[ \\mathsf { J } _ { \\phi } ( 1 ) - \\operatorname { J } _ { \\phi } \\left( \\frac { - 1 } { B - 1 } \\right) \\right] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 351, + 489, + 923, + 590 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Proof. Plug in $\\begin{array} { r } { p = \\frac { B - 1 } { B } } \\end{array}$ and $c = - 1 / ( B - 1 )$ for Thm F.30. ", + "bbox": [ + 173, + 601, + 573, + 619 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Theorem F.32. Let $\\phi : \\mathbb { R } \\mathbb { R }$ be a degree $\\alpha$ positive-homogeneous function. Then for $p \\neq 0 , c \\in$ $\\begin{array} { r } { \\mathbb { R } , \\ \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } } { \\mathrm { d } \\Sigma } \\Big | _ { \\Sigma = \\mathrm { B S B } 1 ( p , c p ) } : \\mathcal { H } _ { B } \\to \\mathcal { H } _ { B } } \\end{array}$ has the following eigendecomposition: ", + "bbox": [ + 171, + 625, + 825, + 660 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "• ${ \\overline { { \\mathbb { M } } } } _ { B }$ has eigenvalue ${ \\mathsf { c } } _ { \\alpha } p ^ { \\alpha - 1 } \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( c \\right)$ \n• $\\overline { { \\mathbb { L } } } _ { B } ( a , b )$ has eigenvalue $\\mathsf { c } _ { \\alpha } \\alpha p ^ { \\alpha - 1 } \\mathrm { J } _ { \\phi } ( 1 )$ where $a \\ = \\ \\mathrm { 2 } \\left( \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( c \\right) - \\alpha \\mathrm { J } _ { \\phi } ( 1 ) \\right)$ and $\\begin{array} { r l } { b } & { { } = } \\end{array}$ $\\alpha \\mathrm { J } _ { \\phi } \\left( c \\right) - c \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( c \\right)$ ", + "bbox": [ + 215, + 671, + 825, + 742 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Proof. Use Thm E.72 with the computations from the proof of $\\mathrm { T h m } \\ \\mathrm { F } . 3 0$ as well as the following computation ", + "bbox": [ + 171, + 755, + 823, + 785 + ], + "page_idx": 64 + }, + { + "type": "equation", + "img_path": "images/cf0ca0a534c40a38257deb302a5586b5c7b1e8d9b4602f28e33b332732fa506e.jpg", + "text": "$$\n\\begin{array} { c } { { w - u = \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\mathsf { J } _ { \\phi } ^ { \\prime } \\left( c \\right) - \\mathsf { c } _ { \\alpha } \\alpha p ^ { \\alpha - 1 } \\mathsf { J } _ { \\phi } ( 1 ) } } \\\\ { { \\displaystyle = \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\left( \\mathsf { J } _ { \\phi } ^ { \\prime } \\left( c \\right) - \\alpha \\mathsf { J } _ { \\phi } ( 1 ) \\right) } } \\\\ { { \\displaystyle v = \\frac { 1 } { 2 } \\mathsf { c } _ { \\alpha } p ^ { \\alpha - 1 } \\left( \\alpha \\mathsf { J } _ { \\phi } \\left( c \\right) - c \\mathsf { J } _ { \\phi } ^ { \\prime } \\left( c \\right) \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 361, + 787, + 638, + 862 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "Theorem F.33. Let $\\phi$ be positive-homogeneous with degree $\\alpha$ . Assume $\\textstyle { \\frac { B - 1 } { 2 } } > \\alpha$ . The operator $\\begin{array} { r } { \\mathcal { U } = G ^ { \\otimes 2 } \\circ \\frac { \\mathrm { d } \\mathbf { V } _ { ( \\phi \\circ \\mathfrak { n } ) } \\left( \\Sigma ^ { G } \\right) } { \\mathrm { d } \\Sigma ^ { G } } \\big | _ { \\Sigma ^ { G } = \\mu ^ { * } G } : \\mathcal { H } _ { B } ^ { G } \\to \\mathcal { H } _ { B } ^ { G } } \\end{array}$ has 3 distinct eigenvalues. They are as follows: ", + "bbox": [ + 173, + 886, + 826, + 928 + ], + "page_idx": 64 + }, + { + "type": "text", + "text": "1. $\\lambda _ { G } ^ { \\uparrow } ( B , \\alpha ) : = 0$ with dimension 1 eigenspace $\\mathbb { R } G$ . ", + "bbox": [ + 212, + 102, + 558, + 119 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "2. ", + "bbox": [ + 210, + 128, + 227, + 141 + ], + "page_idx": 65 + }, + { + "type": "equation", + "img_path": "images/da448001e39ae7cffdb77b6a2f724e31f37b0249516ec40a8cff8f8a49c123af.jpg", + "text": "$$\n\\begin{array} { r l } & { \\lambda _ { \\mathtt { M } } ^ { \\uparrow } ( B , \\alpha ) : = B ^ { \\alpha } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } \\lambda _ { \\mathtt { M } } ^ { G , \\phi } ( B , \\alpha ) } \\\\ & { \\quad \\quad \\quad = \\mathtt { c } _ { \\alpha } B ( B - 1 ) ^ { \\alpha - 1 } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 146, + 764, + 222 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "with dimension $\\frac { B ( B - 3 ) } { 2 }$ eigenspace M. ", + "bbox": [ + 232, + 227, + 488, + 248 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "3. ", + "bbox": [ + 210, + 256, + 227, + 268 + ], + "page_idx": 65 + }, + { + "type": "equation", + "img_path": "images/eeb22a829a0b534b0c129abe2bef3816a69971f4069a2c69aa0bb45a7e0da594.jpg", + "text": "$$\n\\begin{array} { r l } & { \\lambda _ { \\mathbb { L } } ^ { \\uparrow } ( B , \\alpha ) : = B ^ { \\alpha } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\displaystyle \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } \\lambda _ { \\mathbb { L } } ^ { G , \\phi } ( B , \\alpha ) } \\\\ & { \\quad \\quad \\quad = \\mathsf { c } _ { \\alpha } ( B - 1 ) ^ { \\alpha - 1 } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\displaystyle \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\left( ( B - 2 ) \\alpha \\left[ \\mathrm { J } _ { \\phi } ( 1 ) - \\mathrm { J } _ { \\phi } \\left( \\displaystyle \\frac { - 1 } { B - 1 } \\right) \\right] + \\displaystyle \\frac { B } { B - 1 } \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( \\displaystyle \\frac { - 1 } { B - 1 } \\right) \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 273, + 784, + 386 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "with dimension $B - 1$ eigenspace $\\mathbb { L }$ . ", + "bbox": [ + 232, + 390, + 473, + 405 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "Proof. Item 1. The case of $\\lambda _ { G } ^ { \\uparrow } ( B , \\alpha )$ . Since $( \\phi \\circ \\mathsf { n } ) ( \\mu ^ { * } G ) = ( \\phi \\circ \\mathsf { n } ) ( \\mu ^ { * } G + \\omega G )$ for any $\\omega$ , the operator $\\frac { \\mathrm { d } \\mathrm { V } _ { ( \\phi \\circ \\mathsf { n } ) } \\left( \\Sigma ^ { G } \\right) } { \\mathrm { d } \\Sigma ^ { G } } \\Big | _ { \\Sigma ^ { G } = \\mu ^ { * } G }$ ΣG=µ∗G sends G to 0. ", + "bbox": [ + 174, + 420, + 825, + 462 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "Item 2. The case of $\\lambda _ { \\mathbb { M } } ^ { \\uparrow } ( B , \\alpha )$ . Assume $\\dot { \\Sigma } _ { 0 } \\in \\mathbb { M }$ , Thm F.31 gives (with $\\odot$ denoting Hadamard product, i.e. entrywise multiplication) ", + "bbox": [ + 173, + 468, + 823, + 500 + ], + "page_idx": 65 + }, + { + "type": "equation", + "img_path": "images/5109ef9539cd8194cd91b92e5b66f921c5cca3e73d584cf9cd2b1a4a12353db0.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\left. { \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } } \\right| _ { \\Sigma = G } \\{ \\dot { \\Sigma } _ { 0 } \\} = \\lambda _ { \\mathrm { M } } ^ { G , \\phi } ( B , \\alpha ) \\dot { \\Sigma } _ { 0 } } } \\\\ { { \\displaystyle \\qquad = \\mathsf { c } _ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha - 1 } \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) \\dot { \\Sigma } _ { 0 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 503, + 691, + 580 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "Since $\\mathrm { t r } ( \\dot { \\Sigma } _ { 0 } ) = 0$ , Eq. (50) gives ", + "bbox": [ + 173, + 593, + 392, + 609 + ], + "page_idx": 65 + }, + { + "type": "equation", + "img_path": "images/cfc4d3ca71188802c0ac20fbd0682d2ed86a1fc8f7b2386c09bda030ded33e56.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad G ^ { \\otimes 2 } \\simeq \\frac { \\mathrm { d } \\mathrm { V } ( \\phi \\circ \\mathrm { e n } ) ( \\Sigma ^ { G } ) } { \\mathrm { d } \\Sigma ^ { G } } \\Big | _ { \\Sigma ^ { G } = \\mu ^ { - } } \\{ \\dot { \\Sigma } _ { 0 } \\} } \\\\ & { = G ^ { \\otimes 2 } \\circ \\frac { d } { d t } \\mathrm { V } _ { ( \\phi \\circ \\mathrm { e n } ) } ( \\Sigma _ { t } ) \\Big | _ { t = 0 } ^ { \\Sigma } ; } \\\\ & { = B ^ { \\alpha } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } G ^ { \\otimes 2 } \\circ \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\Big | _ { \\Sigma = G } \\left\\{ \\dot { \\Sigma } _ { 0 } \\right\\} } \\\\ & { = B ^ { \\alpha } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } \\mathrm { c } _ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha - 1 } \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) G ^ { \\otimes 2 } \\{ \\dot { \\Sigma } _ { 0 } \\} } \\\\ & { = c _ { \\alpha } B ( B - 1 ) ^ { \\alpha - 1 } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) \\dot { \\Sigma } _ { 0 } } \\\\ & { = c _ { \\alpha } B ( B - 1 ) ^ { \\alpha - 1 } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) \\dot { \\Sigma } _ { 0 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 248, + 616, + 751, + 806 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "So the eigenvalue for $\\mathbb { M }$ is $\\begin{array} { r } { \\lambda _ { \\mathbb { L } } ^ { \\uparrow } = \\mathsf { c } _ { \\alpha } B ( B - 1 ) ^ { \\alpha - 1 } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) . } \\end{array}$ ", + "bbox": [ + 173, + 829, + 736, + 856 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "Item 3. The case of $\\lambda _ { \\mathbb { L } } ^ { \\uparrow } ( B , \\alpha )$ . Let $\\dot { \\Sigma } _ { 0 } = \\mathrm { L } ( B - 2 , 1 )$ . Thm F.31 gives ", + "bbox": [ + 173, + 862, + 642, + 881 + ], + "page_idx": 65 + }, + { + "type": "equation", + "img_path": "images/203f73028c7e7b22e172056ee709ae69f201b369baadfff7343f115dcd43af01.jpg", + "text": "$$\nG ^ { \\otimes 2 } \\circ \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\bigg | _ { \\Sigma = G } \\{ \\dot { \\Sigma } _ { 0 } \\} = \\lambda _ { \\mathbb { L } } ^ { G , \\phi } ( B , \\alpha ) \\dot { \\Sigma } _ { 0 }\n$$", + "text_format": "latex", + "bbox": [ + 351, + 887, + 647, + 922 + ], + "page_idx": 65 + }, + { + "type": "text", + "text": "where ", + "bbox": [ + 173, + 103, + 217, + 117 + ], + "page_idx": 66 + }, + { + "type": "equation", + "img_path": "images/4aae06e55acbd8b83bd8e4345fc00f1c29b4cf1ac0dd4b0db221642bcd3fda56.jpg", + "text": "$$\n\\textsc { \\det { \\detokenize { \\left( B , \\alpha \\right) } } } : = \\frac { 1 } { B } \\mathfrak { c } _ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha - 1 } \\left( ( B - 2 ) \\alpha \\left[ \\mathbf { J } _ { \\phi } ( 1 ) - \\mathbf { J } _ { \\phi } \\left( \\frac { - 1 } { B - 1 } \\right) \\right] + \\frac { B } { B - 1 } \\mathbf { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 121, + 826, + 160 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "Since $\\mathrm { t r } ( \\dot { \\Sigma } _ { 0 } ) = 0$ , Eq. (50) gives ", + "bbox": [ + 173, + 172, + 392, + 190 + ], + "page_idx": 66 + }, + { + "type": "equation", + "img_path": "images/c042e01ab5189325da6400cc21fbf3d723625d2fc2a39ed1c75535f713add312.jpg", + "text": "$$\n\\begin{array} { r l } & { \\bar { C } ^ { \\mathrm { S } ^ { 2 } } \\lesssim \\frac { \\mathrm { d } \\nabla \\langle \\phi \\phi \\rangle \\big ( \\frac { 2 \\xi } { 2 } \\big ) } { \\mathrm { d } \\xi \\mathrm { d } \\xi ^ { \\prime } } \\Big | _ { \\mathrm { c o l } \\sim \\phi } \\xi \\big | _ { \\mathrm { c o l } \\sim \\phi } \\xi \\big | _ { \\mathrm { c o l } } ^ { \\xi } } \\\\ & { = G ^ { \\mathrm { S } ^ { 2 } } \\lesssim \\frac { d } { d \\xi } \\sqrt { \\kappa ( \\mathrm { s m } / \\xi ) } \\Big | _ { \\mathrm { r e c h } } } \\\\ & { = B ^ { \\nu } \\mu ^ { \\epsilon - 1 } 2 ^ { - \\alpha } \\mathrm { P r } ( \\frac { B + 1 } { 2 } , \\alpha ) ^ { - 1 } G ^ { \\mathrm { C } ^ { 2 } } \\lesssim \\frac { \\mathrm { d } \\nabla _ { \\phi } ( \\frac { 2 \\xi } { 2 } ) } { \\mathrm { d } \\xi } \\Big | _ { \\mathrm { s } = \\epsilon } \\big \\{ \\dot { \\Sigma } \\big \\} } \\\\ & { = \\bar { B } ^ { \\nu } \\mu ^ { \\epsilon - 1 } 2 ^ { - \\alpha } \\mathrm { P r } ( \\frac { B + 1 } { 2 } , \\alpha ) ^ { - 1 } \\bar { \\lambda } ^ { \\frac { \\epsilon } { 2 } } \\mathrm { o } } \\\\ & { = ( B - 1 ) ^ { \\alpha - 1 } \\mathrm { c } _ { \\phi } \\mu ^ { \\epsilon - 1 } 2 ^ { - \\alpha } \\mathrm { P r } ( \\frac { B + 1 } { 2 } , \\alpha ) ^ { - 1 } } \\\\ & { \\quad \\times ( ( B - 2 ) \\alpha \\Big [ \\mathbb { L } _ { \\phi } ( 1 ) - \\mathrm { ] } _ { \\phi } ( \\frac { - 1 } { B - 1 } ) ] + \\frac { B } { B - 1 } \\big \\} _ { \\phi } ^ { \\prime } ( \\frac { - 1 } { B - 1 } ) ) } \\\\ & { \\quad \\times \\hat { \\Sigma } _ { \\mathrm { s p } } \\bigg [ \\mathrm { c m } \\Big [ \\mathrm { L } _ { \\phi } ( 1 ) - \\mathrm { ] } _ { \\phi } \\Big ] } \\\\ & { \\quad \\times \\hat { \\Sigma } _ { \\mathrm { s p } } ^ { \\epsilon } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 243, + 195, + 754, + 440 + ], + "page_idx": 66 + }, + { + "type": "equation", + "img_path": "images/c02a755bf86c3e0460ed376b6d8c4ecd4923fc0b8660d4e1fdf33fc21b13621c.jpg", + "text": "$$\n\\begin{array} { r } { \\textnormal { o } \\lambda _ { \\mathrm { L } } ^ { \\uparrow } = ( B - 1 ) ^ { \\alpha - 1 } \\mathfrak { c } _ { \\alpha } \\mu ^ { * - 1 } 2 ^ { - \\alpha } \\mathrm { P } \\left( \\frac { B + 1 } { 2 } , \\alpha \\right) ^ { - 1 } \\left( ( B - 2 ) \\alpha \\left[ \\mathrm { J } _ { \\phi } ( 1 ) - \\mathrm { J } _ { \\phi } \\left( \\frac { - 1 } { B - 1 } \\right) \\right] + \\frac { B } { B - 1 } \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } \\\\ { \\prod } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 184, + 452, + 834, + 487 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "With some routine computation, we obtain ", + "bbox": [ + 176, + 507, + 455, + 521 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "Proposition F.34. With $\\phi$ and $\\alpha$ as above, as $B \\infty$ , ", + "bbox": [ + 173, + 525, + 535, + 540 + ], + "page_idx": 66 + }, + { + "type": "equation", + "img_path": "images/a8c473c8045d633de9e28e4eafaa5a68e9ab2a301bdbadf7de1cc4fb63e71ef6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\lambda _ { \\mathtt { M } } ^ { \\mathtt { T } } ( B , \\alpha ) \\sim \\alpha + B ^ { - 1 } ( 2 \\alpha ^ { 3 } - 4 \\alpha ^ { 2 } - \\alpha + \\frac { \\mathtt { J } _ { \\phi } ^ { \\prime } ( 0 ) } { \\mathtt { J } _ { \\phi } ( 1 ) - \\mathtt { J } _ { \\phi } ( 0 ) } ) + O ( B ^ { - 2 } ) } \\\\ & { \\overset { \\mathtt { J } ^ { \\prime } } { \\underset { \\mathrm { M } } { = } } ( B , \\alpha ) \\sim \\frac { \\mathtt { J } _ { \\phi } ^ { \\prime } ( 0 ) } { \\mathtt { J } _ { \\phi } ( 1 ) - \\mathtt { J } _ { \\phi } ( 0 ) } } \\\\ & { \\quad \\quad \\ + B ^ { - 1 } \\left( ( 2 \\alpha ^ { 2 } - 4 \\alpha + 1 ) \\frac { \\mathtt { J } _ { \\phi } ^ { \\prime } ( 0 ) } { \\mathtt { J } _ { \\phi } ( 1 ) - \\mathtt { J } _ { \\phi } ( 0 ) } - \\frac { \\mathtt { J } _ { \\phi } ^ { \\prime \\prime } ( 0 ) } { \\mathtt { J } _ { \\phi } ( 1 ) - \\mathtt { J } _ { \\phi } ( 0 ) } - \\left( \\frac { \\mathtt { J } _ { \\phi } ^ { \\prime } ( 0 ) } { \\mathtt { J } _ { \\phi } ( 1 ) - \\mathtt { J } _ { \\phi } ( 0 ) } \\right) ^ { 2 } \\right) } \\\\ & { \\quad \\ \\quad + O \\left( B ^ { - 2 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 545, + 828, + 684 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "We can compute, by Proposition E.23, ", + "bbox": [ + 174, + 694, + 423, + 709 + ], + "page_idx": 66 + }, + { + "type": "equation", + "img_path": "images/88c565f890fb7d32cec8a84f7f084a633151d7c9296b7c7a3a2dd8cbf704530a.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\frac { \\mathrm { J } _ { \\phi } ^ { \\prime } ( 0 ) } { \\mathrm { J } _ { \\phi } ( 1 ) - \\mathrm { J } _ { \\phi } ( 0 ) } = \\frac { ( a + b ) ^ { 2 } \\frac { 1 } { \\sqrt { \\pi } } \\frac { \\Gamma ( \\frac { \\alpha } { 2 } + 1 ) ^ { 2 } } { \\Gamma ( \\alpha + \\frac { 1 } { 2 } ) } } { ( a ^ { 2 } + b ^ { 2 } ) - ( a - b ) ^ { 2 } \\frac { 1 } { 2 \\sqrt { \\pi } } \\frac { \\Gamma ( \\frac { \\alpha } { 2 } + \\frac { 1 } { 2 } ) ^ { 2 } } { \\Gamma ( \\alpha + \\frac { 1 } { 2 } ) } } } } \\\\ & { } & { \\leq \\frac { 2 \\Gamma ( \\frac { \\alpha } { 2 } + 1 ) ^ { 2 } } { \\sqrt { \\pi } \\Gamma ( \\alpha + \\frac { 1 } { 2 } ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 717, + 676, + 811 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "where the last part is obtained by optimizing over a and $b$ . On $\\alpha \\in ( - 1 / 2 , \\infty )$ , this is greater than $\\alpha$ iff $\\alpha < 1$ . Thus for $\\alpha \\geq 1$ , the maximum eigenvalue of $\\mathcal { U }$ is always achieved by eigenspace L for large enough $B$ . ", + "bbox": [ + 173, + 815, + 825, + 858 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "G BACKWARD DYNAMICS ", + "text_level": 1, + "bbox": [ + 174, + 877, + 406, + 895 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "Let’s extend the definition of the $\\mathrm { V }$ operator: ", + "bbox": [ + 174, + 909, + 468, + 924 + ], + "page_idx": 66 + }, + { + "type": "text", + "text": "Definition G.1. Suppose $T _ { x } : \\mathbb { R } ^ { n } \\mathbb { R } ^ { m }$ is a linear operator parametrized by $\\boldsymbol { x } \\in \\mathbb { R } ^ { k }$ . Then for a PSD matrix $\\Sigma \\in S _ { k }$ , define $\\mathrm { V } _ { T } ( \\Sigma ) : = \\mathbb { E } [ T _ { x } \\otimes T _ { x } : x \\in \\mathcal { N } ( 0 , \\Sigma ) ] : \\mathbb { R } ^ { n \\times n } \\to \\mathbb { R } ^ { m \\times m }$ , which acts on $n \\times n$ matrices $\\Pi$ by ", + "bbox": [ + 173, + 102, + 825, + 146 + ], + "page_idx": 67 + }, + { + "type": "equation", + "img_path": "images/3835f41f7dbe5dcc07ec0829e6c4b359f5d919a1e6eaf470f7073a98c0b35a61.jpg", + "text": "$$\n\\mathrm { V } _ { T } ( \\Sigma ) \\{ \\Pi \\} = \\mathbb { E } [ T _ { x } \\Pi T _ { x } ^ { T } : x \\in \\mathcal { N } ( 0 , \\Sigma ) ] \\in \\mathbb { R } ^ { m \\times m } .\n$$", + "text_format": "latex", + "bbox": [ + 326, + 151, + 669, + 170 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "Under this definition, $\\mathrm { V } _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma )$ is a linear operator $\\mathbb { R } ^ { B \\times B } \\xrightarrow { } \\mathbb { R } ^ { B \\times B }$ . Recall the notion of adjoint: ", + "bbox": [ + 174, + 180, + 805, + 199 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "Definition G.2. If $V$ is a (real) vector space, then $V ^ { \\dagger }$ , its dual space, is defined as the space of linear functionals $f : V \\to \\mathbb { R }$ on $V$ . If $T : V \\to W$ is a linear operator, then $T ^ { \\dagger }$ , its adjoint, is a linear operator $W ^ { \\dagger } V ^ { \\dagger }$ , defined by $T ^ { \\dagger } ( f ) = v \\mapsto f ( T ( v ) )$ . ", + "bbox": [ + 173, + 204, + 825, + 250 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "If a linear operator is represented by a matrix, with function application represented by matrixvector multiplication (matrix on the left, vector on the right), then the adjoint is represented by matrix transpose. ", + "bbox": [ + 173, + 258, + 825, + 303 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "The backward equation. In this section we are interested in the backward dynamics, given by the following equation ", + "bbox": [ + 173, + 315, + 825, + 345 + ], + "page_idx": 67 + }, + { + "type": "equation", + "img_path": "images/643da00557b9c17a5881bef4aba6d6c4196244b2fd733a88ea3e7048a8911ca5.jpg", + "text": "$$\n\\Pi ^ { l } = \\mathrm { V } _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ^ { l } ) ^ { \\dagger } \\{ \\Pi ^ { l + 1 } \\}\n$$", + "text_format": "latex", + "bbox": [ + 419, + 349, + 578, + 372 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "where $\\Sigma ^ { l }$ is given by the forward dynamics. Particularly, we are interested in the specific case when we have exponential convergence of $\\Sigma ^ { l }$ to a BSB1 fixed point. Thus we will study the asymptotic approximation of the above, namely the linear system. ", + "bbox": [ + 173, + 378, + 823, + 421 + ], + "page_idx": 67 + }, + { + "type": "equation", + "img_path": "images/7404894d2ac41546768c66b8800cb9724cbfc132778ae5dfe9cb3b917e04a270.jpg", + "text": "$$\n\\Pi ^ { l } = \\mathrm { V } _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ^ { * } ) ^ { \\dagger } \\{ \\Pi ^ { l + 1 } \\}\n$$", + "text_format": "latex", + "bbox": [ + 418, + 426, + 580, + 449 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "where $\\Sigma ^ { * }$ is the BSB1 fixed point. Note that after one step of backprop, $\\Pi ^ { L - 1 } = \\mathrm { V } _ { \\mathcal { B } _ { \\phi } ^ { \\prime } } ( \\Sigma ^ { * } ) ^ { \\dagger } \\{ \\Pi ^ { L } \\}$ is in $\\mathcal { H } _ { B } ^ { G }$ . Thus the large $L$ dynamics of Eq. (51) is given by the eigendecomposition of $\\mathrm { V } _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ^ { * } ) ^ { \\dag } \\ \\mathinner { \\uparrow }$ $\\mathcal { H } _ { B } ^ { G } : \\mathcal { H } _ { B } ^ { G } \\to \\mathcal { H } _ { B } ^ { G }$ . It turns out to be much more convenient to study its adjoint $G ^ { \\otimes 2 } \\circ \\mathrm { V } _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ^ { * } )$ , which has the same eigenvalues (in fact, it will turn out that it is self-adjoint). ", + "bbox": [ + 173, + 454, + 825, + 523 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "G.1 EIGENDECOMPOSITION OF $G ^ { \\otimes 2 } \\circ \\mathrm { V } _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ^ { * } )$ ", + "text_level": 1, + "bbox": [ + 173, + 537, + 517, + 556 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "By chain rule, $\\begin{array} { r } { \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) = \\frac { \\mathrm { d } \\phi \\mathsf { o n } ( z ) } { \\mathrm { d } z } \\big | _ { z = G x } \\frac { \\mathrm { d } G x } { \\mathrm { d } x } = \\frac { \\mathrm { d } \\phi \\mathsf { o n } ( z ) } { \\mathrm { d } z } \\big | _ { z = G x } G , \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) \\Delta x = \\frac { \\mathrm { d } \\phi \\mathsf { o n } ( z ) } { \\mathrm { d } z } \\big | _ { z = G x } G \\Delta x } \\end{array}$ Thus ", + "bbox": [ + 174, + 568, + 825, + 601 + ], + "page_idx": 67 + }, + { + "type": "equation", + "img_path": "images/93335f00fc5248de9f83042cee9ecb515657f892f3884ef784e9c47c502587d2.jpg", + "text": "$$\n\\begin{array} { r l } & { \\nabla _ { B _ { \\phi } ^ { \\prime } } ( \\Sigma ) = \\mathbb { E } \\left[ B _ { \\phi } ^ { \\prime } ( x ) ^ { \\otimes 2 } : x \\sim \\mathcal { N } ( 0 , \\Sigma ) \\right] } \\\\ & { \\qquad = \\mathbb { E } \\left[ \\left( \\left. \\frac { \\mathrm { d } \\phi \\circ \\mathrm { n } ( z ) } { \\mathrm { d } z } \\right| _ { z = G x } G \\right) ^ { \\otimes 2 } : x \\sim \\mathcal { N } ( 0 , \\Sigma ) \\right] } \\\\ & { \\qquad = \\mathbb { E } \\left[ \\left. \\frac { \\mathrm { d } \\phi \\circ \\mathrm { n } ( z ) } { \\mathrm { d } z } \\right| _ { z = G x } ^ { \\otimes 2 } \\circ G ^ { \\otimes 2 } : x \\sim \\mathcal { N } ( 0 , \\Sigma ) \\right] ^ { - 1 } } \\\\ & { \\qquad = \\mathbb { E } \\left[ \\left. \\frac { \\mathrm { d } \\phi \\circ \\mathrm { n } ( z ) } { \\mathrm { d } z } \\right| _ { z = G x } ^ { \\otimes 2 } : x \\sim \\mathcal { N } ( 0 , \\Sigma ) \\right] \\circ G ^ { \\otimes 2 } } \\\\ & { \\qquad = \\mathbb { V } \\left[ \\left. \\frac { \\mathrm { d } \\phi \\circ \\mathrm { n } ( \\Sigma ) } { \\mathrm { d } z } \\right| _ { z = G x } ^ { \\otimes 2 } \\right] \\left( \\Sigma \\right) c G ^ { \\otimes 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 603, + 686, + 784 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "If we let $\\mathcal { F } ( \\Sigma ) : = \\mathrm { V } _ { \\left[ x \\mapsto \\frac { \\mathtt { d } \\phi \\circ \\mathsf { n } ( z ) } { \\mathtt { d } z } \\right| _ { z = G x } } \\big ] ( \\Sigma )$ , then $\\mathrm { V } _ { \\mathcal { B } _ { \\phi } ^ { \\prime } } ( \\Sigma ) = \\mathcal { F } ( \\Sigma ) \\circ G ^ { \\otimes 2 }$ . As discussed above, we will seek the eigendecomposition of $G ^ { \\otimes 2 } \\circ { \\mathcal { F } } ( \\Sigma ) \\mid { \\mathcal { H } } _ { B } ^ { G } : { \\mathcal { H } } _ { B } ^ { G } \\to { \\mathcal { H } } _ { B } ^ { G }$ . ", + "bbox": [ + 173, + 792, + 823, + 833 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "We first make some basic calculations. ", + "bbox": [ + 174, + 838, + 428, + 852 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "Proposition G.3. ", + "text_level": 1, + "bbox": [ + 174, + 856, + 289, + 871 + ], + "page_idx": 67 + }, + { + "type": "equation", + "img_path": "images/17557cbc42eac938fd16a4f17133016a95916b975c236f66483bc07d9039f1f1.jpg", + "text": "$$\n\\left. { \\frac { \\operatorname { d n } ( z ) } { \\operatorname { d } z } } \\right| _ { z = y } = { \\sqrt { B } } r ^ { - 1 } ( I - v v ^ { T } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 387, + 868, + 609, + 905 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "where $r = \\| y \\| , v = y / \\| y \\| = \\mathsf { n } ( y ) / \\sqrt { B }$ . ", + "bbox": [ + 174, + 907, + 444, + 926 + ], + "page_idx": 67 + }, + { + "type": "text", + "text": "Proof. We have $\\begin{array} { r } { \\left. \\frac { \\mathrm { d } { \\bf n } ( z ) } { \\mathrm { d } z } \\right| _ { z = y } = \\sqrt { B } \\left. \\frac { \\mathrm { d } z / \\| z \\| } { \\mathrm { d } z } \\right| _ { z = y } , } \\end{array}$ , and ", + "bbox": [ + 173, + 99, + 511, + 127 + ], + "page_idx": 68 + }, + { + "type": "equation", + "img_path": "images/a47196b4c977eef751ae68d468ca1f0fbe90669c41b2065aba32187539e7d49a.jpg", + "text": "$$\n\\begin{array} { r l } { \\frac { \\partial ( y _ { i } / \\| y \\| ) } { \\partial y _ { j } } = \\frac { \\delta _ { i j } \\| y \\| - ( \\partial \\| y \\| / \\partial y _ { j } ) y _ { i } } { \\| y \\| ^ { 2 } } } & { } \\\\ & { \\quad = \\frac { \\delta _ { i j } } { \\| y \\| } - \\frac { y _ { i } } { \\| y \\| ^ { 2 } } \\frac { \\partial \\sqrt { y \\cdot y } } { \\partial y _ { j } } } \\\\ & { \\quad = \\frac { \\delta _ { i j } } { \\| y \\| } - \\frac { y _ { i } } { \\| y \\| ^ { 2 } } \\frac { y _ { j } } { \\| y \\| } } \\\\ & { \\quad = \\frac { \\delta _ { i j } } { \\| y \\| } - \\frac { y _ { i } y _ { j } } { \\| y \\| ^ { 3 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 369, + 131, + 625, + 268 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "so that ", + "bbox": [ + 173, + 267, + 220, + 282 + ], + "page_idx": 68 + }, + { + "type": "equation", + "img_path": "images/b7dcfd2c131f9fd1f4ddb47547b2fd927a3b6eb4d0bef4d2f99e575c82745bbe.jpg", + "text": "$$\n\\left. { \\frac { \\operatorname { d } { \\boldsymbol { z } } / \\| { \\boldsymbol { z } } \\| } { \\operatorname { d } z } } \\right| _ { z = y } = r ^ { - 1 } ( I - v v ^ { T } ) .\n$$", + "text_format": "latex", + "bbox": [ + 397, + 276, + 602, + 314 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "By chain rule, this easily gives ", + "bbox": [ + 173, + 342, + 377, + 357 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "Proposition G.4. ", + "bbox": [ + 173, + 359, + 289, + 373 + ], + "page_idx": 68 + }, + { + "type": "equation", + "img_path": "images/f35c810fe93e0844bfed809540ad27ffda8318d7836ec88ef4738b170989921d.jpg", + "text": "$$\n\\begin{array} { c } { \\displaystyle \\left. \\frac { \\mathrm { d } \\phi \\circ \\mathsf { n } ( z ) } { \\mathrm { d } z } \\right| _ { z = y } = \\sqrt { B } D r ^ { - 1 } ( I - v v ^ { T } ) , } \\\\ { D = \\mathrm { D i a g } ( \\phi ^ { \\prime } ( \\mathsf { n } ( y ) ) ) , r = \\| y \\| , v = y / \\| y \\| = \\mathsf { n } ( y ) / \\sqrt { B } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 173, + 369, + 630, + 428 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "With $v = y / \\| y \\|$ , $r = \\| y \\|$ , and $\\tilde { D } = \\mathrm { D i a g } ( \\phi ^ { \\prime } ( { \\mathsf { n } } ( y ) ) )$ , we then have ", + "bbox": [ + 173, + 434, + 616, + 452 + ], + "page_idx": 68 + }, + { + "type": "equation", + "img_path": "images/54d5e9609a4c1b3f77a73c690712d760a95dac2050e8f3eb13ec031a2749fb81.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { F } ( \\Sigma ) = \\mathbb { E } [ \\tilde { D } ^ { \\otimes 2 } \\circ ( \\sqrt { B } \\frac { \\mathrm { d } z / \\| z \\| } { \\mathrm { d } z } | _ { z = G x } ) ^ { \\otimes 2 } : x \\sim \\mathcal { N } ( 0 , \\Sigma ) ] } \\\\ & { \\quad = \\mathbb { E } [ \\tilde { D } ^ { \\otimes 2 } \\circ ( \\sqrt { B } \\frac { \\mathrm { d } z / \\| z \\| } { \\mathrm { d } z } | _ { z = y } ) ^ { \\otimes 2 } : y \\sim \\mathcal { N } ( 0 , G \\Sigma G ) ] } \\\\ & { \\quad = B \\mathbb { E } [ r ^ { - 2 } ( \\tilde { D } ^ { \\otimes 2 } + ( \\tilde { D } v v ^ { T } ) ^ { \\otimes 2 } - \\tilde { D } \\otimes ( \\tilde { D } v v ^ { T } ) - ( \\tilde { D } v v ^ { T } ) \\otimes \\tilde { D } ) : y \\sim \\mathcal { N } ( 0 , G \\Sigma G ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 453, + 825, + 574 + ], + "page_idx": 68 + }, + { + "type": "equation", + "img_path": "images/b3a85c4609a3696a7101e47a97b7deab5777db4b7cf4a5360110c1ccd1a1fc0c.jpg", + "text": "$$\n\\mathcal { F } ( \\Sigma ) \\{ \\Lambda \\} = B \\mathbb { E } \\left[ r ^ { - 2 } \\left( \\tilde { D } \\Lambda \\tilde { D } + \\tilde { D } v v ^ { T } \\Lambda v v ^ { T } \\tilde { D } - \\tilde { D } \\Lambda v v ^ { T } \\tilde { D } - \\tilde { D } v v ^ { T } \\Lambda \\tilde { D } \\right) : y \\sim \\mathcal { N } ( 0 , G \\Sigma G ) \\right]\n$$", + "text_format": "latex", + "bbox": [ + 176, + 585, + 812, + 616 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "G.1.1 SPHERICAL INTEGRATION ", + "text_level": 1, + "bbox": [ + 174, + 622, + 413, + 637 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "When $\\Sigma = \\Sigma ^ { * }$ is the BSB1 fixed point, one can again easily see that $\\mathcal { F } ( \\Sigma ^ { * } )$ is ultrasymmetric. With $\\begin{array} { r } { \\tau _ { i j } = \\frac { 1 } { 2 } \\left( \\delta _ { i } \\delta _ { j } ^ { T } + \\delta _ { j } \\delta _ { i } ^ { T } \\right) } \\end{array}$ (Defn F.20), we have ", + "bbox": [ + 174, + 646, + 825, + 678 + ], + "page_idx": 68 + }, + { + "type": "equation", + "img_path": "images/e11c4a6e85d4ee8bc536df0fc253357bf3fb60fa37ca4501fffcec4bbedf1fe6.jpg", + "text": "$$\n\\begin{array} { r l } { \\iota ( \\Sigma ^ { * } ) \\langle \\lambda \\rangle = B \\mathbb { E } \\Bigg [ \\Gamma ^ { - 2 } \\Big ( \\tilde { D } \\Lambda \\tilde { D } + \\tilde { D } w ^ { \\top } \\Delta w ^ { \\top } \\tilde { D } - \\tilde { D } \\tilde { D } + \\Delta a w ^ { \\top } \\tilde { D } - \\tilde { D } w w ^ { \\top } \\tilde { D } - \\tilde { D } w ^ { \\top } \\tilde { A } \\tilde { D } \\Big ) : y \\sim N ( 0 , p \\tilde { \\Sigma } \\tilde { U } ) \\Bigg ] } & { } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\quad \\quad \\quad \\quad \\\\ & { \\quad = B ^ { - 1 } B \\mathbb { E } \\Bigg [ - 2 \\Big ( \\tilde { D } \\Lambda \\tilde { D } + \\tilde { D } w ^ { \\top } \\tilde { D } w \\chi \\chi w ^ { \\top } \\tilde { D } - \\tilde { D } \\Lambda w \\pi ^ { \\top } \\tilde { T } - \\tilde { D } w \\pi ^ { \\top } \\tilde { D } - \\tilde { D } w \\pi ^ { \\top } \\tilde { A } \\tilde { D } \\Big ) \\cdot \\tilde { y } \\times N ( 0 , \\mathcal { A } ) \\Bigg ] } & { } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 202, + 680, + 861, + 929 + ], + "page_idx": 68 + }, + { + "type": "text", + "text": "by Lemma E.35, where we assume, WLOG by ultrasymmetry, $k , l \\in \\{ 1 , 2 \\} ; i , j \\in \\{ 1 , \\ldots , 4 \\}$ , and ", + "bbox": [ + 168, + 102, + 818, + 119 + ], + "page_idx": 69 + }, + { + "type": "equation", + "img_path": "images/a491bc7a9bd6737bffa190b301c6c556342f7914624190c66444ff5c17e7f185.jpg", + "text": "$$\n\\begin{array} { l } { { \\mathbf { \\dot { \\tau } } ( v _ { 1 } , \\dots , v _ { 4 } ) : = \\displaystyle \\frac { 1 } { 2 } \\phi ^ { \\prime } ( \\sqrt { B } \\mathfrak { v } v ) _ { k } \\phi ^ { \\prime } ( \\sqrt { B } \\mathfrak { e } v ) _ { l } \\big ( } } \\\\ { { \\mathbf { \\dot { \\tau } } ( k = i \\ \\& \\ l = j ) + \\mathbb { I } ( k = j \\ \\& \\ l = i ) + 2 ( \\mathfrak { e } v ) _ { i } ( \\mathfrak { e } v ) _ { j } ( \\mathfrak { e } v ) _ { k } ( \\mathfrak { e } v ) _ { l } } } \\\\ { { \\mathbf { \\dot { \\tau } } - ( \\mathbb { I } ( i = k ) ( \\mathfrak { e } v ) _ { j } ( \\mathfrak { e } v ) _ { l } + \\mathbb { I } ( j = k ) ( \\mathfrak { e } v ) _ { i } ( \\mathfrak { e } v ) _ { l } + \\mathbb { I } ( i = l ) ( \\mathfrak { e } v ) _ { j } ( \\mathfrak { e } v ) _ { k } + \\mathbb { I } ( j = l ) ( \\mathfrak { e } v ) _ { i } ( \\mathfrak { e } v ) _ { l } } } \\\\ { { \\mathbf { \\dot { \\tau } } } } \\\\ { { \\mathbf { \\dot { \\tau } } _ { 1 } ^ { \\theta } = \\cos { \\theta _ { 1 } } } } \\\\ { { v _ { 2 } ^ { \\theta } = \\sin { \\theta _ { 1 } } \\cos { \\theta _ { 2 } } } } \\\\ { { v _ { 3 } ^ { \\theta } = \\sin { \\theta _ { 1 } } \\sin { \\theta _ { 2 } } \\cos { \\theta _ { 3 } } } } \\\\ { { v _ { 4 } ^ { \\theta } = \\sin { \\theta _ { 1 } } \\sin { \\theta _ { 2 } } \\sin { \\theta _ { 3 } } \\cos { \\theta _ { 4 } } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 155, + 839, + 327 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "and $\\mathbb { P }$ as in Eq. (33). ", + "bbox": [ + 173, + 359, + 308, + 376 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "We can integrate out $\\theta _ { 3 }$ and $\\theta _ { 4 }$ symbolically by Lemma E.29 since their dependence only appear outside of $\\phi ^ { \\prime }$ . This reduces each entry of $\\mathcal { F } ( \\Sigma ^ { * } ) \\{ \\tau _ { i j } \\} _ { k l }$ to 2-dimensional integrals to evaluate numerically. The eigenvalues of $G ^ { \\otimes 2 } \\circ { \\mathcal { F } } ( \\Sigma ^ { * } )$ can then be obtained from Thm E.62. We omit details here and instead focus on results from other methods. ", + "bbox": [ + 173, + 381, + 825, + 440 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "G.1.2 GEGENBAUER EXPANSION ", + "text_level": 1, + "bbox": [ + 174, + 488, + 416, + 503 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "Notice that when $\\Lambda = G$ , the expression for $\\mathcal { F } ( \\Sigma ^ { * } ) \\{ \\Lambda \\}$ significantly simplifies because $G$ acts as identity on $v$ : ", + "bbox": [ + 174, + 526, + 826, + 556 + ], + "page_idx": 69 + }, + { + "type": "equation", + "img_path": "images/37a1cfbc416564f8eab9433fb5c947d23ff17262fd5c717bcce38dc8677ad930.jpg", + "text": "$$\n\\begin{array} { r l } & { \\boldsymbol { \\varepsilon } ( \\Sigma ^ { * } ) \\{ G \\} = \\mu ^ { * } \\ ) ^ { 2 } \\boldsymbol { B } \\mathbb { E } \\left[ r ^ { - 2 } \\left( \\boldsymbol { \\tilde { D } } \\boldsymbol { \\Lambda } \\boldsymbol { \\tilde { D } } + \\boldsymbol { \\tilde { D } } \\boldsymbol { v } \\boldsymbol { v } ^ { T } \\boldsymbol { G } \\boldsymbol { v } \\boldsymbol { v } ^ { T } \\boldsymbol { \\tilde { D } } - \\boldsymbol { \\tilde { D } } \\boldsymbol { G } \\boldsymbol { v } \\boldsymbol { v } ^ { T } \\boldsymbol { \\tilde { D } } - \\boldsymbol { \\tilde { D } } \\boldsymbol { v } \\boldsymbol { v } ^ { T } \\boldsymbol { G } \\boldsymbol { \\tilde { D } } \\right) : \\boldsymbol { y } \\sim \\mathcal { N } ( \\boldsymbol { 0 } , \\boldsymbol { G } ) \\right] } \\\\ & { \\quad \\quad \\quad = \\mu ^ { * } - 1 \\boldsymbol { B } \\mathbb { E } \\left[ r ^ { - 2 } \\left( \\boldsymbol { \\tilde { D } } \\boldsymbol { G } \\boldsymbol { \\tilde { D } } - \\boldsymbol { \\tilde { D } } \\boldsymbol { v } \\boldsymbol { v } ^ { T } \\boldsymbol { \\tilde { D } } \\right) : \\boldsymbol { y } \\sim \\mathcal { N } ( \\boldsymbol { 0 } , \\boldsymbol { G } ) \\right] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 592, + 831, + 648 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "Then the diagonal of the image satisfies ", + "bbox": [ + 174, + 684, + 436, + 699 + ], + "page_idx": 69 + }, + { + "type": "equation", + "img_path": "images/b3f5fddccbe84800e3784946b1ab4e9ee39d156d1496058b8f36a84bd4d295dd.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathcal { F } ( \\Sigma ^ { * } ) \\} G \\int _ { a } } \\\\ & { = \\mu ^ { * - 1 } B \\mathbb { E } \\left[ r ^ { - 2 } \\left( \\frac { B - 1 } { B } \\phi ^ { \\prime } ( \\sqrt { B } v _ { a } ) ^ { 2 } - v _ { a } ^ { 2 } \\phi ^ { \\prime } ( \\sqrt { B } v _ { a } ) ^ { 2 } \\right) : r v \\sim \\mathcal { N } ( 0 , G ) \\right] } \\\\ & { = \\mu ^ { * - 1 } B ( B - 3 ) ^ { - 1 } \\underset { v \\sim \\mathbb { S } ^ { B } } { \\mathbb { E } } \\ \\left( \\frac { B - 1 } { B } \\phi ^ { \\prime } ( \\sqrt { B } \\mathbf { e } _ { a } , v ) ^ { 2 } - ( \\mathbf { e } _ { a } , v ) ^ { 2 } \\phi ^ { \\prime } ( \\sqrt { B } \\mathbf { e } _ { a } , v ) ^ { 2 } \\right) \\mathcal { K } ( v ; G , 2 ) } \\\\ & { \\qquad \\mathrm { ( b y ~ P r o p o s i t i o n ~ E \\lambda ^ { 3 , 3 } ) } } \\\\ & { = \\mu ^ { * - 1 } B ( B - 3 ) ^ { - 1 } \\underset { v \\sim \\mathbb { S } ^ { B - 1 } } { \\underbrace { \\mathbb { E } } } \\left( 1 - ( \\hat { \\mathbf { e } } _ { a } , v ) ^ { 2 } \\right) \\frac { B - 1 } { B } \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathbf { e } } _ { a } , v ) ^ { 2 } } \\\\ & { \\qquad \\quad - \\mu ^ { * - 1 } ( B - 1 ) ( B - 3 ) ^ { - 1 } \\underset { v \\sim \\mathbb { S } ^ { B - 2 } } { \\underbrace { \\mathbb { E } } } \\left( 1 - ( \\hat { \\mathbf { e } } _ { a } , v ) ^ { 2 } \\right) \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathbf { e } } _ { a } , v ) ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 199, + 734, + 803, + 924 + ], + "page_idx": 69 + }, + { + "type": "text", + "text": "And the off-diagonal entries satisfy ", + "bbox": [ + 176, + 103, + 406, + 118 + ], + "page_idx": 70 + }, + { + "type": "equation", + "img_path": "images/5e591431579232ea2086e90407f6ee95dffb71739dd07a72ea9c16800c4cae79.jpg", + "text": "$$\n\\begin{array} { r l } & { \\qquad \\mathcal { F } ( \\Sigma ^ { * } ) \\{ G \\} _ { a b } } \\\\ & { = \\mu ^ { * - 1 } B \\mathbb { E } [ r ^ { - 2 } ( \\cfrac { - 1 } { B } \\phi ^ { \\prime } ( \\sqrt { B } \\mathfrak { e } _ { a , v } ) \\phi ^ { \\prime } ( \\sqrt { B } \\mathfrak { e } _ { b , v } ) } \\\\ & { \\qquad - ( \\mathfrak { e } _ { a , v } ) ( \\mathfrak { e } _ { b , : } v ) \\phi ^ { \\prime } ( \\sqrt { B } \\mathfrak { e } _ { a , : } v ) \\phi ^ { \\prime } ( \\mathfrak { e } _ { a , : } v ) ] : r v \\sim \\mathcal { N } ( 0 , G ) ] } \\\\ & { = \\mu ^ { * - 1 } B ( B - 3 ) ^ { - 1 } \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } ( \\cfrac { - 1 } { B } \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathfrak { e } } _ { a , v } ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathfrak { e } } _ { b , v } ) } \\\\ & { \\qquad - \\cfrac { B - 1 } { B } ( \\hat { \\mathfrak { e } } _ { a , v } ) ( \\hat { \\mathfrak { e } } _ { b , v } ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathfrak { e } } _ { a , : } v ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathfrak { e } } _ { b , : } v ) ) K ( v ; G , 2 ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 309, + 807, + 464 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "(by Proposition E.33) ", + "bbox": [ + 665, + 463, + 808, + 478 + ], + "page_idx": 70 + }, + { + "type": "equation", + "img_path": "images/2b4daed7ea3fa8f71a9a8fa371e86118657305cfee5e61ed3c42843f333f24a7.jpg", + "text": "$$\n\\begin{array} { l } { { = \\mu ^ { * - 1 } B ( B - 3 ) ^ { - 1 } \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\frac { - 1 } { B } ( 1 + ( B - 1 ) ( \\hat { \\mathfrak { e } } _ { a , v } ) ( \\hat { \\mathfrak { e } } _ { b , v } ) ) } } \\\\ { { \\qquad \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathfrak { e } } _ { a , v } ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathfrak { e } } _ { b , v } , v ) } } \\\\ { { \\qquad \\quad } } \\\\ { { = - \\mu ^ { * - 1 } ( B - 3 ) ^ { - 1 } \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } ( 1 + ( B - 1 ) ( \\hat { \\mathfrak { e } } _ { a , v } ) ( \\hat { \\mathfrak { e } } _ { b , v } ) ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathfrak { e } } _ { a , v } ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathfrak { e } } _ { b , v } , v ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 191, + 482, + 808, + 578 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "Lifting Spherical Expectation. We first show a way of expressing $\\mathcal { F } ( \\Sigma ^ { * } ) _ { a a }$ in terms of Gegenbauer basis by lifting the spherical integral over $S ^ { B - 2 }$ to spherical integral over $S ^ { B }$ . While this technique cannot be extended to $\\mathcal { F } ( \\Sigma ^ { * } ) _ { a b } , a \\neq b$ , we believe it could be of use to future related problems. ", + "bbox": [ + 173, + 775, + 825, + 833 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "If $\\phi ( { \\sqrt { B - 1 } } x )$ has Gegenbauer expansion $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ , then $\\phi ^ { \\prime } ( \\sqrt { B - 1 } x ) =$ $\\begin{array} { r c l } { { \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) \\prime } ( x ) } } & { { = } } & { { \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { B - 3 } { c _ { B - 1 , l } } C _ { l - 1 } ^ { ( \\frac { B - 1 } { 2 } ) } ( x ) } } \\end{array}$ $\\begin{array} { r } { \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { B - 3 } { c _ { B - 1 , l } } C _ { l - 1 } ^ { ( \\frac { B - 1 } { 2 } ) } ( \\hat { \\mathbb { e } } _ { a , : } v ) = \\frac { 1 } { \\sqrt { B - 1 } } \\sum _ { l = 0 } ^ { \\infty } a _ { l } ( B - 3 ) \\frac { c _ { B + 1 , l - 1 } } { c _ { B - 1 , l } } Z _ { \\hat { \\mathbb { e } } _ { a , : } } ^ { B , ( l - 1 ) } ( v ) . } \\end{array}$ , so that $\\phi ^ { \\prime } ( \\sqrt { B } \\mathbb { e } _ { a , : } v ) \\ =$ . Here $\\begin{array} { r } { \\hat { \\Phi } _ { a , } } \\end{array}$ : and $v$ are treated as points on the $B$ -dimensional sphere with last 2 coordinates 0. ", + "bbox": [ + 171, + 838, + 826, + 925 + ], + "page_idx": 70 + }, + { + "type": "text", + "text": "Thus by Lemmas E.38 and E.39, ", + "bbox": [ + 173, + 103, + 392, + 118 + ], + "page_idx": 71 + }, + { + "type": "equation", + "img_path": "images/7e70431e43ec193acc6f29546c9bb52dc1fc441bfc7593a295e97d526cb2d3b5.jpg", + "text": "$$\n\\begin{array} { r l } & { \\begin{array} { r l } & { \\rho _ { \\mathrm { e f f } } ^ { ( 1 ) } ( x , y ) } \\\\ & { = \\rho _ { \\mathrm { e f f } } ^ { ( 1 ) } ( x , y ) } \\\\ & { \\rho _ { \\mathrm { e f f } } ^ { ( 1 ) } ( x , y ) } \\end{array} } \\\\ & \\begin{array} { r l } & ( x - y ) ^ { 2 } ( x - 1 ) ^ { 2 } \\left( x - 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ { 2 } \\left( x + 1 \\right) ^ 2 \\end{array} \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 126, + 823, + 382 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": ".45 ", + "bbox": [ + 710, + 385, + 820, + 400 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "Dirichlet-like form of $\\phi$ . For ${ \\mathcal { F } } ( \\Sigma ^ { * } ) \\{ G \\} _ { a b }$ , there’s no obvious way of lifting the spherical expectation to a higher dimension with the weight $( 1 + ( B - 1 ) ( \\hat { \\mathbb { E } } _ { a , : } v ) ( \\hat { \\mathbb { e } } _ { b , : } v ) )$ . We will instead leverage $\\cdot$ . In Thm E.47, setting $u _ { 1 } = \\hat { \\Phi } _ { a , : } = u _ { 2 }$ , we get ", + "bbox": [ + 174, + 606, + 825, + 651 + ], + "page_idx": 71 + }, + { + "type": "equation", + "img_path": "images/d7140a8d9670a11d6cdb5f6a006a1b371be9aedd59e216aacf22f5be0aff056f.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathcal { F } ( \\Sigma ^ { * } ) \\{ G \\} _ { a a } = \\mu ^ { * - 1 } \\frac { B - 1 } { B - 3 } \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } ( ( \\hat { \\mathbb { e } } _ { a , : } \\cdot \\hat { \\mathbb { e } } _ { a , : } ) - ( \\hat { \\mathbb { e } } _ { a , : } v ) ^ { 2 } ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathbb { e } } _ { a , : } v ) ^ { 2 } } \\\\ { \\displaystyle \\quad \\quad = \\mu ^ { * - 1 } \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { l + B - 3 } { B - 3 } l c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( 1 \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 233, + 661, + 763, + 738 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "Likewise, setting $u _ { 1 } = \\widehat { \\Phi } _ { a , : } , u _ { 2 } = \\widehat { \\Phi } _ { b , : }$ , we get ", + "bbox": [ + 173, + 747, + 475, + 762 + ], + "page_idx": 71 + }, + { + "type": "equation", + "img_path": "images/919ba23a700fe3a0159fb5a96715c43593df069b5d8d7afd94ef843449cb0181.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle { } ^ { r } ( \\Sigma ^ { * } ) \\{ G \\} _ { a b } = \\mu ^ { * - 1 } \\displaystyle { \\frac { B - 1 } { B - 3 } \\displaystyle { \\frac { \\mathbb { E } } { v \\sim S ^ { B - 2 } } \\big ( \\big ( \\hat { \\mathbf { e } } _ { a } , \\cdot ~ \\hat { \\mathbf { e } } _ { b , : } \\big ) - \\big ( \\hat { \\mathbf { e } } _ { a , : } v \\big ) \\big ( \\hat { \\mathbf { e } } _ { b , : } v \\big ) \\big ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathbf { e } } _ { a , : } v ) \\phi ^ { \\prime } ( \\sqrt { B - 1 } \\hat { \\mathbf { e } } _ { b , : } v ) } } } \\\\ { { \\displaystyle \\qquad = \\mu ^ { * - 1 } \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\displaystyle { \\frac { l + B - 3 } { B - 3 } l c _ { B - 1 , l } ^ { - 1 } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 772, + 838, + 848 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "Thus, ", + "bbox": [ + 174, + 866, + 214, + 880 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "Theorem G.5. SupposGegenbauer expansion $\\phi ( \\sqrt { B - 1 } x ) , \\phi ^ { \\prime } ( \\sqrt { B - 1 } x ) \\in L ^ { 2 } ( ( 1 - x ^ { 2 } ) ^ { \\frac { B - 3 } { 2 } } )$ and nique $\\phi ( \\sqrt { B - 1 } x )$ hasoint $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ $\\Sigma ^ { * }$ ", + "bbox": [ + 174, + 886, + 826, + 928 + ], + "page_idx": 71 + }, + { + "type": "text", + "text": "of Eq. (43), the eigenvalue of $G ^ { \\otimes 2 } \\circ { \\mathcal { F } } ( \\Sigma ^ { * } )$ corresponding to eigenspace $\\mathbb { R } G$ is ", + "bbox": [ + 171, + 102, + 696, + 119 + ], + "page_idx": 72 + }, + { + "type": "equation", + "img_path": "images/8e411fcb38d1bbf245bd7e481f132b11fb79e8168bbdd4fca325a489b9fba43d.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\lambda _ { G } ^ { \\downarrow } = \\mu ^ { * - 1 } \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { l + B - 3 } { B - 3 } l c _ { B - 1 , l } ^ { - 1 } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( 1 \\right) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } } \\\\ { { \\displaystyle \\quad = \\frac { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { l + B - 3 } { B - 3 } l c _ { B - 1 , l } ^ { - 1 } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( 1 \\right) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } c _ { B - 1 , l } ^ { - 1 } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( 1 \\right) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } } } \\\\ { { \\displaystyle > 1 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 121, + 743, + 236 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "This is minimized (over choices of $\\phi$ ) iff $\\phi$ is linear, in which case $\\begin{array} { r } { \\lambda _ { G } ^ { \\downarrow } = \\frac { B - 2 } { B - 3 } } \\end{array}$ ", + "bbox": [ + 173, + 239, + 683, + 258 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "G.2 LAPLACE METHOD ", + "text_level": 1, + "bbox": [ + 174, + 273, + 352, + 289 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "In this section suppose that $\\phi$ is degree $\\alpha$ positive-homogeneous. Set $D = \\operatorname { D i a g } ( \\phi ^ { \\prime } ( y ) )$ (and recall $v = y / \\| y \\| , r = \\| y \\| )$ . Then $D$ is degree $\\alpha - 1$ positive-homogeneous in $x$ (because $\\phi ^ { \\prime }$ is). Consequently we can rewrite Eq. (52) as follows, ", + "bbox": [ + 173, + 299, + 825, + 342 + ], + "page_idx": 72 + }, + { + "type": "equation", + "img_path": "images/0d0ec51ade02e651d88f6e2f7469443b13e85fb7a5f278cf2b7184e976f101a3.jpg", + "text": "$$\n\\begin{array} { r l r } & { = B _ { \\phantom { \\alpha } y \\sim N ( 0 , G \\Sigma ^ { * } G ) } ^ { \\alpha } \\biggl [ r ^ { - 2 } \\biggl ( r ^ { - 2 ( \\alpha - 1 ) } D ^ { \\otimes 2 } + r ^ { - 2 ( \\alpha + 1 ) } ( D v v ^ { T } ) ^ { \\otimes 2 } \\phantom { x } } \\\\ & { \\phantom { = } - r ^ { - 2 \\alpha } D \\otimes ( D v v ^ { T } ) - r ^ { - 2 \\alpha } ( D v v ^ { T } ) \\otimes D \\biggr ) \\biggr ] } \\\\ & { = B _ { \\phantom { \\alpha } y \\sim N ( 0 , \\mu \\cdot \\mathbf { G } ) } ^ { \\alpha } \\biggl [ r ^ { - 2 \\alpha } D ^ { \\otimes 2 } + r ^ { - 2 ( \\alpha + 2 ) } ( D y ^ { T } ) ^ { \\otimes 2 } - r ^ { - 2 ( \\alpha + 1 ) } \\left( D \\otimes ( D y y ^ { T } ) + ( D y y ^ { T } ) \\otimes D \\right) } \\\\ & { = B _ { \\phantom { \\alpha } ( \\mathbf { A } + \\mathbf { B } - \\mathbf { C } ) } ^ { \\alpha } } & { ( 5 3 \\qquad ( \\mathbf { A } + \\mathbf { B } - \\mathbf { C } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 173, + 361, + 816, + 483 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "where $\\Sigma ^ { * }$ is the BSB1 fixed point of Eq. (43), $G \\Sigma ^ { * } G = \\mu ^ { * } G$ , and ", + "bbox": [ + 174, + 484, + 609, + 501 + ], + "page_idx": 72 + }, + { + "type": "equation", + "img_path": "images/d38cd270457efdaee152b20ade2f6e02bfd5f248415a799767367e2a9df11557.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathsf { A } = \\mathbb { E } \\left[ r ^ { - 2 \\alpha } D ^ { \\otimes 2 } : y \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) \\right] } \\\\ & { \\mathsf { B } = \\mathbb { E } \\left[ r ^ { - 2 ( \\alpha + 2 ) } ( D y y ^ { T } ) ^ { \\otimes 2 } : y \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) \\right] } \\\\ & { \\mathsf { C } = \\mathbb { E } \\left[ r ^ { - 2 ( \\alpha + 1 ) } \\left( D \\otimes ( D y y ^ { T } ) + ( D y y ^ { T } ) \\otimes D \\right) : y \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) \\right] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 267, + 501, + 727, + 577 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "Each term in the sum above is ultrasymmetric, so has the same eigenspaces $\\mathbb { R } G , \\mathbb { M } , \\mathbb { L }$ (this will also become apparent in the computations below without resorting to Thm E.61). We can thus compute the eigenvalues for each of them in order. ", + "bbox": [ + 176, + 578, + 823, + 619 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "In all computation below, we apply Lemma E.2 first to relate the quantity in question to $\\mathrm { V } _ { \\phi }$ ", + "bbox": [ + 169, + 626, + 774, + 642 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "Computing A. For two matrices $\\Sigma , \\Lambda$ , write $\\Sigma \\odot \\Lambda$ for entrywise multiplication. We have by Lemma E.2 ", + "bbox": [ + 173, + 655, + 823, + 684 + ], + "page_idx": 72 + }, + { + "type": "equation", + "img_path": "images/121cc4e0d23601ed26c3e74c8ec45784e9e18a5ed197355a893ca61772b29043.jpg", + "text": "$$\n\\begin{array} { r l } { \\Lambda \\{ \\Lambda \\} } & { = \\mathbb { E } [ r ^ { - 2 \\alpha } D \\Lambda D \\cdot y \\sim { \\cal N } ( 0 , \\mu ^ { * } Q ) ] } \\\\ & { = \\mathbb { E } [ r ^ { - 2 \\alpha } \\Lambda \\odot \\psi ( \\psi ) \\psi ^ { T } \\mid \\psi ^ { T } : y \\sim { \\cal N } ( 0 , \\mu ^ { * } C ) ] } \\\\ & { = \\mathbb { E } [ \\exp ^ { - \\alpha } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ \\mathrm { o } ^ { \\alpha - 1 } \\ \\mathrm { d } \\mathrm { e t } ( f + 2 s \\mu ^ { * } G ) ^ { - 1 / 2 } \\Lambda \\odot \\mathrm { V } _ { \\psi } ( \\frac { \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } G ) } \\\\ & { = \\Gamma ( \\alpha ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ { s } ^ { \\alpha - 1 } ( 1 + 2 s \\mu ^ { * } ) ^ { - ( N - 1 ) / 2 } \\Lambda \\odot ( \\frac { \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } ) ^ { \\alpha - 1 } \\mathrm { V } _ { \\psi } \\langle G \\rangle } \\\\ & { = \\Gamma ( \\alpha ) ^ { - 1 } \\Lambda \\odot \\mathrm { V } _ { \\psi } \\langle G \\rangle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ \\mathrm { o } ( \\mu ^ { * } s ) ^ { \\alpha - 1 } ( 1 + 2 s \\mu ^ { * } ) ^ { - ( N - 1 ) / 2 - \\alpha + 1 } } \\\\ & { = \\Gamma ( \\alpha ) ^ { - 1 } \\Lambda \\odot \\mathrm { V } _ { \\psi } \\langle G \\rangle \\mathrm { d } \\mathrm { e t } ( \\frac { B } { 2 } - 3 ) , } \\\\ & { = \\Lambda \\odot \\mathsf { c } _ { \\alpha - 1 } ( \\frac { B - 1 } { B } ) ^ { \\alpha - 1 } \\mathrm { R S B } ( \\mathrm { J } _ { \\psi } ( 1 ) , \\mathrm { J } _ { \\psi } ( \\frac { - 1 } { B - 1 } ) ) \\mathrm { P } ( \\frac { B - 3 } { 2 } , \\alpha ) ^ { - 1 } \\mid \\mu ^ { * } ^ { - 1 } 2 ^ { - \\alpha } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 186, + 685, + 812, + 909 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "Then $\\cdot$ gives the eigendecomposition for $B ^ { \\alpha } G ^ { \\otimes 2 } \\circ \\mathsf { A } \\mathrm { ~ } \\uparrow \\mathcal { H } _ { B } ^ { G }$ ", + "bbox": [ + 173, + 909, + 624, + 925 + ], + "page_idx": 72 + }, + { + "type": "text", + "text": "Theorem G.6. Let $\\begin{array} { r } { R _ { B , \\alpha } : = ( 2 \\alpha - 1 ) \\mathsf { c } _ { \\alpha - 1 } B ( B - 1 ) ^ { \\alpha - 1 } \\mathrm { P } \\left( \\frac { B - 3 } { 2 } , \\alpha \\right) ^ { - 1 } 2 ^ { - \\alpha } } \\end{array}$ . Then $B ^ { \\alpha } G ^ { \\otimes 2 } \\circ \\mathsf { A } \\mid \\mathcal { H } _ { B } ^ { G }$ has the following eigendecomposition. ", + "bbox": [ + 173, + 101, + 823, + 133 + ], + "page_idx": 73 + }, + { + "type": "equation", + "img_path": "images/6feab62db100642a4455593d3563f81577eef605f64d9e2b4eaa25e9e98fa074.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\lambda _ { \\delta \\_ { - } } ^ { \\delta \\_ { - } } - R _ { \\beta , \\omega \\_ \\mu ^ { - } } ^ { \\delta \\_ { - } - 1 } \\left( \\begin{array} { l l l } { R } & { 1 , } & { 1 , y _ { \\delta } ^ { \\delta } \\left( 1 \\right) } \\\\ { R } & { 1 , } & { 2 y _ { \\delta } ^ { \\delta } \\left( \\_ { S \\_ { - } } - 1 \\right) } \\end{array} \\right) , } \\\\ & { \\quad \\quad = \\frac { \\left( 2 \\alpha \\right) \\cdot \\left( R - 3 \\right) \\cdot \\left( \\alpha ^ { 2 } ( R - 1 ) y _ { \\delta } ( y _ { \\delta } ( 1 ) + ( 2 \\alpha - 1 ) y _ { \\delta } ( \\frac { 1 } { 1 - \\beta } ) \\right) } { \\left( 2 \\alpha - 1 \\right) \\left( 3 R - 3 \\right) \\cdot \\left( R - 1 \\right) \\left( \\beta ( 1 ) - 1 \\right) } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\lambda _ { \\delta \\_ { - } } ^ { \\delta \\_ { - } } = R _ { \\beta , \\omega \\_ \\mu ^ { - } } ^ { \\delta \\_ { - } - 1 } \\left( \\frac { R - 2 } { B } , y _ { \\varepsilon } ^ { \\varepsilon } ( 1 ) + \\frac { 2 } { B } , y _ { \\delta } ^ { \\varepsilon } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) , } \\\\ & { \\quad \\quad \\quad - \\frac { \\left( 2 \\alpha + B - 3 \\right) \\cdot \\left( \\alpha ^ { 2 } ( B - 2 ) y _ { \\delta } ( x _ { \\delta } ( 1 ) ) + 2 ( 2 \\alpha - 1 ) y _ { \\delta } ^ { \\varepsilon } \\left( \\frac { - 1 } { 1 - B } \\right) \\right) } { \\left( 2 \\alpha - 1 \\right) \\left( B - 3 \\right) \\cdot \\left( \\beta ( 1 ) - 1 \\right) \\left( \\beta ( 1 ) - 1 \\right) } } \\\\ & { \\quad \\quad \\quad \\quad \\lambda _ { \\delta \\_ { - } } ^ { \\delta \\_ { - } } = R _ { \\beta , \\omega ^ { \\varepsilon - } } ^ { \\delta \\_ { - } - 1 } \\gamma _ { \\varepsilon } ^ { \\delta } \\left( \\frac { - 1 } { B - 1 } \\right) } \\\\ & { \\quad \\quad \\quad - \\frac { B \\cdot \\left( 2 \\alpha + B - 3 \\right) \\cdot \\varepsilon _ { 0 } ^ { \\varepsilon } \\left( \\frac { - 1 } { 1 - \\beta } \\right) } { \\left( B - 4 \\right) \\left( B - 1 \\right) \\cdot \\left( 1 - \\beta \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 281, + 136, + 718, + 404 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "Proof. We have $\\begin{array} { r } { B ^ { \\alpha } \\mathsf { A } = ( 2 \\alpha - 1 ) ^ { - 1 } R _ { B , \\alpha } \\mu ^ { * } \\overset { - 1 } { \\underset { \\ r { \\infty } } { } } \\mathrm { D O S } ( \\mathrm { J } _ { \\phi ^ { \\prime } } ( 1 ) , 0 , \\mathrm { J } _ { \\phi ^ { \\prime } } ( \\frac { - 1 } { B - 1 } ) ) . } \\end{array}$ . This allows us to apply $\\cdot$ . We make a further simplification $\\mathrm { J } _ { \\phi ^ { \\prime } } ( c ) = ( 2 \\alpha - 1 ) \\mathrm { J } _ { \\phi } ^ { \\prime } ( c )$ via Proposition E.22. ", + "bbox": [ + 173, + 415, + 825, + 455 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "Computing B. We simplify ", + "bbox": [ + 174, + 469, + 370, + 484 + ], + "page_idx": 73 + }, + { + "type": "equation", + "img_path": "images/f981d0a13411e6d2900a8b732e0720369c8ba1a5274e757e9e73c4f9c9b2de2d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathsf { B } \\{ \\Lambda \\} = \\mathbb E \\left[ r ^ { - 2 ( \\alpha + 2 ) } D y y ^ { T } \\Lambda y y ^ { T } D : y \\sim \\mathcal N ( 0 , \\mu ^ { * } G ) \\right] } \\\\ & { \\qquad = \\mathbb E \\left[ r ^ { - 2 ( \\alpha + 2 ) } \\psi ( y ) y ^ { T } \\Lambda y \\psi ( y ) ^ { T } : y \\sim \\mathcal N ( 0 , \\mu ^ { * } G ) \\right] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 486, + 692, + 541 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "where $\\psi ( y ) : = y \\phi ^ { \\prime } ( y )$ , which, in this case of $\\phi$ being degree $\\alpha$ positive-homogeneous, is also equal to $\\alpha \\phi ( y )$ . ", + "bbox": [ + 174, + 541, + 823, + 571 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "By Lemma E.2, $\\mathsf { B } \\{ \\Lambda \\}$ equals ", + "bbox": [ + 174, + 577, + 369, + 592 + ], + "page_idx": 73 + }, + { + "type": "equation", + "img_path": "images/9f0727f5eabf2306cd73441200405969487f6388fbdb15b55e6b7fddcb73364a.jpg", + "text": "$$\n\\Gamma ( \\alpha + 2 ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { \\alpha + 1 } \\operatorname * { d e t } ( I + 2 s \\mu ^ { * } G ) ^ { - 1 / 2 } \\mathbb { E } [ \\psi ( y ) y ^ { T } \\Lambda y \\psi ( y ) ^ { T } : y \\sim \\mathcal { N } ( 0 , \\frac { \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } G ) ]\n$$", + "text_format": "latex", + "bbox": [ + 199, + 593, + 820, + 627 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "This expression naturally leads us to consider the following definition. ", + "bbox": [ + 173, + 635, + 635, + 650 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "Definition G.7. Let $\\psi : \\mathbb { R } \\mathbb { R }$ be measurable and $\\Sigma \\in S _ { B }$ . Define $\\mathrm { V } _ { \\psi } ^ { ( 4 ) } ( \\Sigma ) : { \\mathcal { H } } _ { B } { \\mathcal { H } } _ { B }$ by ", + "bbox": [ + 173, + 654, + 789, + 672 + ], + "page_idx": 73 + }, + { + "type": "equation", + "img_path": "images/ebe31ef1755bef989732389e3d057cfaf9b877c4a5ec7eb8db83747703d6fd9a.jpg", + "text": "$$\n\\nabla _ { \\psi } ^ { ( 4 ) } ( \\Sigma ) \\{ \\Lambda \\} = \\mathbb { E } [ \\psi ( y ) y ^ { T } \\Lambda y \\psi ( y ) ^ { T } : y \\sim \\mathcal { N } ( 0 , \\Sigma ) ]\n$$", + "text_format": "latex", + "bbox": [ + 326, + 675, + 669, + 698 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "For two matrices $\\Sigma , \\Lambda$ , write $\\langle \\Sigma , \\Lambda \\rangle : = \\mathrm { t r } ( \\Sigma ^ { T } \\Lambda )$ . We have the following identity. ", + "bbox": [ + 174, + 705, + 702, + 723 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "Proposition G.8. $\\begin{array} { r } { \\mathrm { V } _ { \\phi } ^ { ( 4 ) } ( \\Sigma ) \\{ \\Lambda \\} = \\mathrm { V } _ { \\phi } ( \\Sigma ) \\langle \\Sigma , \\Lambda \\rangle + 2 \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\{ \\Sigma \\Lambda \\Sigma \\} . } \\end{array}$ . ", + "bbox": [ + 171, + 724, + 620, + 746 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "Proof. We have ", + "bbox": [ + 173, + 758, + 281, + 773 + ], + "page_idx": 73 + }, + { + "type": "equation", + "img_path": "images/73732505211d31c8c788d2d9ce41d1481b97bc4232d374c325659cd87eb2c5b6.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { \\mathrm { I V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\{ \\Lambda \\} = ( 2 \\pi ) ^ { - B / 2 } \\displaystyle \\int \\mathrm { d } z \\phi ( z ) ^ { \\otimes 2 } \\left( \\frac { \\mathrm { d } } { \\mathrm { d } \\Sigma } \\mathrm { d e t } ( \\Sigma ) ^ { - 1 / 2 } e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\right) \\{ \\Lambda \\} } \\\\ & { \\quad \\quad \\quad = ( 2 \\pi ) ^ { - B / 2 } \\displaystyle \\int \\mathrm { d } z \\phi ( z ) ^ { \\otimes 2 } \\left. \\left[ \\frac { - 1 } { 2 } \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } \\Sigma ^ { - 1 } + \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } \\frac { 1 } { 2 } \\Sigma ^ { - 1 } z z z ^ { T } \\Sigma ^ { - 1 } \\right] e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } } \\right. } \\\\ & { \\quad \\quad \\quad = \\displaystyle \\frac { 1 } { 2 } ( 2 \\pi ) ^ { - B / 2 } \\mathrm { d e t } \\Sigma ^ { - 1 / 2 } \\displaystyle \\int \\mathrm { d } z \\phi ( z ) ^ { \\otimes 2 } e ^ { - \\frac { 1 } { 2 } z ^ { T } \\Sigma ^ { - 1 } z } \\left[ \\langle z z ^ { T } , \\Sigma ^ { - 1 } \\Lambda \\Sigma ^ { - 1 } \\rangle - \\langle \\Sigma ^ { - 1 } , \\Lambda \\rangle \\right] } \\\\ & { \\quad \\quad \\quad = \\displaystyle \\frac { 1 } { 2 } \\mathrm { V } _ { \\phi } ^ { ( 4 ) } ( \\Sigma ) \\{ \\Sigma ^ { - 1 } \\Lambda \\Sigma ^ { - 1 } \\} - \\displaystyle \\frac { 1 } { 2 } \\mathrm { V } _ { \\phi } ( \\Sigma ) \\langle \\Sigma ^ { - 1 } , \\Lambda \\rangle . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 776, + 839, + 910 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "Making the substitution $\\Lambda \\Sigma \\Lambda \\Sigma$ , we get the desired result. ", + "bbox": [ + 173, + 909, + 578, + 924 + ], + "page_idx": 73 + }, + { + "type": "text", + "text": "If $\\varphi$ is degree $\\alpha$ positive-homogeneous, then $\\mathrm { V } _ { \\varphi } ^ { ( 4 ) }$ is degree $\\alpha + 1$ positive-homogeneous. Thus, ", + "bbox": [ + 171, + 101, + 802, + 121 + ], + "page_idx": 74 + }, + { + "type": "equation", + "img_path": "images/4fe98fe6959dd6a1aba36f05cee930854691de5c774207b928265f234dda934f.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { B } \\{ \\Lambda \\} = \\Gamma ( \\alpha + 2 ) ^ { - 1 } \\displaystyle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { \\alpha + 1 } \\operatorname* { d e t } ( I + 2 s \\mu ^ { * } G ) ^ { - 1 / 2 } \\mathbb { V } _ { \\psi } ^ { ( 4 ) } \\left( \\frac { \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } G \\right) \\{ \\Lambda \\} } \\\\ & { \\quad \\quad = \\Gamma ( \\alpha + 2 ) ^ { - 1 } \\displaystyle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ s ^ { \\alpha + 1 } ( 1 + 2 s \\mu ^ { * } ) ^ { - ( B - 1 ) / 2 } \\left( \\frac { \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } \\right) ^ { \\alpha + 1 } \\mathbb { V } _ { \\psi } ^ { ( 4 ) } ( G ) \\{ \\Lambda \\} } \\\\ & { \\quad \\quad = \\Gamma ( \\alpha + 2 ) ^ { - 1 } \\mathbb { V } _ { \\psi } ^ { ( 4 ) } ( G ) \\{ \\Lambda \\} \\displaystyle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ ( \\mu ^ { * } s ) ^ { \\alpha + 1 } ( 1 + 2 s \\mu ^ { * } ) ^ { - ( B - 1 ) / 2 - \\alpha - 1 } } \\\\ & { \\quad \\quad = \\Gamma ( \\alpha + 2 ) ^ { - 1 } \\mathbb { V } _ { \\psi } ^ { ( 4 ) } ( G ) \\{ \\Lambda \\} \\mathrm { B e t a } ( \\frac { B - 3 } { 2 } , \\alpha + 2 ) 2 ^ { - \\alpha - 2 } \\mu ^ { * - 1 } } \\\\ & { \\quad \\quad = \\mathrm { P } \\left( \\frac { B - 3 } { 2 } , \\alpha + 2 \\right) ^ { - 1 } 2 ^ { - \\alpha - 2 } \\mu ^ { * - 1 } \\mathbb { V } _ { \\psi } ^ { ( 4 ) } ( G ) \\{ \\Lambda \\} . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 209, + 157, + 787, + 334 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "Theorem G.9. $B ^ { \\alpha } G ^ { \\otimes 2 } \\circ \\mathsf { B } \\ \\uparrow \\ \\mathcal { H } _ { B } ^ { G }$ has the following eigendecomposition (note that here $\\mu ^ { * }$ is still with respect to $\\phi$ , not $\\psi = \\alpha \\phi$ ) ", + "bbox": [ + 173, + 410, + 823, + 440 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "1. Eigenspace $\\mathbb { R } G$ with eigenvalue ", + "bbox": [ + 214, + 467, + 444, + 483 + ], + "page_idx": 74 + }, + { + "type": "equation", + "img_path": "images/01fa8c044eec88da4074ef271252941d370b4a624226d0d2d5748d331e227d4a.jpg", + "text": "$$\n\\lambda _ { G } ^ { \\mathsf { B } } : = \\frac { \\alpha ^ { 2 } } { B - 3 }\n$$", + "text_format": "latex", + "bbox": [ + 480, + 520, + 576, + 553 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "2. Eigenspace L with eigenvalue ", + "bbox": [ + 212, + 597, + 431, + 612 + ], + "page_idx": 74 + }, + { + "type": "equation", + "img_path": "images/ba3155c927e5f67d351203d428feeefd1d8f7c3a3b62aef21643f5a8142323dd.jpg", + "text": "$$\n\\begin{array} { r l } & { \\lambda _ { \\mathbb { L } } ^ { \\mathbb { B } } : = B ^ { \\alpha } \\mathrm { P } \\left( \\frac { B - 3 } { 2 } , \\alpha + 2 \\right) ^ { - 1 } 2 ^ { - \\alpha - 2 } \\mu ^ { * - 1 } 2 \\lambda _ { \\mathbb { L } } ^ { G , \\psi } ( B , \\alpha ) } \\\\ & { \\quad = \\frac { 2 \\alpha ^ { 3 } ( B - 2 ) } { ( B - 3 ) ( B - 1 ) ( B - 1 + 2 \\alpha ) } + \\frac { 2 \\alpha ^ { 2 } B J _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) } { ( B - 3 ) ( B - 1 ) ^ { 2 } ( B - 1 + 2 \\alpha ) ( J _ { \\phi } ( 1 ) - \\mathrm { J } _ { \\phi } \\left( \\frac { - 1 } { B - 1 } \\right) ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 230, + 648, + 852, + 742 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "3. Eigenspace M with eigenvalue ", + "bbox": [ + 212, + 785, + 436, + 800 + ], + "page_idx": 74 + }, + { + "type": "equation", + "img_path": "images/22e2dc36218c8989c9ba3ee5e1d0919f81df02e253801f828b588b60cd4c8542.jpg", + "text": "$$\n\\begin{array} { c } { { \\lambda _ { \\mathbb { M } } ^ { \\mathtt { B } } : = B ^ { \\alpha } \\mathrm { P } \\left( \\frac { B - 3 } { 2 } , \\alpha + 2 \\right) ^ { - 1 } 2 ^ { - \\alpha - 2 } \\mu ^ { * - 1 } 2 \\lambda _ { \\mathbb { M } } ^ { G , \\psi } ( B , \\alpha ) } } \\\\ { { = \\frac { 2 \\alpha ^ { 2 } B \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) } { ( B - 3 ) ( B - 1 ) ( B - 1 + 2 \\alpha ) ( \\mathrm { J } _ { \\phi } ( 1 ) - \\mathrm { J } _ { \\phi } \\left( \\frac { - 1 } { B - 1 } \\right) ) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 838, + 722, + 930 + ], + "page_idx": 74 + }, + { + "type": "text", + "text": "Proof. The only thing to justify is the value of $\\lambda _ { G } ^ { \\mathsf { B } }$ . We have ", + "bbox": [ + 173, + 102, + 570, + 119 + ], + "page_idx": 75 + }, + { + "type": "equation", + "img_path": "images/f02366a23579a37b4dadf92c0d6d37f03138d9febb01bcc0f2fda0d73e9e46cb.jpg", + "text": "$$\n\\begin{array} { r l } { \\lambda _ { C } ^ { \\mathbb { B } } = B ^ { \\alpha } \\mathrm { P } \\Bigg ( \\frac { B - 3 } { 2 } , \\alpha + 2 \\Bigg ) ^ { - 1 } 2 ^ { - \\alpha - 2 } \\mu ^ { * - 1 } } & { } \\\\ & { \\qquad \\times \\left( ( B - 1 ) \\mathrm { c } _ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha } \\left( \\mathrm { J } _ { \\mathcal { S } } ( 1 ) - \\mathrm { J } _ { \\Psi } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) + 2 \\lambda _ { \\Re } ^ { \\mathcal { A } , \\Re } ( B , \\alpha ) \\right) } \\\\ & { = B ^ { \\alpha } \\mathrm { P } \\Bigg ( \\frac { B - 3 } { 2 } , \\alpha + 2 \\Bigg ) ^ { - 1 } 2 ^ { - \\alpha - 2 } \\mu ^ { * - 1 } ( B - 1 + 2 \\alpha ) \\mathrm { c } _ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha } \\left( \\mathrm { J } _ { \\Phi } ( 1 ) - \\mathrm { J } _ { \\Psi } \\left( \\frac { - 1 } { B - 1 } \\right) \\right. } \\\\ & { = \\mathrm { c } _ { \\alpha } ( B - 1 ) ^ { \\alpha } ( B - 1 + 2 \\alpha ) \\mathrm { P } \\left( \\frac { B - 3 } { 2 } , \\alpha + 2 \\right) ^ { - 1 } 2 ^ { - \\alpha - 2 } \\mu ^ { * - 1 } \\left( \\mathrm { J } _ { \\Psi } ( 1 ) - \\mathrm { J } _ { \\Psi } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } \\\\ & { = \\mathrm { c } _ { \\alpha } ( B - 1 ) ^ { \\alpha } \\mathrm { P } \\left( \\frac { B - 3 } { 2 } , \\alpha + 1 \\right) ^ { - 1 } 2 ^ { - \\alpha - 1 } \\mu ^ { * - 1 } \\left( \\mathrm { J } _ { \\Phi } ( 1 ) - \\mathrm { J } _ { \\Psi } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } \\\\ & { = \\mathrm { c } _ { \\alpha } ( B - 1 ) ^ { \\alpha } \\mathrm { P } \\left( \\frac { B - 3 } { 2 } , \\alpha + 1 \\right) ^ { - 1 } 2 ^ { - \\alpha - 1 } \\mathrm { K } _ { \\alpha , B } ^ { - 1 } \\alpha ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 166, + 131, + 816, + 349 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "Computing C. By Lemma E.2, ", + "text_level": 1, + "bbox": [ + 173, + 564, + 393, + 580 + ], + "page_idx": 75 + }, + { + "type": "equation", + "img_path": "images/6199bae60706fff551cd0836e7f5198210868f8ca62e5f3c19e2e1471dde951e.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\sum _ { i } } \\{ \\Lambda \\} = \\mathbb { E } \\left[ r ^ { - 2 ( \\alpha + 1 ) } \\left( D \\Lambda g ^ { \\tau } D + D y g ^ { \\tau } \\Lambda D \\right) : y \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) \\right] } } \\\\ { { \\displaystyle \\quad = \\Gamma ( \\alpha + 1 ) ^ { - 1 } \\int \\mathrm { d } s ~ s ^ { \\alpha } \\mathrm { d e t } ( I + 2 s \\mu ^ { * } G ) ^ { - 1 / 2 } \\mathbb { E } [ D \\Lambda y g ^ { \\tau } D + D y g ^ { \\tau } \\Lambda D : y \\sim \\mathcal { N } ( 0 , \\frac { \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } \\zeta ^ { * } } } \\\\ { { \\displaystyle \\quad \\quad = \\Gamma ( \\alpha + 1 ) ^ { - 1 } \\int \\mathrm { d } s ~ s ^ { \\alpha } ( 1 + 2 s \\mu ^ { * } ) ^ { - ( B - 1 ) / 2 } \\left( \\frac { \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } \\right) ^ { \\alpha } \\mathbb { E } [ D \\Lambda g y ^ { \\tau } D + D y g ^ { \\tau } \\Lambda D : y > \\mathcal { N } } } \\\\ { { \\displaystyle \\quad = \\Gamma ( \\alpha + 1 ) ^ { - 1 } \\int \\mathrm { d } s ~ ( \\mu ^ { * } s ) ^ { \\alpha } ( 1 + 2 s \\mu ^ { * } ) ^ { - ( B - 1 ) / 2 - \\alpha } \\mathbb { E } [ D \\Lambda y g ^ { \\tau } D + D y g ^ { \\tau } \\Lambda D : y \\sim \\mathcal { N } ( 0 , G ) ] } } \\\\ { { \\displaystyle \\quad = \\Gamma ( \\alpha + 1 ) ^ { - 1 } \\mathrm { B e t a } \\left( \\frac { B - 3 } { 2 } , \\alpha + 1 \\right) 2 ^ { - \\alpha - 1 } \\mu ^ { * - 1 } \\mathbb { E } [ D \\Lambda y g ^ { \\tau } D + D y g ^ { \\tau } \\Lambda D : y \\sim \\mathcal { N } ( 0 , G ) ] } } \\\\ { { \\displaystyle \\quad = \\mathbb { P } \\left( \\frac { B - 3 } { 2 } , \\alpha + 1 \\right) ^ { - 1 } 2 ^ { - \\alpha - 1 } \\mu ^ { * - 1 } \\mathbb { E } [ D \\Lambda g y ^ { T } D + D y g ^ { \\tau } \\Lambda D : y \\sim \\mathcal { N } ( 0 , G ) ] } } \\\\ { { \\displaystyle \\quad = \\mathbb { P } \\left( \\frac { B - 3 } { 2 } , \\alpha + 1 \\right) ^ { - 1 } 2 ^ { - \\alpha - 1 } \\mu ^ { * - 1 } \\mathbb { E } [ D \\Lambda g ^ { \\tau } D + D y g ^ { \\tau } \\Lambda D : y \\sim \\mathcal { N } ( 0 , G ) ] } } \\end \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 590, + 839, + 795 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "Lemma G.10. Suppose $\\phi$ is degree $\\alpha$ positive-homogeneous. Then for $\\Lambda \\in \\mathcal { H } _ { B }$ , ", + "bbox": [ + 171, + 804, + 702, + 820 + ], + "page_idx": 75 + }, + { + "type": "equation", + "img_path": "images/6baab1e5c4b30d08fa86e75434d25b718d55bc4845e64c7eaefbd2fc94d8d01c.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\mathbb { E } [ D \\otimes ( D y y ^ { T } ) + D y y ^ { T } \\otimes D : y \\sim \\mathcal { N } ( 0 , \\Sigma ) ] = \\alpha \\left. \\frac { \\mathrm { d } { \\mathrm { V } _ { \\phi } } ( \\Pi \\Sigma \\Pi ) } { \\mathrm { d } \\Pi } \\right. _ { \\Pi = I } } } \\\\ { { \\displaystyle \\mathbb { E } [ D \\Lambda y y ^ { T } D + D y y ^ { T } \\Lambda D : y \\sim \\mathcal { N } ( 0 , \\Sigma ) ] = \\alpha \\frac { \\mathrm { d } { \\mathrm { V } _ { \\phi } } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\{ \\Sigma \\Lambda + \\Lambda \\Sigma \\} } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 258, + 829, + 740, + 900 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "where $D = \\operatorname { D i a g } ( \\phi ^ { \\prime } ( y ) )$ . ", + "bbox": [ + 173, + 909, + 339, + 925 + ], + "page_idx": 75 + }, + { + "type": "text", + "text": "Proof. Let $\\Pi _ { t } , t \\in ( - \\epsilon , \\epsilon )$ be a smooth path in $\\mathcal { H } _ { B }$ with $\\Pi _ { 0 } = I$ . Write $D _ { t } = \\mathrm { D i a g } ( \\phi ^ { \\prime } ( \\Pi _ { t } y ) )$ , so that $D _ { 0 } = D$ . Then, using $\\dot { \\Pi } _ { t }$ to denote $t$ derivative, ", + "bbox": [ + 171, + 102, + 825, + 135 + ], + "page_idx": 76 + }, + { + "type": "equation", + "img_path": "images/30d5a0a9ddbd0436cbf2bb576f57832de8b11866f0c8cac67a37c1abc8d25995.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { d } { d t } \\mathrm { V } _ { \\phi } ( \\Pi _ { t } \\Sigma \\Pi _ { t } ) = \\frac { d } { d t } \\mathbb { E } [ \\phi ( \\Pi _ { t } y ) \\phi ( \\Pi _ { t } y ) ^ { T } : y \\sim \\mathcal { N } ( 0 , \\Sigma ) ] } \\\\ { \\displaystyle \\qquad = \\mathbb { E } [ D _ { t } \\dot { \\Pi } _ { t } y \\phi ( \\Pi _ { t } y ) ^ { T } + \\phi ( \\Pi _ { t } y ) y ^ { T } \\dot { \\Pi } _ { t } D _ { t } : y \\sim \\mathcal { N } ( 0 , \\Sigma ) ] } \\\\ { \\displaystyle \\frac { d } { d t } \\mathrm { V } _ { \\phi } ( \\Pi _ { t } \\Sigma \\Pi _ { t } ) \\Bigg | _ { t = 0 } = \\mathbb { E } [ D \\dot { \\Pi } _ { 0 } y \\phi ( y ) ^ { T } + \\phi ( y ) y ^ { T } \\dot { \\Pi } _ { 0 } D : y \\sim \\mathcal { N } ( 0 , \\Sigma ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 250, + 138, + 751, + 228 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "Because for $x \\in \\mathbb { R }$ , $\\alpha \\phi ( x ) = x \\phi ^ { \\prime } ( x )$ , for $\\boldsymbol { y } \\in \\mathbb { R } ^ { B }$ we can write $\\phi ( y ) = \\alpha ^ { - 1 } \\mathrm { D i a g } ( \\phi ^ { \\prime } ( y ) ) y ,$ . Then ", + "bbox": [ + 169, + 231, + 803, + 250 + ], + "page_idx": 76 + }, + { + "type": "equation", + "img_path": "images/46036cc6fa043de27aaed2f06f151b96aa344b0ba6815ec1a0b686fd4f0a3a02.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle \\alpha \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\{ \\Sigma \\dot { \\Pi } _ { 0 } + \\dot { \\Pi } _ { 0 } \\Sigma \\} = \\alpha \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } ( \\Sigma ) } { \\mathrm { d } \\Sigma } \\left\\{ \\frac { d \\Pi _ { t } \\Sigma \\Pi _ { t } } { d t } \\right\\} } \\\\ { \\displaystyle \\qquad = \\alpha \\left. \\frac { d } { d t } \\mathrm { V } _ { \\phi } ( \\Pi _ { t } \\Sigma \\Pi _ { t } ) \\right. _ { t = 0 } } \\\\ { \\displaystyle \\qquad = \\mathbb { E } [ D \\dot { \\Pi } _ { 0 } y y ^ { T } D + D y y ^ { T } \\dot { \\Pi } _ { 0 } D : y \\sim \\mathcal { N } ( 0 , \\Sigma ) ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 256, + 253, + 741, + 347 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "Therefore, for $\\Lambda \\in \\mathcal { H } _ { B } ^ { G }$ , ", + "bbox": [ + 174, + 386, + 333, + 402 + ], + "page_idx": 76 + }, + { + "type": "equation", + "img_path": "images/165514c81718b8ff680524a2f35d5ae2f108b3ca9f9be11d49c0c33d3dabe1b0.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\mathsf { C } } \\{ \\Lambda \\} = { \\mathsf { P } } \\left( \\frac { B - 3 } { 2 } , \\alpha + 1 \\right) ^ { - 1 } 2 ^ { - \\alpha - 1 } \\mu ^ { * - 1 } \\alpha \\left. \\frac { \\mathrm { d } { \\mathsf { V } } _ { \\phi } ( { \\Sigma } ) } { \\mathrm { d } \\Sigma } \\right. _ { { \\Sigma } = G } \\left\\{ G \\Lambda + \\Lambda G \\right\\} } } \\\\ { { \\displaystyle ~ = 2 \\alpha { \\mathsf { P } } \\left( \\frac { B - 3 } { 2 } , \\alpha + 1 \\right) ^ { - 1 } 2 ^ { - \\alpha - 1 } \\mu ^ { * - 1 } \\left. \\frac { \\mathrm { d } { \\mathsf { V } } _ { \\phi } ( { \\Sigma } ) } { \\mathrm { d } \\Sigma } \\right. _ { { \\Sigma } = G } \\left\\{ \\Lambda \\right\\} } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 253, + 409, + 743, + 486 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "So Thm F.31 gives ", + "text_level": 1, + "bbox": [ + 173, + 515, + 300, + 530 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "Theorem G.11. $B ^ { \\alpha } G ^ { \\otimes 2 } \\circ \\mathsf C$ has the following eigendecomposition ", + "bbox": [ + 174, + 532, + 619, + 549 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "1. eigenspace $\\mathbb { R } G$ with eigenvalue ", + "bbox": [ + 214, + 559, + 442, + 574 + ], + "page_idx": 76 + }, + { + "type": "equation", + "img_path": "images/4b0f4d58e53978d7cd2bed8a9a8faa7b78eca9af5b14cb631070300a42cb9a4d.jpg", + "text": "$$\n\\begin{array} { l } { { \\lambda _ { G } ^ { \\mathrm { \\tiny { C } } } : = 2 B ^ { \\alpha } \\alpha \\mathrm { P } \\left( \\displaystyle \\frac { B - 3 } { 2 } , \\alpha + 1 \\right) ^ { - 1 } 2 ^ { - \\alpha - 1 } \\mu ^ { * - 1 } \\lambda _ { G } ^ { G , \\phi } ( B , \\alpha ) } } \\\\ { { \\mathrm { \\tiny { ~ \\alpha ~ } } = \\displaystyle \\frac { 2 \\alpha ^ { 2 } } { B - 3 } \\qquad } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 328, + 579, + 725, + 652 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "2. eigenspace L with eigenvalue ", + "bbox": [ + 212, + 662, + 429, + 678 + ], + "page_idx": 76 + }, + { + "type": "equation", + "img_path": "images/ff1f9feeec4e115d25890d7fdd11ebcc6a78c6f1510c0cc3dcb393a672bf5197.jpg", + "text": "$$\n\\begin{array} { r l } & { \\lambda _ { \\perp } ^ { \\mathsf { C } } : = 2 B ^ { \\alpha } \\alpha \\mathrm { P } \\left( \\frac { B - 3 } { 2 } , \\alpha + 1 \\right) ^ { - 1 } 2 ^ { - \\alpha - 1 } \\mu ^ { * - 1 } \\lambda _ { \\perp } ^ { G , \\phi } ( B , \\alpha ) } \\\\ & { \\quad = \\frac { 2 \\alpha ^ { 2 } ( B - 2 ) } { ( B - 3 ) ( B - 1 ) } + \\frac { 2 \\alpha B \\mathrm { J } _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) } { ( B - 3 ) ( B - 1 ) ^ { 2 } \\left( \\mathrm { J } _ { \\phi } ( 1 ) - \\mathrm { J } _ { \\phi } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 308, + 683, + 746, + 775 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "3. eigenspace M with eigenvalue ", + "bbox": [ + 212, + 786, + 433, + 801 + ], + "page_idx": 76 + }, + { + "type": "equation", + "img_path": "images/13f73be182ccf42f55f3f966f1c0995e1f4455b1d41d618a3c1933d721a8408a.jpg", + "text": "$$\n\\begin{array} { l } { { \\lambda _ { \\mathbb { M } } ^ { \\mathbb { C } } : = 2 B ^ { \\alpha } \\alpha \\mathrm { P } \\left( \\frac { B - 3 } { 2 } , \\alpha + 1 \\right) ^ { - 1 } 2 ^ { - \\alpha - 1 } \\mu ^ { * - 1 } \\lambda _ { \\mathbb { M } } ^ { G , \\phi } ( B , \\alpha ) } } \\\\ { { \\mathrm { ~ } = \\frac { 2 \\alpha B J _ { \\phi } ^ { \\prime } \\left( \\frac { - 1 } { B - 1 } \\right) } { ( B - 3 ) ( B - 1 ) ( \\mathrm { J } _ { \\phi } ( 1 ) - \\mathrm { J } _ { \\phi } \\left( \\frac { - 1 } { B - 1 } \\right) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 328, + 806, + 728, + 898 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "Altogether, by Eq. (53) and Thms G.6, G.9 and G.11, this implies ", + "bbox": [ + 173, + 909, + 604, + 924 + ], + "page_idx": 76 + }, + { + "type": "text", + "text": "Theorem G.12. $G ^ { \\otimes 2 } \\circ \\mathcal { F } ( \\Sigma ^ { * } ) : \\mathcal { H } _ { B } ^ { G } \\to \\mathcal { H } _ { B } ^ { G }$ has eigenspaces $\\mathbb { R } G , \\mathbb { M } , \\mathbb { L }$ respectively with the following eigenvalues ", + "bbox": [ + 173, + 101, + 823, + 132 + ], + "page_idx": 77 + }, + { + "type": "equation", + "img_path": "images/6aaacb50e7d03758a59ac5c6b6b4abaf75c93005a3b2a0f5dff036ec531a1abe.jpg", + "text": "$$\n\\begin{array} { r l } { \\lambda _ { 1 } ^ { \\mu } = } & { \\frac { \\lambda ^ { \\mu } } { 2 } ( \\mu - 3 ) + 2 \\lambda _ { 1 } ^ { \\mu } ( \\frac { \\lambda _ { 2 } } { \\mu } ( \\frac { \\lambda _ { 1 } ^ { \\mu } } { \\mu } ) + \\frac { \\lambda _ { 2 } ^ { \\mu } } { \\mu } ( \\frac { \\lambda _ { 2 } ^ { \\mu } } { \\mu } ) + \\frac { \\lambda _ { 3 } ^ { \\mu } } { 2 } ( \\mu - 3 ) \\lambda _ { 2 } ^ { \\mu } ) , \\quad \\psi _ { 2 } ^ { \\mu } = 2 } \\\\ & { - \\frac { \\lambda _ { 3 } ^ { \\mu } } { 2 } ( \\mu - 3 ) ( \\mu - 5 ) \\beta _ { 3 } ^ { \\mu } - 1 5 \\zeta _ { 2 } ^ { \\mu } ( \\mu - 1 ) ( \\frac { \\lambda _ { 2 } ^ { \\mu } } { \\mu } ) ^ { \\mu } } \\\\ & { - c ^ { \\mu } ( \\mu - 3 ) c ^ { \\mu } ( \\mu - 3 ) c ^ { \\mu } ( \\mu - 1 ) c ^ { \\mu } ( \\frac { \\lambda _ { 3 } ^ { \\mu } } { \\mu } ) ^ { \\mu } } \\\\ & { - \\frac { \\lambda _ { 3 } ^ { \\mu } } { 2 } ( \\mu - 3 ) ( \\lambda _ { 3 } ^ { \\mu } + 3 ) - \\frac { \\lambda _ { 3 } ^ { \\mu } } { 2 } ( \\mu - 1 ) c ^ { \\mu } ( \\frac { \\lambda _ { 2 } ^ { \\mu } } { \\mu } ) ^ { \\mu } } \\\\ & { - c ^ { \\mu } ( \\frac { \\lambda _ { 3 } ^ { \\mu } } { \\mu } - 3 ) ( \\lambda _ { 3 } ^ { \\mu } + \\frac { \\lambda _ { 3 } ^ { \\mu } } { 2 } ) ^ { \\mu } , \\quad \\psi _ { 2 } ^ { \\mu } = 2 } \\\\ & - c ^ { \\mu } ( \\frac { \\lambda _ { 3 } ^ { \\mu } } { 2 } c ^ { \\mu } - 3 ) ( \\frac { \\lambda _ { 3 } ^ { \\mu } } { 2 } \\lambda _ { 3 } ^ { \\mu } ) ^ { \\mu } ( \\frac { \\lambda _ { 3 } ^ { \\mu } } { \\mu } ) ^ { \\mu } ( \\frac { \\lambda _ { 2 } ^ { \\mu } } { \\mu } ) ^ { \\mu } ( \\frac { \\lambda _ { 3 } ^ { \\mu } } { \\mu } ) ^ { \\mu } \\psi _ { 2 } ^ { \\mu } ( \\frac \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 178, + 136, + 816, + 488 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "H CROSS BATCH: FORWARD DYNAMICS ", + "text_level": 1, + "bbox": [ + 173, + 496, + 526, + 513 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "In this section, we study the generalization of Eq. (43) to multiple batches. ", + "bbox": [ + 174, + 526, + 663, + 541 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "Definition H.1. For linear operators $\\mathcal T _ { i } : \\mathcal X _ { i } \\mathcal y _ { i } , i = 1 , . . . , k$ , we write $\\oplus _ { i } \\mathcal { T } _ { i } : \\oplus _ { i } \\mathcal { X } _ { i } \\to \\oplus _ { i } \\mathcal { V } _ { i }$ for the operator $( x _ { 1 } , \\ldots , x _ { k } ) \\mapsto ( { \\mathcal { T } } _ { 1 } ( x _ { 1 } ) , \\ldots , { \\mathcal { T } } _ { k } ( x _ { k } ) )$ . We also write $\\mathcal { T } _ { 1 } ^ { \\oplus n } : = \\oplus _ { j = 1 } ^ { n } \\mathcal { T } _ { 1 }$ for the direct sum of $n$ copies of $\\mathcal { T } _ { 1 }$ . ", + "bbox": [ + 173, + 542, + 826, + 589 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "For $k \\geq 2$ , now consider the extended ( $^ { 6 6 } k$ -batch”) dynamics on $\\widetilde { \\Sigma } \\in { S } _ { k B }$ defined by ", + "bbox": [ + 171, + 598, + 725, + 616 + ], + "page_idx": 77 + }, + { + "type": "equation", + "img_path": "images/06d2c25e42a6dd2ec9c4f5ef9655145f1e814bf2c26c202225d5c8ac65e3f78a.jpg", + "text": "$$\n\\widetilde { \\Sigma } ^ { l } = \\mathrm { V } _ { B _ { \\phi } ^ { \\oplus k } } ( \\widetilde { \\Sigma } ^ { l - 1 } )\n$$", + "text_format": "latex", + "bbox": [ + 232, + 616, + 357, + 641 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "If we restrict the dynamics to just the upper left $B \\times B$ submatrix (as well as any of the diagonal $B \\times B$ blocks) of $\\widetilde { \\Sigma } ^ { l }$ , then we recover Eq. (43). ", + "bbox": [ + 173, + 662, + 823, + 694 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "H.1 LIMIT POINTS ", + "text_level": 1, + "bbox": [ + 174, + 708, + 318, + 723 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "In general, like in the case of Eq. (43), it is difficult to prove global convergence behavior. Thus we manually look for fixed points and the local convergence behaviors around them. A very natural extension of the notion of BSB1 matrices is the following ", + "bbox": [ + 173, + 734, + 825, + 776 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "Definition H.2. We say a matrix $\\widetilde { \\Sigma } \\in { \\cal S } _ { k B }$ is CBSB1 (short for “1-Step Cross-Batch Symmetry Breaking”) if $\\widetilde { \\Sigma }$ in block form $k \\times k$ blocks, each of size $B \\times B )$ has one common BSB1 block on the diagonal and one common constant block on the off-diagonal, i.e. ", + "bbox": [ + 173, + 779, + 825, + 825 + ], + "page_idx": 77 + }, + { + "type": "equation", + "img_path": "images/827620e42fc3c1f83e1c010c2c3b14083f5a5d8db8d2b7a822ac60b661981304.jpg", + "text": "$$\n\\begin{array} { r } { \\left( \\begin{array} { c c c c } { \\mathrm { B S B 1 } ( a , b ) } & { c { \\bf 1 1 } ^ { T } } & { c { \\bf 1 1 } ^ { T } } & { \\cdots } \\\\ { c { \\bf 1 1 } ^ { T } } & { \\mathrm { B S B 1 } ( a , b ) } & { c { \\bf 1 1 } ^ { T } } & { \\cdots } \\\\ { c { \\bf 1 1 } ^ { T } } & { c { \\bf 1 1 } ^ { T } } & { \\mathrm { B S B 1 } ( a , b ) } & { \\cdots } \\\\ { \\vdots } & { \\vdots } & { \\vdots } & { \\ddots } \\end{array} \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 333, + 825, + 663, + 898 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "We will study the fixed point $\\widetilde { \\Sigma } ^ { * }$ to Eq. (54) of CBSB1 form. ", + "bbox": [ + 173, + 907, + 575, + 924 + ], + "page_idx": 77 + }, + { + "type": "text", + "text": "H.1.1 SPHERICAL INTEGRATION AND GEGENBAUER EXPANSION ", + "bbox": [ + 173, + 102, + 637, + 119 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "Theorem H.3. Let $\\widetilde { \\Sigma } \\ \\in \\ S _ { k B }$ . If $\\widetilde { \\Sigma }$ is CBSB1 then $\\nabla _ { \\boldsymbol { B } _ { \\phi } ^ { \\oplus k } } ( \\widetilde { \\Sigma } )$ is CBSB1 and equals $\\widetilde { \\Sigma } ^ { * } : =$ $\\left( \\begin{array} { c c } { { \\Sigma ^ { * } } } & { { c ^ { * } } } \\\\ { { c ^ { * } } } & { { \\Sigma ^ { * } } } \\end{array} \\right)$ where $\\Sigma ^ { * }$ is the BSB1 fixed point of Thm F.5 and $c ^ { * } \\geq 0$ with $\\begin{array} { r l } { \\sqrt { c ^ { * } } } & { { } = } \\end{array}$ $\\begin{array} { r } { \\frac { \\dot { \\Gamma ( ( B - 1 ) / 2 ) } } { \\Gamma ( ( B - 2 ) / 2 ) } \\pi ^ { - 1 / 2 } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta \\ \\phi ( - \\sqrt { B - 1 } \\cos \\theta ) \\sin ^ { B - 3 } \\theta } \\end{array}$ . If $\\phi ( { \\sqrt { B - 1 } } x )$ has Gegenbauer expansion $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ , then $c ^ { * } = a _ { 0 } ^ { 2 }$ . ", + "bbox": [ + 173, + 123, + 826, + 223 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "Proof. We will prove for $k = 2$ . The general $k$ cases follow from the same reasoning. ", + "bbox": [ + 173, + 236, + 735, + 253 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "Let $\\widetilde { \\Sigma } = \\left( \\begin{array} { c c } { { \\Sigma } } & { { c \\mathbf { 1 } \\mathbf { 1 } ^ { T } } } \\\\ { { c \\mathbf { 1 } \\mathbf { 1 } ^ { T } } } & { { \\Sigma } } \\end{array} \\right)$ where $\\Sigma = \\mathrm { B S B } 1 ( a , b )$ . As remarked below Eq. (54), restricting to any diagonal blocks just recovers the dynamics of Eq. (43), which gives the claim about the diagonal blocks being $\\Sigma ^ { * }$ through Thm F.5. ", + "bbox": [ + 173, + 258, + 826, + 319 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "We now look at the off-diagonal blocks. ", + "bbox": [ + 174, + 324, + 436, + 339 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "where $\\begin{array} { r l } { \\nabla _ { B _ { \\phi } ^ { \\oplus 2 } } ( \\widetilde { \\Sigma } ) _ { 1 : B , B + 1 : 2 B } = \\mathbb { E } [ \\mathcal { B } _ { \\phi } ( z _ { 1 : B } ) \\otimes \\mathcal { B } _ { \\phi } ( z _ { B + 1 : 2 B } ) : z \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ) ] } & { } \\\\ & { = \\mathbb { E } [ \\phi \\circ \\boldsymbol { \\mathrm { n } } ( y _ { 1 : B } ) \\otimes \\phi \\circ \\boldsymbol { \\mathrm { n } } ( y _ { B + 1 : 2 B } ) : y \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { G } ) ] } \\\\ { \\widetilde { \\Sigma } ^ { G } } & : = \\begin{array} { r l } { G ^ { \\oplus 2 } \\widetilde { \\Sigma } G ^ { \\oplus 2 } } & { = \\begin{array} { r l } { \\binom { G } { 0 } \\widetilde { \\Sigma } \\left( \\begin{array} { c c } { G } & { 0 } \\\\ { 0 } & { G } \\end{array} \\right) \\widetilde { \\Sigma } \\left( \\begin{array} { c c } { G } & { 0 } \\\\ { 0 } & { G } \\end{array} \\right) } & { = \\begin{array} { r l } { \\binom { G \\Sigma G } { c 4 1 ^ { T } G } } & { G \\Sigma G \\pmb { \\mathrm { \\Delta } } } \\end{array} } \\end{array} } \\end{array} \\end{array}$ $\\left( \\begin{array} { c c } { { ( a - b ) G } } & { { 0 } } \\\\ { { 0 } } & { { ( a - b ) G } } \\end{array} \\right)$ (the last step follows from Lemma E.52). Thus $y _ { 1 : B }$ is independent from $y _ { B + 1 : 2 B }$ , and ", + "bbox": [ + 171, + 339, + 828, + 465 + ], + "page_idx": 78 + }, + { + "type": "equation", + "img_path": "images/1935c016aaebecd115379903eff270e1700c95ffc03060c4afc75811e65db836.jpg", + "text": "$$\n\\begin{array} { r } { \\mathsf { V } _ { \\mathcal { B } _ { \\phi } ^ { \\oplus 2 } } \\bigl ( \\widetilde { \\Sigma } \\bigr ) _ { 1 : B , B + 1 : 2 B } = \\mathbb { E } [ \\phi \\circ \\mathsf { n } ( x ) : x \\sim \\mathcal { N } ( 0 , ( a - b ) G ) ] ^ { \\otimes 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 465, + 699, + 491 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "By symmetry, ", + "bbox": [ + 173, + 492, + 267, + 507 + ], + "page_idx": 78 + }, + { + "type": "equation", + "img_path": "images/e4fb62351742542bbd5b040573e9689fbeaaba3fc4eeee31a86da1aa311995a6.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbb { E } [ \\phi \\circ \\mathsf { n } ( x ) : x \\sim \\mathcal { N } ( 0 , ( a - b ) G ) ] = \\sqrt { c ^ { * } } \\mathbf { 1 } } \\\\ { \\mathbb { E } [ \\phi \\circ \\mathsf { n } ( x ) : x \\sim \\mathcal { N } ( 0 , ( a - b ) G ) ] ^ { \\otimes 2 } = c ^ { * } \\mathbf { 1 } \\mathbf { 1 } ^ { T } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 343, + 507, + 655, + 547 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "where $\\sqrt { c ^ { * } } : = \\mathbb { E } [ \\phi ( \\mathfrak { n } ( x ) _ { 1 } ) : x \\sim \\mathcal { N } ( 0 , ( a - b ) G ) ]$ . We can compute ", + "bbox": [ + 174, + 550, + 622, + 568 + ], + "page_idx": 78 + }, + { + "type": "equation", + "img_path": "images/58235b701b18e6bb58da8a08df23655d61d40a9fe21c1c166b214a36e02349c0.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { = \\mathbb { E } [ \\phi ( { \\mathsf { n } } ( \\mathfrak { e x } ) _ { 1 } ) : x \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] \\qquad \\mathrm { w h e r e ~ e i s ~ a s ~ i n ~ E q . ~ } ( 3 3 ) } \\\\ & { } & { = \\mathbb { E } [ \\phi ( ( \\mathsf { e n } ( x ) ) _ { 1 } ) : x \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] } \\\\ & { } & { = \\mathbb { E } [ \\phi ( \\sqrt { B } ( \\mathfrak { e v } ) _ { 1 } ) : x \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] \\qquad \\mathrm { b e c a u s e ~ \\bar { e } ~ i s ~ a n ~ i s o m e t r y } } \\\\ & { } & \\\\ & { } & { = \\mathbb { E } [ \\phi ( \\sqrt { B } ( \\mathfrak { e v } ) _ { 1 } ) : x \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] \\qquad \\mathrm { w i t h ~ } v = x / \\| x \\| } \\\\ & { } & { = \\frac { \\Gamma ( ( B - 1 ) / 2 ) } { \\Gamma ( ( B - 2 ) / 2 ) } \\pi ^ { - 1 / 2 } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta \\phi ( - \\sqrt { B - 1 } \\cos \\theta ) \\sin ^ { B - 3 } \\theta } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 604, + 714, + 751 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "by Lemma E.37. At the same time, by Proposition E.33, we can obtain the Gegenbauer expansion ", + "bbox": [ + 173, + 751, + 810, + 766 + ], + "page_idx": 78 + }, + { + "type": "equation", + "img_path": "images/e057e85509e76202f3ffb44e49a5ab4e623ae2315e5299809efd3d0ddc9aeadd.jpg", + "text": "$$\n\\begin{array} { r l } & { \\sqrt { c ^ { * } } = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B - 1 } \\hat { \\mathbf { e } } _ { 1 , \\tau } ) } \\\\ & { \\quad = \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\displaystyle \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( \\mathbf { e } _ { 1 , \\tau } v ) } \\\\ & { \\quad = \\left. \\displaystyle \\sum _ { l = 0 } ^ { \\infty } a _ { l } Z _ { \\hat { \\mathbf { e } } _ { 1 , z } } ^ { B - 2 , ( l ) } , 1 \\right. } \\\\ & { \\quad = a _ { 0 } Z _ { \\hat { \\mathbf { e } } _ { 1 , z } } ^ { B - 2 , ( 0 ) } ( \\hat { \\mathbf { e } } _ { 1 , z } ) } \\\\ & { \\quad = a _ { 0 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 767, + 643, + 922 + ], + "page_idx": 78 + }, + { + "type": "text", + "text": "Corollary H.4. With the notation as in Thm $\\cdot$ , if $\\phi$ is positive-homogeneous of degree $\\alpha$ and $\\phi ( x ) = a \\rho _ { \\alpha } ( x ) - b \\rho _ { \\alpha } ( - x )$ , then $\\begin{array} { r } { \\sqrt { c ^ { * } } = ( a - b ) \\frac { 1 } { 2 \\sqrt { \\pi } } ( B - 1 ) ^ { \\alpha / 2 } \\frac { \\Gamma ( ( B - 1 ) / 2 ) \\Gamma ( ( \\alpha + 1 ) / 2 ) } { \\Gamma ( ( \\alpha + B - 1 ) / 2 ) } } \\end{array}$ . ", + "bbox": [ + 173, + 148, + 826, + 185 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "Proof. We compute ", + "bbox": [ + 173, + 223, + 307, + 239 + ], + "page_idx": 79 + }, + { + "type": "equation", + "img_path": "images/6e569902a95700db60468811b54717d8af56ffd40f2591a8a1209fd743285f17.jpg", + "text": "$$\n\\int _ { 0 } ^ { \\pi / 2 } \\mathrm { d } \\theta ( \\cos \\theta ) ^ { \\alpha } \\sin ^ { B - 3 } \\theta = \\frac { 1 } { 2 } \\mathrm { B e t a } \\left( 0 , 1 ; \\frac { \\alpha + 1 } { 2 } , \\frac { B - 2 } { 2 } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 299, + 252, + 704, + 290 + ], + "page_idx": 79 + }, + { + "type": "equation", + "img_path": "images/20d2767afdd9bf35f7891432cb99e5f5502dd00d07e136d2264d4c618d715a64.jpg", + "text": "$$\n\\begin{array} { r l r } & { } & { = \\displaystyle \\frac { 1 } { 2 } \\mathrm { B e t a } \\left( \\frac { \\alpha + 1 } { 2 } , \\frac { B - 2 } { 2 } \\right) } \\\\ & { } & { \\displaystyle \\int _ { \\pi / 2 } ^ { \\pi } \\mathrm { d } \\theta \\left( - \\cos \\theta \\right) ^ { \\alpha } \\sin ^ { B - 3 } \\theta = \\int _ { 0 } ^ { \\pi / 2 } \\mathrm { d } \\theta \\left( \\cos \\theta \\right) ^ { \\alpha } \\sin ^ { B - 3 } \\theta } \\\\ & { } & { = \\displaystyle \\frac { 1 } { 2 } \\mathrm { B e t a } \\left( \\frac { \\alpha + 1 } { 2 } , \\frac { B - 2 } { 2 } \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 290, + 309, + 684, + 420 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "So for a positive homogeneous function $\\phi ( x ) = a \\rho _ { \\alpha } ( x ) - b \\rho _ { \\alpha } ( - x )$ , ", + "bbox": [ + 173, + 431, + 625, + 448 + ], + "page_idx": 79 + }, + { + "type": "equation", + "img_path": "images/7701b0b087d74bbe2674d481f9f25d11ea06d1d77ca10263eba43c1596f42299.jpg", + "text": "$$\n{ \\begin{array} { r l } & { { \\sqrt { c ^ { * } } } = { \\frac { \\Gamma ( ( B - 1 ) / 2 ) } { \\Gamma \\left( ( B - 2 ) / 2 \\right) } } \\pi ^ { - 1 / 2 } \\left( a \\int _ { \\pi / 2 } ^ { \\pi } \\mathrm { d } \\theta \\left( - { \\sqrt { B - 1 } } \\cos \\theta \\right) ^ { \\alpha } \\sin ^ { B - 3 } \\theta \\right. } \\\\ & { \\qquad \\left. - b \\int _ { 0 } ^ { \\pi / 2 } \\mathrm { d } \\theta \\left( { \\sqrt { B - 1 } } \\cos \\theta \\right) ^ { \\alpha } \\sin ^ { B - 3 } \\theta \\right) } \\\\ & { = ( a - b ) { \\frac { \\Gamma ( ( B - 1 ) / 2 ) } { \\Gamma ( ( B - 2 ) / 2 ) } } \\pi ^ { - 1 / 2 } ( B - 1 ) ^ { \\alpha / 2 } { \\frac { 1 } { 2 } } \\operatorname { B e t a } \\left( { \\frac { \\alpha + 1 } { 2 } } , { \\frac { B - 2 } { 2 } } \\right) } \\\\ & { = ( a - b ) { \\frac { 1 } { 2 { \\sqrt { \\pi } } } } ( B - 1 ) ^ { \\alpha / 2 } { \\frac { \\Gamma ( ( B - 1 ) / 2 ) \\Gamma ( ( \\alpha + 1 ) / 2 ) } { \\Gamma ( ( \\alpha + B - 1 ) / 2 ) } } } \\end{array} }\n$$", + "text_format": "latex", + "bbox": [ + 253, + 462, + 743, + 608 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "Expanding the beta function and combining with Thm H.3 gives the desired result. ", + "bbox": [ + 171, + 625, + 714, + 641 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "H.1.2 LAPLACE METHOD ", + "text_level": 1, + "bbox": [ + 174, + 680, + 366, + 695 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "While we don’t need to use the Laplace method to compute $c ^ { * }$ for positive homogeneous functions (since it is already given by Corollary H.4), going through the computation is instructive for the machinery for computing the eigenvalues in a later section. ", + "bbox": [ + 173, + 707, + 825, + 751 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "Lemma H.5 (The Laplace Method Cross Batch Master Equation). For $A , B , C \\ \\in \\ \\mathbb { N } ,$ , let $f :$ $\\mathbb { R } ^ { A + B } \\mathbb { R } ^ { C }$ and let $a , b \\geq 0$ . Suppose for any $y \\in \\mathbb { R } ^ { A } , z \\in \\mathbb { R } ^ { B }$ , $\\| f ( y , z ) \\| \\leq h ( \\sqrt { \\| y \\| ^ { 2 } + \\| z \\| ^ { 2 } } )$ for some nondecreasing function $h : \\mathbb { R } ^ { \\geq 0 } \\mathbb { R } ^ { \\geq 0 }$ such that $\\mathbb { E } [ h ( r \\| z \\| ) : z \\in \\mathcal { N } ( 0 , I _ { A + B } ) ]$ exists for every $r \\geq 0$ . Define $\\wp ( \\Sigma ) : = \\mathbb { E } [ \\| y \\| ^ { - 2 a } \\| z \\| ^ { - 2 b } f ( y , z ) : ( y , z ) \\sim \\mathcal { N } ( 0 , \\Sigma ) ]$ ]. Then on $\\{ \\Sigma \\in { \\mathcal { S } } _ { A + B } : \\operatorname { r a n k } \\Sigma > 2 ( a + b ) \\}$ , $\\wp ( \\Sigma )$ is well-defined and continuous, and furthermore satisfies ", + "bbox": [ + 173, + 757, + 825, + 834 + ], + "page_idx": 79 + }, + { + "type": "equation", + "img_path": "images/9cf0f498cffc0bca92efd518a42bfa85bed033be902952beafe87f0b883d9f05.jpg", + "text": "$$\n\\wp ( \\Sigma ) = \\Gamma ( a ) ^ { - 1 } \\Gamma ( b ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } \\mathrm { d } t s ^ { a - 1 } t ^ { b - 1 } \\operatorname * { d e t } ( I _ { A + B } + 2 \\Omega ) ^ { - 1 / 2 } \\underset { ( y , z ) \\sim \\mathcal { N } ( 0 , \\Pi ) } { \\mathbb { E } } f ( y , z )\n$$", + "text_format": "latex", + "bbox": [ + 179, + 845, + 789, + 881 + ], + "page_idx": 79 + }, + { + "type": "text", + "text": "Proof. If $\\Sigma$ is full rank, we can show Fubini-Tonelli theorem is valid in the following computation by the same arguments of the proof of Lemma E.2. ", + "bbox": [ + 171, + 102, + 825, + 132 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/7294911ddcda6e5bc4cd0a1fd9eb37a8767860bf8004b8971756c8e66737c5ae.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathbb { E } [ \\| y \\| ^ { - 2 a } \\| z \\| ^ { - 2 b } f ( y , z ) : ( y , z ) \\sim \\mathcal { N } ( 0 , \\Sigma ) ] } \\\\ & { = \\mathbb { E } [ f ( y , z ) \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\Gamma ( a ) ^ { - 1 } s ^ { a - 1 } e ^ { - \\| y \\| ^ { 2 } s } \\int _ { 0 } ^ { \\infty } \\mathrm { d } t \\Gamma ( b ) ^ { - 1 } t ^ { b - 1 } e ^ { - \\| y \\| ^ { 2 } t } : ( y , z ) \\sim \\mathcal { N } ( 0 , \\Sigma ) ] } \\\\ & { = ( 2 \\pi ) ^ { - \\frac { A + B } { 2 } } \\Gamma ( a ) ^ { - 1 } \\Gamma ( b ) ^ { - 1 } \\operatorname* { d e t } \\Sigma ^ { - 1 / 2 } } \\\\ & { \\qquad \\times \\displaystyle \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } \\mathrm { d } t s ^ { a - 1 } t ^ { b - 1 } \\int _ { \\mathbb { R } ^ { A + B } } \\mathrm { d } y \\mathrm { ~ d } z f ( y , z ) e ^ { - \\frac { 1 } { 2 } ( y , z ) ( \\Sigma ^ { - 1 } + 2 D ^ { 2 } ) ( y , z ) ^ { T } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 207, + 132, + 795, + 247 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/a2230ecfd24feb5a67045369f28c8f8e955d4e5f3f2ca768bf1c30160de56502.jpg", + "text": "$$\n= \\Gamma ( a ) ^ { - 1 } \\Gamma ( b ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ \\int _ { 0 } ^ { \\infty } \\mathrm { d } t \\ s ^ { a - 1 } t ^ { b - 1 } \\operatorname * { d e t } ( \\Sigma ( \\Sigma ^ { - 1 } + 2 D ^ { 2 } ) ) ^ { - 1 / 2 } \\mathbb { E } f ( y , z )\n$$", + "text_format": "latex", + "bbox": [ + 207, + 263, + 732, + 299 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "where in the last line, $( y , z ) \\sim \\mathcal { N } ( 0 , ( \\Sigma ^ { - 1 } + 2 D ^ { 2 } ) ^ { - 1 } )$ . We recover the equation in question with the following simplifications. ", + "bbox": [ + 173, + 299, + 828, + 329 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/4e5d07cfab7c9084a2fc86236e2f33d917e3fe277ed07458f0d6746bf816cb30.jpg", + "text": "$$\n\\begin{array} { r l } & { ( \\Sigma ^ { - 1 } + 2 D ^ { 2 } ) ^ { - 1 } = \\Sigma ( I _ { A + B } + 2 D ^ { 2 } \\Sigma ) ^ { - 1 } } \\\\ & { \\qquad = \\Sigma ( D ^ { - 1 } + 2 D \\Sigma ) ^ { - 1 } D ^ { - 1 } } \\\\ & { \\qquad = \\Sigma D ( I _ { A + B } + 2 D \\Sigma D ) ^ { - 1 } D ^ { - 1 } } \\\\ & { \\qquad = D ^ { - 1 } \\Omega ( I _ { A + B } + 2 \\Omega ) ^ { - 1 } D ^ { - 1 } } \\\\ & { \\qquad \\mathrm { d e t } ( \\Sigma ( \\Sigma ^ { - 1 } + 2 D ^ { 2 } ) ) = \\mathrm { d e t } ( I _ { A + B } + 2 \\Sigma D ^ { 2 } ) } \\\\ & { \\qquad = \\mathrm { d e t } ( I _ { A + B } + 2 D \\Sigma D ) } \\\\ & { \\qquad = \\mathrm { d e t } ( I _ { A + B } + 2 \\Omega ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 328, + 676, + 465 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "The case of general $\\Sigma$ with $\\operatorname { r a n k } \\Sigma > 2 ( a + b )$ is given by the same continuity arguments as in Lemma E.2. ", + "bbox": [ + 171, + 472, + 825, + 502 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "Let $\\widetilde { \\Sigma } \\in { \\cal S } _ { 2 B } , \\widetilde { \\Sigma } = \\left( \\begin{array} { c c } { { \\Sigma } } & { { \\Xi } } \\\\ { { \\Xi ^ { T } } } & { { \\Sigma ^ { \\prime } } } \\end{array} \\right)$ where $\\Sigma , \\Sigma ^ { \\prime } \\in { \\cal S } _ { B }$ and $\\Xi \\in \\mathbb { R } ^ { B \\times B }$ . Consider the off-diagonal block of $\\nabla _ { \\boldsymbol { B } _ { \\phi } ^ { \\oplus 2 } } ( \\widetilde { \\Sigma } )$ . ", + "bbox": [ + 173, + 513, + 826, + 566 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/a6ca79aa385e7a1de847871b5aac0dc46fbf585635fdc3150f3180d5df67ddc2.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\mathbb { E } [ \\mathcal { B } _ { \\phi } ( z ) \\otimes \\mathcal { B } _ { \\phi } ( z ^ { \\prime } ) : ( z , z ^ { \\prime } ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ) ] } } \\\\ & { = \\mathbb { E } [ \\phi ( \\mathbf { n } ( y ) ) \\otimes \\phi ( \\mathbf { n } ( y ^ { \\prime } ) ) : ( y , y ^ { \\prime } ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { G } ) ] } \\\\ & { = B ^ { \\alpha } \\mathbb { E } [ \\| y \\| ^ { - \\alpha } \\| y ^ { \\prime } \\| ^ { - \\alpha } \\phi ( y ) \\otimes \\phi ( y ^ { \\prime } ) : ( y , y ^ { \\prime } ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { G } ) ] } \\\\ & { = B ^ { \\alpha } \\Gamma ( \\alpha / 2 ) ^ { - 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } \\mathrm { d } t ( s t ) ^ { \\alpha / 2 - 1 } \\mathrm { d e t } ( I _ { 2 B } + 2 \\Omega ) ^ { - 1 / 2 } \\underset { ( y , y ^ { \\prime } ) \\sim \\mathcal { N } ( 0 , \\Pi ) } { \\mathbb { E } } \\phi ( y ) \\otimes \\phi ( y ^ { \\prime } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 200, + 568, + 803, + 670 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/0acfa1cda84219024b8abdd72c65e50108ba4b9b0932ef1603aebe37c4aaba9c.jpg", + "text": "$$\n= B ^ { \\alpha } \\Gamma ( \\alpha / 2 ) ^ { - 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\ \\int _ { 0 } ^ { \\infty } \\mathrm { d } t \\ ( s t ) ^ { \\alpha / 2 - 1 } \\operatorname * { d e t } ( I _ { 2 B } + 2 \\Omega ) ^ { - 1 / 2 } \\mathrm { V } _ { \\phi } ( \\Pi ) _ { 1 : B , B + 1 : 2 B }\n$$", + "text_format": "latex", + "bbox": [ + 199, + 688, + 740, + 722 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "$\\Omega = \\left( \\begin{array} { c c } { { s \\Sigma ^ { G } } } & { { \\sqrt { s t } \\Xi ^ { G } } } \\\\ { { \\sqrt { s t } ( \\Xi ^ { G } ) ^ { T } } } & { { t \\Sigma ^ { \\prime G } } } \\end{array} \\right)$ and $\\Pi = D ^ { - 1 } \\Omega ( I + 2 \\Omega ) ^ { - 1 } D ^ { - 1 }$ with $D = \\sqrt { s } I _ { B } \\oplus \\sqrt { t } I _ { B } =$ $\\left( \\begin{array} { c c } { { \\sqrt { s } I _ { B } } } & { { \\mathrm { ~ ~ 0 ~ } } } \\\\ { { 0 } } & { { \\sqrt { t } I _ { B } } } \\end{array} \\right) .$ ", + "bbox": [ + 173, + 723, + 825, + 787 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "Theorem H.6 (Rephrasing and adding to Corollary H.4). Let $\\widetilde { \\Sigma } \\in { S } _ { k B }$ and $\\phi$ be positive homogeneous of degree $\\alpha$ . If $\\widetilde { \\Sigma }$ is CBSB1 then $\\nabla _ { \\boldsymbol { B } _ { \\phi } ^ { \\oplus k } } ( \\widetilde { \\Sigma } )$ is CBSB1 and equals $\\widetilde { \\Sigma } ^ { * } = \\left( \\begin{array} { c c } { { \\Sigma ^ { * } } } & { { c ^ { * } { \\bf 1 1 } ^ { T } } } \\\\ { { c ^ { * } { \\bf 1 1 } ^ { T } } } & { { \\Sigma ^ { * } } } \\end{array} \\right)$ where $\\Sigma ^ { * }$ is the BSB1 fixed point of Thm F.5 and ", + "bbox": [ + 174, + 790, + 823, + 852 + ], + "page_idx": 80 + }, + { + "type": "equation", + "img_path": "images/77f9f789de21ba9dc5c7c0cef769ba2a1afd68676bf3507933e68596860fccac.jpg", + "text": "$$\n\\begin{array} { l } { { c ^ { * } = { \\tt c } _ { \\alpha } \\left( \\frac { B - 1 } { 2 } \\right) ^ { \\alpha } \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\frac { \\alpha } { 2 } \\right) ^ { - 2 } { \\tt J } _ { \\phi } ( 0 ) } } \\\\ { { \\mathrm { = } ( a - b ) ^ { 2 } \\displaystyle \\frac { 1 } { 4 \\pi } ( B - 1 ) ^ { \\alpha } \\left( \\displaystyle \\frac { \\Gamma ( ( B - 1 ) / 2 ) \\Gamma ( ( \\alpha + 1 ) / 2 ) } { \\Gamma ( ( \\alpha + B - 1 ) / 2 ) } \\right) ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 297, + 854, + 699, + 930 + ], + "page_idx": 80 + }, + { + "type": "text", + "text": "Proof. Like in the proof of Corollary H.4, we only need to compute the cross-batch block, which is given above by Eq. (56). Recall that Σ = $\\widetilde { \\Sigma } = \\left( \\begin{array} { c c } { { \\Sigma } } & { { c \\mathbf { 1 } \\mathbf { 1 } ^ { T } } } \\\\ { { c \\mathbf { 1 } \\mathbf { 1 } ^ { T } } } & { { \\Sigma } } \\end{array} \\right)$ where $\\Sigma = \\mathrm { B S B } 1 ( a , b )$ . Set $\\mu : = a - b$ . Note that because the off-diagonal block $\\boldsymbol { \\Xi } = c \\mathbf { 1 1 } ^ { T }$ by assumption, $\\Xi ^ { G } = 0$ , so that ", + "bbox": [ + 173, + 102, + 826, + 165 + ], + "page_idx": 81 + }, + { + "type": "equation", + "img_path": "images/80611d67da4c5f996e070abbc93740bf075304e65d10e41ece065f7e7e6b72c8.jpg", + "text": "$$\n\\begin{array} { c } { { \\Omega = s \\Sigma ^ { G } \\oplus t \\Sigma ^ { G } } } \\\\ { { = s \\mu G \\oplus t \\mu G } } \\\\ { { \\Omega ( I + 2 \\Omega ) ^ { - 1 } = \\displaystyle \\frac { s \\mu } { 1 + 2 s \\mu } G \\oplus \\displaystyle \\frac { t \\mu } { 1 + 2 t \\mu } G } } \\\\ { { \\Pi = \\displaystyle \\frac { \\mu } { 1 + 2 s \\mu } G \\oplus \\displaystyle \\frac { \\mu } { 1 + 2 t \\mu } G } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 361, + 170, + 637, + 275 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Therefore, ", + "bbox": [ + 173, + 297, + 245, + 313 + ], + "page_idx": 81 + }, + { + "type": "equation", + "img_path": "images/3773886d5a3bb499b77a7b7f128b158717f27d5f8fdc6fe3b9f710a4b1c8efc5.jpg", + "text": "$$\n\\begin{array} { c c } { { \\operatorname * { d e t } ( I + 2 \\Omega ) = ( 1 + 2 s \\mu ) ^ { B - 1 } ( 1 + 2 t \\mu ) ^ { B - 1 } } } & { { \\mathrm { ~ } } } \\\\ { { \\mathrm { V } _ { \\phi } ( \\Pi ) = \\mu ^ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha } \\Delta \\mathrm { V } _ { \\phi } \\left( \\begin{array} { c c } { { \\mathrm { B S B 1 } ( 1 , \\frac { - 1 } { B - 1 } ) } } & { { 0 } } \\\\ { { 0 } } & { { \\mathrm { B S B 1 } ( 1 , \\frac { - 1 } { B - 1 } ) } } \\end{array} \\right) \\Delta \\mathrm { V } _ { \\phi } ( \\Pi ) V _ { \\phi } ( \\Pi ) } } & { { \\mathrm { ~ a ~ n ~ d ~ } } } \\\\ { { \\mathrm { ~ } } } & { { \\mathrm { ~ a ~ n ~ d ~ } \\left( \\frac { B - 1 } { B - 1 } \\right) ^ { \\alpha } \\Delta \\mathrm { V } _ { \\phi } \\left( \\begin{array} { c c } { { \\mathrm { B S B 1 } ( 1 , \\frac { - 1 } { B - 1 } ) } } & { { 0 } } \\\\ { { 0 } } & { { \\mathrm { B S B 1 } ( 1 , \\frac { - 1 } { B - 1 } ) } } \\end{array} \\right) \\Delta \\mathrm { V } _ { \\phi } ( \\Pi ) V _ { \\phi } ( \\Pi ) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 245, + 319, + 753, + 377 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "$\\Delta = \\left( \\begin{array} { c c } { { ( 1 + 2 s \\mu ) ^ { - \\alpha / 2 } I _ { B } } } & { { 0 } } \\\\ { { 0 } } & { { ( 1 + 2 t \\mu ) ^ { - \\alpha / 2 } I _ { B } } } \\end{array} \\right)$ . In particular, the off-diagonal block is constant with entry $\\begin{array} { r } { \\mathsf { c } _ { \\alpha } \\mu ^ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha } ( 1 + 2 s \\mu ) ^ { - \\alpha / 2 } ( 1 + 2 t \\mu ) ^ { - \\alpha / 2 } \\mathsf { J } _ { \\phi } ( 0 ) . } \\end{array}$ ", + "bbox": [ + 174, + 386, + 821, + 436 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Finally, by Eq. (56), ", + "bbox": [ + 173, + 440, + 308, + 455 + ], + "page_idx": 81 + }, + { + "type": "equation", + "img_path": "images/b8596628db4deb2eda9aa7c2ffde59c3b38259677091f70f006796e7a48c65ed.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\mathbb { E } [ B _ { \\phi } ( z ) \\otimes B _ { \\phi } ( z ^ { \\prime } ) : ( z , z ^ { \\prime } ) \\sim \\mathcal { N } ( 0 , \\tilde { \\Sigma } ) ] } } \\\\ { { \\displaystyle = B ^ { \\alpha } \\Gamma ( \\alpha / 2 ) ^ { - 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } \\mathrm { d } t ( s t ) ^ { \\alpha / 2 - 1 } \\mathrm { d e t } ( I _ { 2 B } + 2 \\Omega ) ^ { - 1 / 2 } \\mathrm { V } _ { \\phi } ( \\Pi ) _ { 1 : B , B + 1 : 2 B } } } \\\\ { { \\displaystyle = B ^ { \\alpha } \\Gamma ( \\alpha / 2 ) ^ { - 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } \\mathrm { d } t ( s t ) ^ { \\alpha / 2 - 1 } [ ( 1 + 2 s \\mu ) ( 1 + 2 t \\mu ) ] ^ { - \\frac { B - 1 } { 2 } } } } \\\\ { { \\displaystyle ~ \\times \\mathrm { c } _ { \\alpha } \\mu ^ { \\alpha } \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha } [ ( 1 + 2 s \\mu ) ( 1 + 2 t \\mu ) ] ^ { - \\alpha / 2 } \\mathrm { J } _ { \\phi } ( 0 ) \\mathbf { 1 } ^ { T } } } \\\\ { { \\displaystyle = \\mathrm { c } _ { \\alpha } ( B - 1 ) ^ { \\alpha } \\Gamma ( \\alpha / 2 ) ^ { - 2 } \\mathrm { J } _ { \\phi } ( 0 ) \\mathbf { 1 } ^ { 1 T } \\int _ { 0 } ^ { \\infty } \\mathrm { d } \\sigma \\int _ { 0 } ^ { \\infty } \\mathrm { d } \\sigma \\left( \\sigma \\tau \\right) ^ { \\alpha / 2 - 1 } ( 1 + 2 \\sigma ) ^ { - \\frac { B - 1 + \\alpha } { 2 } } ( 1 + 2 \\tau ) ^ { - \\frac { B - 1 + \\alpha } { 2 } } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 462, + 825, + 625 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Simplification with Defn E.7 and Proposition E.21 gives the result. ", + "bbox": [ + 173, + 724, + 612, + 741 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "H.2 LOCAL CONVERGENCE ", + "text_level": 1, + "bbox": [ + 174, + 757, + 382, + 772 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "We now look at the linearized dynamics around CBSB1 fixed points of Eq. (54), and investigate the eigen-properties of $\\left. \\frac { \\mathrm { d } \\mathrm { V } _ { B _ { \\phi } ^ { \\oplus 2 } } } { \\mathrm { d } \\widetilde { \\Sigma } } \\right| _ { \\widetilde { \\Sigma } = \\widetilde { \\Sigma } ^ { * } }$ . By Lemma F.17, its nonzero eigenvalues are exactly those of $\\begin{array} { r } { \\tilde { \\mathcal { F } } : = ( G ^ { \\oplus 2 } ) ^ { \\otimes 2 } \\circ \\frac { \\mathrm { d } \\mathrm { V } _ { ( \\phi \\circ \\mathrm { m } ) \\oplus 2 } ( \\tilde { \\Sigma } ^ { \\sharp G } ) } { \\mathrm { d } \\tilde { \\Sigma } ^ { G } } \\bigg | _ { \\tilde { \\Sigma } ^ { G } = \\mu ^ { * } G ^ { \\oplus 2 } } , \\tilde { \\mathcal { F } } \\{ \\Lambda \\} = G ^ { \\oplus 2 } \\left( \\left. \\frac { \\mathrm { d } \\mathrm { V } _ { ( \\phi \\circ \\mathrm { m } ) \\oplus 2 } } { \\mathrm { d } \\tilde { \\Sigma } ^ { G } } \\right| _ { \\mu ^ { * } G ^ { \\oplus 2 } } \\{ \\Lambda \\} \\right) G ^ { \\oplus 2 } . } \\end{array}$ . Here $\\widetilde { \\Sigma } ^ { G } = G ^ { \\oplus 2 } \\widetilde { \\Sigma } G ^ { \\oplus 2 }$ e. Thus it suffices to obtain the eigendecomposition of $\\tilde { \\mathcal { F } }$ . ", + "bbox": [ + 171, + 784, + 826, + 883 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "rst, notice that $\\tilde { \\mathcal { F } }$ acts on the diagonal b s $G ^ { \\otimes 2 } \\circ \\frac { \\mathrm { d V } _ { \\phi \\circ \\mathsf { n } } } { \\mathrm { d } \\Sigma ^ { G } }$ , and the off-diagonal components of $\\Lambda$ ${ \\tilde { \\mathcal { F } } } \\{ \\Lambda \\}$ ", + "bbox": [ + 173, + 888, + 825, + 925 + ], + "page_idx": 81 + }, + { + "type": "text", + "text": "Definition H.7. Let $\\mathcal { T } : \\mathcal { H } _ { k B } \\mathcal { H } _ { k B }$ be a linear operator such that for any block matrix $\\widetilde { \\Sigma } \\in \\mathcal { H } _ { k B }$ with $k \\times k$ blocks $\\widetilde { \\Sigma } ^ { i j } , i , j \\in [ k ]$ , of size $B$ , we have ", + "bbox": [ + 171, + 101, + 823, + 135 + ], + "page_idx": 82 + }, + { + "type": "equation", + "img_path": "images/ca27a200cf88f5fd8e29526bedb3b8fd9464bad1350895f4e20e371d444c5577.jpg", + "text": "$$\n\\begin{array} { l } { T \\{ \\widetilde { \\Sigma } \\} ^ { i i } = { \\mathcal { U } } \\{ \\widetilde { \\Sigma } ^ { i i } \\} } \\\\ { T \\{ \\widetilde { \\Sigma } \\} ^ { i j } = \\mathcal { V } \\{ \\widetilde { \\Sigma } ^ { i i } \\} + \\mathcal { V } \\{ \\widetilde { \\Sigma } ^ { j j } \\} ^ { T } + \\mathcal { W } \\{ \\widetilde { \\Sigma } ^ { i j } \\} , \\forall i \\not = j , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 326, + 136, + 669, + 178 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "where $\\mathcal { U } : \\mathcal { H } _ { B } \\mathcal { H } _ { B } , \\mathcal { V } : \\mathcal { H } _ { B } \\mathbb { R } ^ { B \\times B }$ , and $\\mathcal { W } : \\mathbb { R } ^ { B \\times B } \\mathbb { R } ^ { B \\times B }$ are linear operators, and ${ \\mathcal { T } } \\{ { \\widetilde { \\Sigma } } \\} ^ { i j }$ denotes the $( i , j )$ th block of $\\mathcal { T } \\{ \\widetilde { \\Sigma } \\}$ . We say that $\\tau$ is blockwise diagonal-off-diagonal semidirect, or BDOS for short. We write more specifically $\\mathcal { T } = \\mathrm { B D O S } ( \\mathcal { U } , \\mathcal { V } , \\mathcal { W } )$ . ", + "bbox": [ + 173, + 178, + 826, + 226 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "We have $\\tilde { \\mathcal { F } } = \\mathrm { B D O S } ( \\mathcal { U } , \\mathcal { V } , \\mathcal { W } )$ where ", + "bbox": [ + 174, + 234, + 429, + 251 + ], + "page_idx": 82 + }, + { + "type": "equation", + "img_path": "images/237962c91ca20df52b4a4f3ea71c314e92e8fe153fc00130f9f7e2032690838d.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\mathcal { U } = G ^ { \\otimes 2 } \\circ \\left. \\frac { \\mathrm { d } \\nabla _ { \\phi \\circ \\mathbf { n } } } { \\mathrm { d } \\Sigma ^ { G } } \\right. _ { \\Sigma ^ { G } = \\mu ^ { * } G } } } \\\\ { { \\displaystyle \\mathcal { V } = G ^ { \\otimes 2 } \\circ \\left. \\frac { \\mathrm { d } } { \\mathrm { d } ( \\widetilde { \\Sigma } ^ { G } ) ^ { 1 1 } } \\underset { ( h , h ^ { \\prime } ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { G } ) } { \\mathbb { E } } \\phi ( \\mathbf { n } ( h ) ) \\otimes \\phi ( \\mathbf { n } ( h ^ { \\prime } ) ) \\right. _ { \\widetilde { \\Sigma } ^ { G } = \\mu ^ { * } G ^ { \\oplus 2 } } } } \\\\ { { \\displaystyle \\mathcal { W } = 2 G ^ { \\otimes 2 } \\circ \\left. \\frac { \\mathrm { d } } { \\mathrm { d } ( \\widetilde { \\Sigma } ^ { G } ) ^ { 1 2 } } \\underset { ( h , h ^ { \\prime } ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { G } ) } { \\mathbb { E } } \\phi ( \\mathbf { n } ( h ) ) \\otimes \\phi ( \\mathbf { n } ( h ^ { \\prime } ) ) \\right. _ { \\widetilde { \\Sigma } ^ { G } = \\mu ^ { * } G ^ { \\oplus 2 } } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 252, + 736, + 380 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Note that the factor 2 in $\\mathcal { W }$ is due to the contribution of $( \\widetilde { \\Sigma } ^ { G } ) ^ { 2 1 }$ by symmetry. We will first show that $\\mathcal { W }$ is multiplication by a constant. ", + "bbox": [ + 176, + 381, + 823, + 412 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "H.2.1 SPHERICAL INTEGRATION AND GEGENBAUER EXPANSION ", + "bbox": [ + 173, + 424, + 637, + 440 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Theorem H.8. Suppose $\\phi ( { \\sqrt { B - 1 } } x )$ has Gegenbauer expansion $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ . Then satisfying $G \\Lambda G = \\Lambda ,$ ", + "bbox": [ + 173, + 448, + 825, + 487 + ], + "page_idx": 82 + }, + { + "type": "equation", + "img_path": "images/2ca8a1e4ffaf6dd2533b9bb3ff84d641e8147f7f280b6d9c19a81d0c6c215927.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\mathcal { W } \\{ \\Lambda \\} = 2 a _ { 1 } ^ { 2 } { \\mu ^ { * } } ^ { \\mathrm { - 1 } } \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\frac 1 2 \\right) ^ { 2 } \\frac { B } { B - 1 } \\Lambda } } \\\\ { { \\displaystyle \\quad = \\frac { a _ { 1 } ^ { 2 } \\frac { 1 } { 2 } \\mathrm { P } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { - 2 } B ( B - 1 ) } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } \\Lambda } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 488, + 700, + 577 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Proof. Let $\\begin{array} { r } { J = \\left. \\frac { \\mathsf { d } } { \\mathsf { d } ( \\widetilde { \\Sigma } ^ { G } ) ^ { 1 2 } } \\mathbb { E } _ { ( h , h ^ { \\prime } ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { G } ) } \\phi ( \\mathsf { n } ( h ) ) \\otimes \\phi ( \\mathsf { n } ( h ^ { \\prime } ) ) \\right. _ { \\widetilde { \\Sigma } ^ { G } = \\mu ^ { * } G ^ { \\oplus 2 } } . \\mathrm { C o m b i n i n g } . } \\end{array}$ Lemma F.18 with projection by $\\mathbb { P }$ , we get ", + "bbox": [ + 174, + 589, + 821, + 628 + ], + "page_idx": 82 + }, + { + "type": "equation", + "img_path": "images/a4e7634fae298dec6aa13a1db6a4de384623b04937b6cc4c59ef60f433ab1c82.jpg", + "text": "$$\n\\begin{array} { r l } & { J \\{ \\Lambda \\} = \\displaystyle \\frac { 1 } { 2 } _ { y , y ^ { \\prime } \\sim N \\mathbb { 0 } , \\mu ^ { * } I B = y ) } \\phi \\left( \\mathfrak { n } ( \\mathtt { e } y ) \\right) \\otimes \\phi \\left( \\mathfrak { n } ( \\mathtt { e } y ^ { \\prime } ) \\right) \\langle \\mu ^ { * - 2 } y y ^ { \\prime T } - \\mu ^ { * - 1 } I _ { B - 1 } , \\mathtt { e } ^ { T } \\Lambda \\mathtt { e } \\rangle } \\\\ & { \\qquad = \\displaystyle \\frac { 1 } { 2 } \\mu ^ { * - 1 } _ { y , y ^ { \\prime } \\sim N \\mathbb { 0 } , I _ { B - 1 } ) } \\overline { { \\Phi } } \\left( \\mathfrak { n } ( \\mathtt { e } y ) \\right) \\otimes \\phi \\left( \\mathfrak { n } ( \\mathtt { e } y ^ { \\prime } ) \\right) \\left( y ^ { \\prime T } \\mathfrak { e } ^ { T } \\Lambda \\mathtt { e } y - \\mathrm { t r } \\Lambda \\right) } \\\\ & { \\qquad = \\displaystyle \\frac { 1 } { 2 } \\mu ^ { * - 1 } \\bigg ( \\left( \\sqrt { 2 } \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\frac { 1 } { 2 } \\right) \\right) ^ { 2 } _ { v , v ^ { \\prime } \\sim S ^ { B - 2 } } \\phi \\left( \\sqrt { B } \\mathfrak { e } v \\right) \\otimes \\phi \\left( \\sqrt { B } \\mathfrak { e } v ^ { \\prime } \\right) \\left( v ^ { \\prime T } \\mathfrak { e } ^ { T } \\Lambda \\mathtt { e } v \\right) } \\\\ & { \\qquad - \\left( \\mathrm { t r } \\Lambda \\right) _ { v , v ^ { \\prime } \\sim S ^ { B - 2 } } \\phi \\left( \\sqrt { B } \\mathfrak { e } v \\right) \\otimes \\phi \\left( \\sqrt { B } \\mathfrak { e } v ^ { \\prime } \\right) \\bigg ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 630, + 808, + 771 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "by Proposition E.33. ", + "text_level": 1, + "bbox": [ + 666, + 772, + 805, + 786 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Note that Ev,v0∼SB−2 φ \u0010√Bev\u0011 ⊗ φ \u0010√Bev0\u0011 = \u0010Ev∼SB−2 φ \u0010√Bev\u0011\u0011⊗2 is a constant matrix, because of the independence of $v$ from and the spherical symmetry of the integrand. Thus this term drops out in $G ^ { \\mathrm { \\hat { \\otimes } 2 } } \\circ J \\{ \\Lambda \\}$ . ", + "bbox": [ + 173, + 791, + 826, + 845 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "Next, we analyze $H \\{ \\Lambda \\} = \\mathbb { E } _ { v , v ^ { \\prime } \\sim S ^ { B - 2 } } \\phi \\left( { \\sqrt { B } } \\Phi v \\right) \\otimes \\phi \\left( { \\sqrt { B } } \\Phi v ^ { \\prime } \\right) \\left( v ^ { \\prime T } \\Phi ^ { T } \\Lambda \\mathbb { e } v \\right)$ . This is a linear function in $\\Lambda$ with coefficients ", + "bbox": [ + 174, + 851, + 823, + 888 + ], + "page_idx": 82 + }, + { + "type": "equation", + "img_path": "images/efc60107a6e958dd4739a59d808bbbfb836a2e0ee3208d69f5a0eac6a1bec7f1.jpg", + "text": "$$\nH _ { a b | c d } = \\frac { B - 1 } { B } \\left( \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B - 1 } \\hat { \\mathbf { e } } _ { b , : } v ) ( \\hat { \\mathbf { e } } _ { c , : } v ) \\right) \\left( \\underset { v \\sim S ^ { B - 2 } } { \\mathbb { E } } \\phi ( \\sqrt { B - 1 } \\hat { \\mathbf { e } } _ { a , : } v ) ( \\hat { \\mathbf { e } } _ { d , : } v ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 204, + 887, + 794, + 922 + ], + "page_idx": 82 + }, + { + "type": "text", + "text": "By the usual zonal spherical harmonics argument, ", + "bbox": [ + 174, + 103, + 501, + 118 + ], + "page_idx": 83 + }, + { + "type": "equation", + "img_path": "images/b7a3c3c5e9fb0879913a285a4df0438c0b0c6e80038e6f98050aa162459e5e53.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\underset { s \\sim \\delta = 2 } { \\overset { \\mathbb { R } } { \\sim } } \\phi ( \\sqrt { B } - 1 \\hat { \\Phi } _ { b , \\sigma } ) \\big ( \\hat { e } _ { e , \\sigma } , y \\big ) } \\\\ & { = \\bigg \\langle \\overset { \\sum } { i = 1 } a _ { 1 } \\frac { 1 } { c B - 1 , i } C _ { 1 } ^ { ( \\frac { 1 - \\alpha - 3 } { 2 } ) } \\big ( \\langle \\hat { e } _ { b , \\sigma } , \\cdot \\rangle \\big ) , \\langle \\hat { e } _ { e , \\sigma } , \\cdot \\rangle \\bigg \\rangle _ { S ^ { B - 1 } } } \\\\ & { = \\bigg \\langle \\overset { \\sum } { i = 1 } a _ { 1 } Z _ { k , \\sigma } ^ { B - 2 , ( b ) } , \\frac { 1 } { B - 1 } Z _ { e , \\sigma } ^ { B - 2 , ( b ) } \\bigg \\rangle _ { S ^ { B - 2 } } } \\\\ & { = \\frac { a _ { 1 } } { B - 1 } Z _ { e , \\sigma } ^ { B - 2 , ( 1 ) } ( \\hat { e } _ { b , \\sigma } ) } \\\\ & { = \\frac { a _ { 1 } } { B - 1 } \\frac { 1 } { c B - 1 , 1 } C _ { 1 } ^ { ( \\frac { 1 - \\alpha - 3 } { 2 } ) } \\big ( \\langle \\hat { e } _ { b , \\sigma } , \\hat { e } _ { e , \\cdot } \\rangle \\big ) } \\\\ & { = \\bigg \\{ \\frac { a _ { 1 } } { B - 1 } \\frac { 1 } { c B - 1 , 1 } \\big ( \\mathrm { f i } b = c } \\\\ & { = \\bigg \\{ \\frac { a _ { 1 } } { B - 1 } , \\quad \\mathrm { f t b } = c } \\\\ & { \\geq \\frac { a _ { 1 } } { \\delta - 1 } , \\quad \\mathrm { o t h e r w i s e } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 121, + 669, + 342 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Thus, ", + "bbox": [ + 173, + 342, + 214, + 356 + ], + "page_idx": 83 + }, + { + "type": "equation", + "img_path": "images/386fe2551ef6196659cbf48d102e974d929d114a1ede1fdeec1c5942974ffe83.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle H \\{ \\Lambda \\} _ { a b } = \\frac { B - 1 } { B } \\Bigg ( \\left( \\frac { - a _ { 1 } } { B - 1 } \\right) ^ { 2 } \\sum _ { c , d \\in [ B ] } ^ { \\mathrm { ~ \\sum ~ \\sum ~ } _ { h < d } } } } \\\\ { { \\displaystyle ~ + \\left( a _ { 1 } - \\frac { - a _ { 1 } } { B - 1 } \\right) \\left( \\frac { - a _ { 1 } } { B - 1 } \\right) \\sum _ { c - b \\in \\mathbb { S } ^ { d - a } } \\Lambda _ { c d } } } \\\\ { { \\displaystyle ~ + \\left( a _ { 1 } ^ { 2 } - \\frac { a _ { 1 } ^ { 2 } } { ( B - 1 ) ^ { 2 } } \\right) \\Lambda _ { a b } \\Bigg ) } } \\\\ { { \\displaystyle = \\frac { B - 1 } { B } \\left( - 2 \\left( a _ { 1 } - \\frac { - a _ { 1 } } { B - 1 } \\right) \\left( \\frac { - a _ { 1 } } { B - 1 } \\right) \\Lambda _ { a b } + \\left( a _ { 1 } ^ { 2 } - \\frac { a _ { 1 } ^ { 2 } } { ( B - 1 ) ^ { 2 } } \\right) \\Lambda _ { a b } \\right) } } \\\\ { { \\displaystyle ~ \\mathrm { b e c a u s c r o w s ~ a n d ~ c o l u m s ~ o f ~ A s ~ s u m ~ t o ~ } } } \\\\ { { \\displaystyle = a _ { 1 } ^ { 2 } \\frac { B } { B - 1 } \\Lambda _ { a b } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 220, + 357, + 782, + 569 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Therefore, ", + "bbox": [ + 173, + 565, + 245, + 580 + ], + "page_idx": 83 + }, + { + "type": "equation", + "img_path": "images/63c6de0f04df5c2761a06a0fb0909e6d43eb220b5a049f50eab4346a97550b26.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathcal { W } \\{ \\Lambda \\} = 2 G ^ { \\otimes 2 } \\circ J \\{ \\Lambda \\} } \\\\ & { \\qquad = 2 a _ { 1 } ^ { 2 } \\mu ^ { * - 1 } \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\frac { 1 } { 2 } \\right) ^ { 2 } \\frac { B } { B - 1 } \\Lambda } \\\\ & { \\qquad = \\frac { a _ { 1 } ^ { 2 } \\frac { 1 } { 2 } \\mathrm { P } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { - 2 } B ( B - 1 ) } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } \\Lambda } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 295, + 580, + 700, + 689 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "where we used the identity $\\begin{array} { r } { \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\frac { 1 } { 2 } \\right) \\left( \\frac { 2 } { B - 1 } \\right) = \\mathrm { P } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { - 1 } } \\end{array}$ . ", + "bbox": [ + 173, + 693, + 591, + 717 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Next, we step back a bit to give a full treatment of the eigen-properties of BDOS operators. Then we specialize back to our case and give the result for $\\tilde { \\mathcal { F } }$ . ", + "bbox": [ + 173, + 729, + 823, + 761 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Definition H.9. Let $\\mathcal { H } _ { k B } ^ { G } : = \\{ \\widetilde { \\Sigma } \\in \\mathcal { H } _ { k B } : G ^ { \\oplus k } \\widetilde { \\Sigma } G ^ { \\oplus k } = \\widetilde { \\Sigma } \\}$ be the subspace of $\\mathcal { H } _ { k B }$ stable under ", + "bbox": [ + 171, + 763, + 823, + 796 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Definition H.10. Let $\\widetilde { \\mathbb { M } } _ { k B } \\in \\mathcal { H } _ { k B } ^ { G }$ be the space of block matrices $\\widetilde { \\Sigma } \\in \\mathcal { H } _ { k B } ^ { G }$ where the diagonal blocks $\\widetilde { \\Sigma } ^ { i i }$ are all zero. Likewise, let $\\overline { { \\mathbb { M } } } _ { k B } \\in \\mathcal { H } _ { k B }$ be the space of block matrices $\\widetilde { \\Sigma } \\in \\mathcal { H } _ { k B }$ where the diagonal blocks $\\widetilde { \\Sigma } ^ { i i }$ are all zero. We suppress the subscript $k B$ when it is clear from the context. ", + "bbox": [ + 173, + 799, + 826, + 853 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "Definition H.11. Define the block $L$ -shaped matrix ", + "bbox": [ + 176, + 854, + 514, + 868 + ], + "page_idx": 83 + }, + { + "type": "equation", + "img_path": "images/ff61d766a67c11173c4134c61e2326f8270b5ff22673b80ece33448fde477cee.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathrm { L } } } _ { k B } ( \\Sigma , \\Lambda ) : = \\left( \\begin{array} { c c c } { \\Sigma } & { - \\Lambda } & { \\cdots } \\\\ { - \\Lambda ^ { T } } & { 0 } & { \\cdots } \\\\ { \\vdots } & { \\vdots } & { \\ddots } \\end{array} \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 379, + 871, + 619, + 929 + ], + "page_idx": 83 + }, + { + "type": "text", + "text": "BDOS operators of the form $\\mathrm { B D O S } ( \\mathcal { U } , \\mathcal { V } , w )$ , where $w$ stands for multiplication by a constant $w$ , like $\\tilde { \\mathcal { F } }$ , have simple eigendecomposition, just like DOS operators. ", + "bbox": [ + 173, + 102, + 823, + 135 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Theorem H.12. If $\\gamma - w I _ { B }$ is not singular, then $\\operatorname { B D O S } ( \\mathcal { V } , \\mathcal { U } , w ) : \\mathcal { H } _ { k B } \\to \\mathcal { H } _ { k B }$ has the following eigenspaces and eigenvectors ", + "bbox": [ + 173, + 137, + 825, + 166 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "1. Me with associated eigenvalue $w$ . ", + "bbox": [ + 214, + 176, + 449, + 198 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "2. For each eigenvalue $\\lambda$ and eigenvector $\\Sigma \\in \\mathcal { H } _ { B }$ of $\\mathcal { U }$ , the block matrix ", + "bbox": [ + 210, + 204, + 696, + 222 + ], + "page_idx": 84 + }, + { + "type": "equation", + "img_path": "images/237cb238b888a9a9fe7d4bc7ffb09673b69040b04ba4bc29a56a11b8cc0aa760.jpg", + "text": "$$\n\\begin{array} { r } { \\overline { { \\mathbf { L } } } _ { k B } ( \\Sigma , - ( \\lambda - w ) ^ { - 1 } \\mathcal { V } \\{ \\Sigma \\} ) = \\left( ( \\lambda - w ) ^ { - 1 } \\mathcal { V } \\{ \\Sigma \\} ^ { T } \\begin{array} { c c c } { ( \\lambda - w ) ^ { - 1 } \\mathcal { V } \\{ \\Sigma \\} } & { \\cdots } \\\\ { ( \\lambda - w ) ^ { - 1 } \\mathcal { V } \\{ \\Sigma \\} ^ { T } } & { 0 } & { \\cdots } \\\\ { \\vdots } & { \\vdots } & { \\ddots } \\end{array} \\right) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 224, + 789, + 284 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "along with any matrix obtained from permutating its block columns and rows simultaneously. The associated eigenvalue is $\\lambda$ . ", + "bbox": [ + 230, + 287, + 821, + 318 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "$I f \\mathrm { B D O S } ( \\mathcal { V } , \\mathcal { U } , w )$ also restricts to $\\mathcal { H } _ { k B } ^ { G } \\mathcal { H } _ { k B } ^ { G }$ , then with respect to this signature, the eigendecomposition is obtained by replacing $\\mathcal { H } _ { k B }$ with $\\mathcal { H } _ { k B } ^ { G }$ in the above: ", + "bbox": [ + 171, + 325, + 823, + 359 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "1. Block diagonal matrices $\\widetilde { \\Sigma } \\in \\mathcal { H } _ { k B } ^ { G }$ where the diagonal blocks $\\widetilde { \\Sigma } ^ { i i }$ are all zero. The associated eigenvalue is $w$ . ", + "bbox": [ + 209, + 368, + 821, + 401 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "2. For each eigenvalue $\\lambda$ and eigenvector $\\begin{array} { r l r l } { \\Sigma } & { { } \\in } & { { \\mathcal { H } } _ { B } ^ { G } } \\end{array}$ of $\\mathcal { U }$ , the matrix $\\overline { { \\mathrm { L } } } _ { k B } \\big ( \\Sigma , - ( \\lambda - w ) ^ { - 1 } \\mathcal { V } \\{ \\Sigma \\} \\big )$ along with any matrix obtained from permutating its block columns and rows simultaneously. The associated eigenvalue is $\\lambda$ . ", + "bbox": [ + 212, + 409, + 825, + 455 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Proof. The proofs are similar in both cases: One can easily verify that these are eigenvectors and eigenvalues. Then by dimensionality considerations, these must be all of them. □ ", + "bbox": [ + 174, + 469, + 825, + 500 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "By Thms H.8 and H.12, we easily obtain ", + "bbox": [ + 176, + 513, + 444, + 529 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Theorem H.13. Suppose $\\phi ( { \\sqrt { B - 1 } } x )$ has Gegenbauer expansion $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ . The eigendecomposition of $\\tilde { \\mathcal { F } } : \\mathcal { H } _ { 2 B } ^ { G } \\to \\mathcal { H } _ { 2 B } ^ { G }$ is given below, as long as $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } \\neq \\lambda _ { \\mathbb { L } } ^ { \\uparrow } , \\lambda _ { \\mathbb { M } } ^ { \\uparrow }$ . ", + "bbox": [ + 174, + 534, + 826, + 577 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "1. $\\smash { \\widetilde { \\mathbb { M } } }$ with associated eigenvalue ", + "bbox": [ + 212, + 588, + 431, + 604 + ], + "page_idx": 84 + }, + { + "type": "equation", + "img_path": "images/9fedefebdf0b3cedf894e7ac1cce2deaaf2353595c22dbd8019a79e1aece2e5c.jpg", + "text": "$$\n\\begin{array} { r l } & { \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } : = 2 a _ { 1 } ^ { 2 } \\mu ^ { * } ^ { - 1 } \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\frac { 1 } { 2 } \\right) ^ { 2 } \\frac { B } { B - 1 } } \\\\ & { \\quad \\quad = \\frac { a _ { 1 } ^ { 2 } \\frac { 1 } { 2 } \\mathrm { P } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { - 2 } B ( B - 1 ) } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 339, + 609, + 715, + 699 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "2. The space generated by $\\overline { { \\mathrm { L } } } _ { 2 B } \\big ( G , - \\big ( \\lambda _ { \\mathbb { L } } ^ { \\uparrow } - \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } \\big ) ^ { - 1 } \\mathcal { V } \\{ G \\} \\big )$ along with any matrix obtained efrom permutating its block columns and rows simultaneously. The associated eigenvalue is 0. ", + "bbox": [ + 212, + 709, + 823, + 758 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "3. The space generated by $\\overline { { \\mathrm { L } } } _ { 2 B } ( L , - ( \\lambda _ { \\mathbb { L } } ^ { \\uparrow } - \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } ) ^ { - 1 } \\mathcal { V } \\{ L \\} )$ , where $L = \\mathrm { L } _ { B } ( B - 2 , 1 )$ , along ewith any matrix obtained from permutating its block columns and rows simultaneously. The associated eigenvalue is $\\lambda _ { \\mathbb { L } } ^ { \\uparrow }$ . ", + "bbox": [ + 212, + 765, + 825, + 819 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "4. The space generated by $\\overline { { \\mathrm { L } } } _ { 2 B } \\big ( M , - \\big ( \\lambda _ { \\mathbb { M } } ^ { \\uparrow } - \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } \\big ) ^ { - 1 } \\mathcal { V } \\{ M \\} \\big )$ , for any $M \\in \\mathbb { M } ,$ , along with eany matrix obtained from permutating its block columns and rows simultaneously. The associated eigenvalue is $\\lambda _ { \\mathbb { M } } ^ { \\uparrow }$ . ", + "bbox": [ + 212, + 827, + 825, + 881 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Here, $\\nu$ is as in Eq. (57). The local convergence dynamics of Eq. (54) to the CBSB1 fixed point $\\widetilde { \\Sigma } ^ { * }$ has the same eigenvalues $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } , \\lambda _ { \\mathbb { L } } ^ { \\uparrow } , \\lambda _ { \\mathbb { M } } ^ { \\uparrow }$ , and $\\boldsymbol { \\theta }$ . ", + "bbox": [ + 174, + 891, + 820, + 928 + ], + "page_idx": 84 + }, + { + "type": "text", + "text": "Note that, to reason about $\\tilde { \\mathcal { F } }$ extended over $k$ batches, one can replace $2 B$ with $k B$ everywhere in $\\cdot$ and the theorem will still hold. As we will see below, $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow }$ is the same as the eigenvalue egoverning the dynamics of the cross batch correlations in the gradient propagation. ", + "bbox": [ + 173, + 102, + 825, + 151 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "H.2.2 LAPLACE METHOD ", + "text_level": 1, + "bbox": [ + 174, + 166, + 367, + 183 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "As with the single batch case, with positive homogeneous $\\phi$ , we can obtain a more direct answer of the new eigenvalue $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow }$ in terms of the J function of $\\phi$ . ", + "bbox": [ + 173, + 193, + 825, + 226 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "Theorem H.14. Let $\\phi$ be degree α positive homogeneous. Then the eigenvalue of $\\tilde { \\mathcal { F } }$ on M is ", + "bbox": [ + 168, + 233, + 781, + 251 + ], + "page_idx": 85 + }, + { + "type": "equation", + "img_path": "images/d6b8315501d9dc4217e83bd6931a9edcbc790e99fe9180dc7244058fd94cb70d.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } = \\frac { B } { 2 } \\left( \\frac { B - 1 } { 2 } \\right) ^ { \\alpha - 1 } \\mathrm { P } \\left( \\frac { B } { 2 } , \\frac { \\alpha } { 2 } \\right) ^ { - 2 } \\mu ^ { * - 1 } \\mathrm { c } _ { \\alpha } \\mathrm { J } _ { \\phi } ^ { \\prime } ( 0 ) } } \\\\ { { \\displaystyle \\quad = \\frac { B } { B - 1 } \\frac { \\mathrm { P } \\left( \\frac { B - 1 } { 2 } , \\alpha \\right) } { \\mathrm { P } \\left( \\frac { B } { 2 } , \\frac { \\alpha } { 2 } \\right) ^ { 2 } } \\frac { \\mathrm { J } _ { \\phi } ^ { \\prime } ( 0 ) } { \\mathrm { J } _ { \\phi } ( 1 ) - \\mathrm { J } _ { \\phi } \\left( \\frac { - 1 } { B - 1 } \\right) } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 321, + 257, + 676, + 343 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "Proof. Let $\\widetilde { \\Sigma } ^ { * } = \\left( \\begin{array} { c c } { { \\Sigma ^ { * } } } & { { c ^ { * } { \\bf 1 1 } ^ { T } } } \\\\ { { c ^ { * } { \\bf 1 1 } ^ { T } } } & { { \\Sigma ^ { * } } } \\end{array} \\right)$ be the CBSB1 fixed point in $\\mathrm { T h m } \\mathrm { H } . 6$ . Take a smooth path $\\widetilde { \\Sigma } _ { \\tau } = \\left( \\begin{array} { l l } { \\Sigma _ { \\tau } } & { \\Xi _ { \\tau } } \\\\ { \\Xi _ { \\tau } ^ { T } } & { \\Sigma _ { \\tau } ^ { \\prime } } \\end{array} \\right) \\in \\mathcal { S } _ { 2 B } , \\tau \\in ( - \\epsilon , \\epsilon )$ such that $\\begin{array} { r } { \\widetilde { \\Sigma } _ { 0 } = \\widetilde { \\Sigma } ^ { * } , \\dot { \\widetilde { \\Sigma } } _ { 0 } : = \\frac { d } { d \\tau } \\widetilde { \\Sigma } = \\left( \\begin{array} { r l } { 0 } & { \\Upsilon } \\\\ { \\Upsilon ^ { T } } & { 0 } \\end{array} \\right) } \\end{array}$ for some Υ ∈ RB×B. Then with Ωτ = \u0012 sΣGτ√ st(ΞGτ )T and $\\Pi _ { \\tau } = D ^ { - 1 } \\Omega _ { \\tau } ( I + 2 \\Omega _ { \\tau } ) ^ { - 1 } D ^ { - 1 }$ with $D = \\sqrt { s } I _ { B } \\oplus \\sqrt { t } I _ { B } = \\left( \\begin{array} { c c } { { \\sqrt { s } I _ { B } } } & { { 0 } } \\\\ { { 0 } } & { { \\sqrt { t } I _ { B } } } \\end{array} \\right)$ ", + "bbox": [ + 171, + 361, + 826, + 491 + ], + "page_idx": 85 + }, + { + "type": "equation", + "img_path": "images/5a7c22bf2bc6eb05d8ce25c21a6d44d89fdef83d8e5a361bbb40a9c682cfb338.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\displaystyle \\frac { d } { d \\tau } \\mathbb { E } [ B _ { \\phi } ( z ) \\otimes B _ { \\phi } ( z ^ { \\prime } ) : ( z , z ^ { \\prime } ) \\sim \\mathcal { N } ( 0 , \\bar { \\Sigma } ) ] _ { \\tau = 0 } } \\\\ & { = \\displaystyle \\frac { d } { d \\tau } b ^ { \\alpha } \\Gamma ( \\alpha / 2 ) ^ { - 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } \\mathrm { d } t \\left( s \\right) ^ { \\alpha / 2 - 1 } \\mathrm { d } \\mathrm { e t } ( ( 2 g _ { R } + 2 \\Omega ) ^ { - 1 / 2 } \\mathbb { V } _ { \\phi } ( \\mathrm { H } ) _ { 1 : B , R + 1 : 2 R } \\bigg | _ { \\tau = 0 } } \\\\ & { = B ^ { \\alpha } \\Gamma ( \\alpha / 2 ) ^ { - 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } \\mathrm { d } t \\left( s \\right) ^ { \\alpha / 2 - 1 } \\bigg ( \\displaystyle \\frac { d } { d \\tau } \\mathrm { d } \\mathrm { e t } ( I _ { 2 B } + 2 \\Omega ) ^ { - 1 / 2 } \\bigg | _ { \\tau = 0 } V _ { \\phi } ( \\mathrm { H } ) _ { 1 : B , R + 1 : 2 B } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & & { = \\displaystyle \\mathrm { d } \\mathrm { e t } ( I _ { 2 B } + 2 \\Omega ) ^ { - 1 / 2 } \\frac { d } { d \\tau } \\displaystyle \\frac { V _ { \\phi } ( \\mathrm { H } ) _ { 1 : B , B + 1 : 2 B } } { d \\tau } \\bigg | _ { \\tau = 0 } \\bigg ) } \\\\ & { = B ^ { \\alpha } \\Gamma ( \\alpha / 2 ) ^ { - 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } \\mathrm { d } t \\left( s \\right) ^ { \\alpha / 2 - 1 } \\bigg ( - \\mathrm { d } \\mathrm { e t } ( I + 2 \\Omega _ { 0 } ) ^ { - 1 / 2 } \\mathrm { t r } \\left( ( I + 2 \\Omega _ { 0 } ) ^ { - 1 } \\hat { \\Omega } _ { 0 } \\right) \\mathbb { V } _ { \\phi } ( \\mathrm { H } ) } \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 497, + 821, + 715 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "by chain rule and Lemma F.26, where $\\begin{array} { r } { \\mathcal { I } = \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } } { \\mathrm { d } \\Sigma } \\bigg | _ { \\Sigma = \\Omega _ { 0 } ( I + 2 \\Omega _ { 0 } ) ^ { - 1 } } } \\end{array}$ . We compute ", + "bbox": [ + 171, + 741, + 692, + 777 + ], + "page_idx": 85 + }, + { + "type": "equation", + "img_path": "images/be69cdf1e91a13870b470c5f7737e9d693cf23082b0ec9bfa7edb42690bdc42b.jpg", + "text": "$$\n\\begin{array} { c } { { \\Omega _ { 0 } = \\mu ^ { * } ( s G \\circledast t G ) } } \\\\ { { \\dot { \\Omega } _ { 0 } = \\sqrt { s t } \\left( \\begin{array} { c c } { { 0 } } & { { \\Upsilon ^ { G } } } \\\\ { { ( \\Upsilon ^ { G } ) ^ { T } } } & { { 0 } } \\end{array} \\right) } } \\\\ { { \\dot { ( I + 2 \\Omega _ { 0 } ) ^ { - 1 } = ( ( 1 + 2 s \\mu ^ { * } ) ^ { - 1 } G + \\frac { 1 } { B } { \\bf 1 1 } ^ { T } ) \\circledast ( ( 1 + 2 t \\mu ^ { * } ) ^ { - 1 } G + \\frac { 1 } { B } { \\bf 1 1 } ^ { T } ) } } } \\\\ { { \\operatorname* { d e t } ( I + 2 \\Omega _ { 0 } ) = ( 1 + 2 s \\mu ^ { * } ) ^ { B - 1 } ( 1 + 2 t \\mu ^ { * } ) ^ { B - 1 } } } \\\\ { { \\Omega _ { 0 } ( I + 2 \\Omega _ { 0 } ) ^ { - 1 } = \\frac { s \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } G \\circledast \\frac { t \\mu ^ { * } } { 1 + 2 t \\mu ^ { * } } G . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 240, + 784, + 756, + 925 + ], + "page_idx": 85 + }, + { + "type": "text", + "text": "So then ", + "bbox": [ + 173, + 103, + 227, + 118 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/1f19aa5e3e1f6578640a2bd1f403455cfd127286603a0f06fa4464bcd931faf7.jpg", + "text": "$$\n7 = \\left( \\left( \\frac { s \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } } \\right) ^ { ( \\alpha - 1 ) / 2 } I _ { B } \\oplus \\left( \\frac { t \\mu ^ { * } } { 1 + 2 t \\mu ^ { * } } \\right) ^ { ( \\alpha - 1 ) / 2 } I _ { B } \\right) ^ { \\otimes 2 } \\circ \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha - 1 } \\frac { \\mathrm { d } \\mathrm { V } _ { \\phi } } { \\mathrm { d } \\Sigma } \\bigg | _ { \\Sigma = \\mathrm { B S B } 1 ( 1 , \\frac { - 1 } { B - 1 } ) } ,\n$$", + "text_format": "latex", + "bbox": [ + 181, + 121, + 839, + 165 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "With ", + "bbox": [ + 173, + 170, + 209, + 184 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/07a4ab9505024412b8f1091779ededbed587439e06a29197b676f9c0e69c176a.jpg", + "text": "$$\n\\begin{array} { c c } { { \\displaystyle \\left. \\frac { d } { d \\tau } \\Omega ( I + 2 \\Omega ) ^ { - 1 } \\right| _ { \\tau = 0 } = ( I + 2 \\Omega _ { 0 } ) ^ { - 1 } \\dot { \\Omega } _ { 0 } ( I + 2 \\Omega _ { 0 } ) ^ { - 1 } } } & { { } } \\\\ { { } } & { { ( \\mathrm { b y } } } \\\\ { { } } & { { = \\displaystyle \\frac { \\sqrt { s t } } { ( 1 + 2 s \\mu ^ { * } ) ( 1 + 2 t \\mu ^ { * } ) } \\left( { \\begin{array} { c c } { { 0 } } & { { \\Upsilon ^ { G } } } \\\\ { { ( \\Upsilon ^ { G } ) ^ { T } } } & { { 0 } } \\end{array} } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 284, + 188, + 715, + 280 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "we have ", + "bbox": [ + 173, + 281, + 230, + 295 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/b0478421f81d722ebce6fdfacd983bc4486733f6e8c1ed848d2d1579f04c48e4.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad \\mathcal { I } \\left\\{ \\displaystyle \\frac { d } { d \\tau } \\Omega ( I + 2 \\Omega ) ^ { - 1 } \\bigg | _ { \\tau = 0 } \\right\\} } \\\\ & { = \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha - 1 } \\displaystyle \\frac { \\sqrt { s t } } { ( 1 + 2 s \\mu ^ { * } ) ( 1 + 2 t \\mu ^ { * } ) } \\mathrm { c } _ { \\alpha } J _ { \\phi } ^ { \\prime } ( 0 ) } \\\\ & { \\qquad \\times \\left( \\frac { s \\mu ^ { * } } { 1 + 2 s \\mu ^ { * } ) } ^ { ( \\alpha - 1 ) / 2 } \\left( \\frac { t \\mu ^ { * } } { 1 + 2 t \\mu ^ { * } } \\right) ^ { ( \\alpha - 1 ) / 2 } \\left( { ( \\Upsilon ^ { G } ) } ^ { T } \\right) ^ { G } \\right) } \\\\ & { = \\left( \\frac { B - 1 } { B } \\right) ^ { \\alpha - 1 } \\displaystyle \\frac { \\left( s t \\right) ^ { \\alpha / 2 } \\mu ^ { * \\alpha - 1 } \\mathrm { c } _ { \\alpha } J _ { \\phi } ^ { \\prime } ( 0 ) } { ( 1 + 2 s \\mu ^ { * } ) ( \\alpha + 1 ) ^ { \\prime } 2 ( 1 + 2 t \\mu ^ { * } ) ^ { ( \\alpha + 1 ) / 2 } \\left( ( \\Upsilon ^ { G } ) ^ { T } \\right. \\left. \\begin{array} { l l } { \\mathrm { T } ^ { G } } \\\\ { \\mathrm { 0 } } \\end{array} \\right) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 256, + 297, + 746, + 450 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Thus the product $( I + 2 \\Omega _ { 0 } ) ^ { - 1 } \\dot { \\Omega } _ { 0 }$ has zero diagonal blocks so that its trace is 0. Therefore, only the second term in the sum in Eq. (58) above survives, and we have ", + "bbox": [ + 171, + 454, + 823, + 483 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/9c557f45da4c8c6404c71ae18ca0c5fbcd74d369f500c64ab03e9a77ae6c9b51.jpg", + "text": "$$\n\\begin{array} { r l } & { \\frac { d } { d \\tau } \\mathbb { E } [ B _ { + } | \\hat { z } | \\mathcal { E } \\theta _ { + } | \\hat { z } \\rangle \\otimes B _ { + } ( z ) | \\hat { z } \\rangle \\langle z , z | ^ { 2 } \\rangle \\sim \\mathcal { N } ( 0 , \\hat { z } ) ] _ { | \\omega | } } \\\\ & = B ^ { \\ast } \\Gamma ( \\alpha / 2 ) ^ { - 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } d \\tau \\mathrm { d } s ^ { \\mathrm { i n } / 2 - 1 } \\{ 1 + 2 g \\mu ^ { s } \\} ^ { - \\frac { D - 1 } { 2 } ( 1 + 2 g \\mu ^ { s } ) ^ { - \\frac { D - 1 } { 2 } } ( 1 + 2 g \\mu ^ { s } ) ^ { - \\frac { D - 1 } { 2 } } \\times } \\\\ & { \\qquad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & & = B ( B - 1 ) ^ { \\ast - 1 } \\Gamma ( \\kappa / 2 ) ^ { - 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } d \\tau ( s ) ^ { \\omega / 2 - 1 } \\frac { ( \\omega ^ { \\omega - 1 } ) ^ { \\omega - 1 } ( \\omega ^ { \\omega - 1 } ) ^ { \\omega - 1 } ( \\omega ^ { \\omega - 1 } ) ^ { \\omega } ( 1 + 2 g \\mu ^ { s } ) ^ { - \\frac { D - 1 } { 2 } } ( 1 + 2 g \\mu ^ { s } ) ^ { - \\frac { D - 1 } { 2 } } } \\\\ & = B ( B - 1 ) ^ { \\ast - 1 } \\Gamma ( \\kappa / 2 ) ^ { - 2 } \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } \\mathrm { d } s \\int _ { 0 } ^ { \\infty } d \\tau ( s ) ^ { \\omega / 2 - 1 } \\frac { ( \\omega ^ { \\omega - 1 } - \\omega \\hat { \\omega } ^ { \\omega } ) ( 0 ) } ( 1 + 2 g \\mu ^ { s } ) ^ { 1 / 2 } ( 1 - 2 g \\mu ^ { s } ) ^ { \\omega - 1 } ( 1 + 2 g \\mu ^ { s } ) ^ { \\omega - 1 } ( \\frac \\ \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 491, + 821, + 751 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "I CROSS BATCH: BACKWARD DYNAMICS ", + "text_level": 1, + "bbox": [ + 173, + 792, + 532, + 810 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "For $k \\geq 2$ , we wish to study the following generalization of Eq. (51), ", + "bbox": [ + 173, + 824, + 627, + 840 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/b23b53ec45f2bd3eb8176655db9f0763f3c0ac14a595ef7df82dc52d1571a1f3.jpg", + "text": "$$\n\\tilde { \\Pi } ^ { l } = \\mathrm { V } _ { { B _ { \\phi } ^ { \\oplus k ^ { \\prime } } } } ( \\widetilde { \\Sigma } ^ { l } ) ^ { \\dagger } \\big \\{ \\tilde { \\Pi } ^ { l + 1 } \\big \\} .\n$$", + "text_format": "latex", + "bbox": [ + 411, + 844, + 586, + 869 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "As in the single batch case, we approximate it by taking $\\widetilde { \\Sigma } ^ { l }$ to its CBSB1 limit $\\widetilde { \\Sigma } ^ { * }$ , so that we analyze ", + "bbox": [ + 174, + 875, + 823, + 893 + ], + "page_idx": 86 + }, + { + "type": "equation", + "img_path": "images/739a18e3394364c4ff5f93123d5e0c8fa71326c5c84e394cfc83dcd197f745c6.jpg", + "text": "$$\n\\tilde { \\Pi } ^ { l } = \\mathrm { V } _ { B _ { \\phi } ^ { \\oplus k ^ { \\prime } } } ( \\widetilde { \\Sigma } ^ { * } ) ^ { \\dagger } \\{ \\tilde { \\Pi } ^ { l + 1 } \\}\n$$", + "text_format": "latex", + "bbox": [ + 411, + 896, + 584, + 922 + ], + "page_idx": 86 + }, + { + "type": "text", + "text": "Its dynamics is then given by the eigendecomposition of $\\nabla _ { B _ { \\phi } ^ { \\oplus _ { k ^ { \\prime } } } } ( \\widetilde { \\Sigma } ^ { * } )$ , in parallel to the single batch scenario. ", + "bbox": [ + 174, + 101, + 826, + 137 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "WLOG, we only need to consider the case $k = 2$ . ", + "bbox": [ + 173, + 142, + 500, + 159 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/4cf2bcb6353580d7c4fbab86c0faa3062537dd4f6e19fe19f838da20e141ad3f.jpg", + "text": "$$\n\\begin{array} { r l } { \\mathrm { V } _ { B _ { \\phi } ^ { \\Phi ^ { 2 } } } / \\widetilde { \\Sigma } ^ { * } \\big ) = \\underset { ( x , y ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { * } ) } { \\mathbb { E } } [ ( B _ { \\phi } ^ { \\prime } ( x ) \\oplus B _ { \\phi } ^ { \\prime } ( y ) ) ^ { \\otimes 2 } ] } & { } \\\\ { \\mathrm { V } _ { B _ { \\phi } ^ { \\Phi ^ { 2 } \\prime } } ( \\widetilde { \\Sigma } ^ { * } ) \\left\\{ \\left( \\begin{array} { l l } { \\Sigma } & { \\Xi } \\\\ { \\Xi ^ { T } } & { \\Lambda } \\end{array} \\right) \\right\\} = \\underset { ( x , y ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { * } ) } { \\mathbb { E } } \\left[ \\left( B _ { \\phi } ^ { \\prime } ( x ) \\Sigma B _ { \\phi } ^ { \\prime } ( x ) ^ { T } \\quad B _ { \\phi } ^ { \\prime } ( x ) \\Xi B _ { \\phi } ^ { \\prime } ( y ) ^ { T } \\right) \\right] } & { } \\\\ { \\mathrm { V } _ { B _ { \\phi } ^ { \\Phi ^ { 2 } \\prime } } ( \\widetilde { \\Sigma } ^ { * } ) \\left\\{ \\left( \\begin{array} { l l } { \\Sigma } & { \\Lambda } \\\\ { \\Xi ^ { T } } & { \\Lambda } \\end{array} \\right) \\right\\} = \\underset { ( x , y ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { * } ) } { \\mathbb { E } } \\left[ \\left( B _ { \\phi } ^ { \\prime } ( y ) \\Xi ^ { T } B _ { \\phi } ^ { \\prime } ( x ) ^ { T } \\quad B _ { \\phi } ^ { \\prime } ( y ) \\Lambda B _ { \\phi } ^ { \\prime } ( y ) ^ { T } \\right) \\right] } & { } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 215, + 165, + 782, + 236 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "From this one sees that $\\mathrm { V } _ { \\left[ ( B _ { \\phi } ^ { \\oplus 2 } ) ^ { \\prime } \\right] }$ acts independently on each block, and consequently so does its adjoint. The diagonal blocks of Eq. (59) evolves according to Eq. (51) which we studied in Appendix G. In this section we will study the evolution of the off-diagonal blocks, which is given by ", + "bbox": [ + 173, + 241, + 825, + 303 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/bd7d022ffa1d8c318ef1723ef60754d23a905b863ce3a2f227b9b7f654ee3bc8.jpg", + "text": "$$\n\\begin{array} { r l r } { \\Xi ^ { l } = \\underset { ( x , y ) \\sim \\mathcal { N } ( 0 , \\tilde { \\Sigma } ^ { * } ) } { \\mathbb { E } } [ \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) \\otimes \\mathcal { B } _ { \\phi } ^ { \\prime } ( y ) ] ^ { \\dagger } \\{ \\Xi ^ { l + 1 } \\} } & { = \\underset { ( x , y ) \\sim \\mathcal { N } ( 0 , \\tilde { \\Sigma } ^ { * } ) } { \\mathbb { E } } [ \\mathcal { B } _ { \\phi } ^ { \\prime } ( y ) ^ { T } \\otimes \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) ^ { T } ] \\{ \\Xi ^ { l + 1 } \\} } \\\\ & { = \\underset { ( x , y ) \\sim \\mathcal { N } ( 0 , \\tilde { \\Sigma } ^ { * } ) } { \\mathbb { E } } [ \\mathcal { B } _ { \\phi } ^ { \\prime } ( y ) ^ { T } \\Xi ^ { l + 1 } \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) ^ { T } ] } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 186, + 308, + 784, + 369 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "Definition I.1. Let $\\mathcal { M } _ { B } : = \\mathbb { R } ^ { B \\times B }$ be the set of all $B \\times B$ real matrices, and let $\\mathcal { M } _ { B } ^ { G } : = \\{ \\Xi \\in$ $\\mathcal { M } _ { B } : G \\Xi G = \\Xi \\}$ . ", + "bbox": [ + 166, + 376, + 825, + 409 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "Define $\\mathcal { T } : = \\mathbb { E } _ { ( x , y ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { * } ) } [ \\mathcal { B } _ { \\phi } ^ { \\prime } ( x ) \\otimes \\mathcal { B } _ { \\phi } ^ { \\prime } ( y ) ]$ . As in the single batch case, after one step of backprop, $\\Xi ^ { L - 1 } = \\mathcal { V } ^ { \\dagger } \\{ \\Xi ^ { L } \\}$ is in $\\mathcal { M } _ { B } ^ { G }$ . Thus it suffices to study the eigendecomposition of $G ^ { \\otimes 2 } \\circ T ^ { \\dagger } \\mid { \\mathcal { M } } _ { B } ^ { G } :$ $\\mathcal { M } _ { B } ^ { G } \\mathcal { M } _ { B } ^ { G }$ . Below, we will show that its adjoint $\\mathcal { T } \\circ G ^ { \\otimes 2 }$ acting on $\\mathcal { M } _ { B } ^ { G }$ is in fact multiplication by a constant, and thus so is it. ", + "bbox": [ + 173, + 416, + 826, + 482 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "We first compute ", + "bbox": [ + 174, + 488, + 289, + 502 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/98a4cb6733a05849f224c6e7a0d5366510679fd64f600fa77775734c7c384199.jpg", + "text": "$$\n\\begin{array} { r } { \\mathcal { T } = \\underset { ( \\xi , \\eta ) \\sim \\mathcal { N } ( 0 , G ^ { \\oplus 2 } \\widetilde { \\Sigma } ^ { * } G ^ { \\oplus 2 } ) } { \\mathbb { E } } \\left[ \\left. \\frac { \\mathrm { d } \\phi \\circ \\mathsf { n } ( z ) } { \\mathrm { d } z } \\right| _ { z = \\xi } \\otimes \\left. \\frac { \\mathrm { d } \\phi \\circ \\mathsf { n } ( z ) } { \\mathrm { d } z } \\right| _ { z = \\eta } \\right] \\circ G ^ { \\otimes 2 } } \\\\ { = \\underset { ( \\xi , \\eta ) \\sim \\mathcal { N } ( 0 , \\mu ^ { * } \\left( \\begin{array} { l l } { G } & { G } \\\\ { 0 } & { G } \\end{array} \\right) ) } { \\mathbb { E } } \\left[ \\left. \\frac { \\mathrm { d } \\phi \\circ \\mathsf { n } ( z ) } { \\mathrm { d } z } \\right| _ { z = \\xi } \\otimes \\left. \\frac { \\mathrm { d } \\phi \\circ \\mathsf { n } ( z ) } { \\mathrm { d } z } \\right| _ { z = \\eta } \\right] \\circ G ^ { \\otimes 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 266, + 508, + 730, + 597 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "since $\\widetilde { \\Sigma } ^ { * }$ is CBSB1 with diagonal blocks $\\Sigma ^ { * }$ (Corollary H.4). Then $\\xi$ is independent from $\\eta$ , so this is just ", + "bbox": [ + 173, + 606, + 823, + 636 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/11a336088b309599b663ac2af8d4b871b90414403df61c57e43674cd0ab80e55.jpg", + "text": "$$\n\\begin{array} { c } { { { \\displaystyle \\mathbb { E } } } } \\\\ { { { \\displaystyle ( x , y ) \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { * } ) } } } \\\\ { { \\displaystyle = \\sum _ { \\xi \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) } \\mathbb { E } _ { \\xi } ^ { \\prime } ( y ) ] = \\sum _ { \\xi \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) \\atop \\xi \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) } [ \\frac { \\mathrm { d } \\phi \\circ \\mathsf { n } ( z ) } { \\mathrm { d } z } | _ { z = \\xi } ] ^ { \\otimes 2 } \\circ G ^ { \\otimes 2 } \\ } } \\\\ { { { \\displaystyle = \\sum _ { \\xi \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) } \\mathbb { E } [ \\sqrt { B } D r ^ { - 1 } ( I - v v v ^ { T } ) ] ^ { \\otimes 2 } \\circ G ^ { \\otimes 2 } \\ } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 240, + 642, + 758, + 722 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "(by Proposition G.4) ", + "text_level": 1, + "bbox": [ + 617, + 723, + 754, + 738 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "where $\\begin{array} { r c l r c l } { D } & { = } & { \\mathrm { D i a g } ( \\phi ^ { \\prime } ( \\sqrt { B } v ) ) , r } & { = } & { \\| \\xi \\| , v } & { = } & { \\xi / \\| \\xi \\| } \\end{array}$ . But notice that $T : =$ $\\mathbb { E } _ { \\xi \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) } \\left[ D r ^ { - 1 } ( I - v v ^ { T } ) \\right]$ is actually BSB1, with diagonal ", + "bbox": [ + 173, + 747, + 823, + 781 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/8c6c31c90e6d07be176bbc61faa8a9b00d90ec67772a7e1d673d078cc88c4461.jpg", + "text": "$$\n_ { r v \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) } r ^ { - 1 } ( 1 - v _ { i } ^ { 2 } ) \\phi ^ { \\prime } ( \\sqrt { B } v _ { i } ) = \\underset { r v \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) } { \\mathbb { E } } r ^ { - 1 } ( 1 - v _ { 1 } ^ { 2 } ) \\phi ^ { \\prime } ( \\sqrt { B } v _ { 1 } ) , \\forall i \\in [ B ]\n$$", + "text_format": "latex", + "bbox": [ + 218, + 787, + 777, + 815 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "and off-diagonal ", + "bbox": [ + 173, + 824, + 285, + 838 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/f211317ea3807501cd67aafdde0b9d3e885069537930f66b5073855375006abb.jpg", + "text": "$$\n\\begin{array} { r } { \\underset { r v \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) } { \\mathbb { E } } r ^ { - 1 } ( - v _ { i } v _ { j } ) \\phi ^ { \\prime } ( \\sqrt { B } v _ { i } ) = \\underset { r v \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) } { \\mathbb { E } } r ^ { - 1 } ( - v _ { 1 } v _ { 2 } ) \\phi ^ { \\prime } ( \\sqrt { B } v _ { 1 } ) , \\forall i \\neq j \\in [ B ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 205, + 844, + 792, + 873 + ], + "page_idx": 87 + }, + { + "type": "text", + "text": "Thus by Lemma E.51, $T$ has two eigenspaces, $\\{ x : G x = x \\}$ and $\\mathbb { R } \\mathbf { 1 }$ , with respective eigenvalues $T _ { 1 1 } - T _ { 1 2 }$ and $T _ { 1 1 } + ( B - 1 ) T _ { 1 2 }$ . However, because of the composition with $G ^ { \\overleftarrow { \\otimes } 2 }$ , only the former eigenspace survives with nonzero eigenvalue in $G ^ { \\otimes 2 } \\circ \\tau$ . ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 87 + }, + { + "type": "equation", + "img_path": "images/cfa0b7fbcbed60039b2732daffdaf5f6e6f55d2a416f792b979636ad2bf34f63.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { T h e o r e m } \\qquad \\mathbf { I . 2 . } \\qquad \\tau \\qquad \\ o n \\qquad \\mathcal { M } _ { B } ^ { G } \\qquad i s \\qquad j u s t \\qquad m u l t i p l i c a t i o n \\qquad b y \\qquad \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\pm } } \\\\ & { \\frac { 1 } { 2 \\pi } B \\mu ^ { * - 1 } \\left( \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta \\ \\sin ^ { B - 1 } \\theta \\phi ^ { \\prime } \\bigl ( - \\sqrt { B - 1 } \\cos \\theta \\bigr ) \\right) ^ { 2 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 171, + 99, + 772, + 145 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "Proof. By the above reasoning, the sole eigenspace with nonzero eigenvalue is $\\{ x : G x = x \\} ^ { \\otimes 2 } =$ $\\mathcal { M } _ { B } ^ { G }$ . It has eigenvalue ", + "bbox": [ + 174, + 156, + 825, + 186 + ], + "page_idx": 88 + }, + { + "type": "equation", + "img_path": "images/b8e77de4d0bfcb0fcd82460963cc5b1590c4d607c9822161a8ec89bbf92e0334.jpg", + "text": "$$\n\\begin{array} { r l } & { \\quad B \\left( \\mathbb { E } [ r ^ { - 1 } ( 1 - v _ { 1 } ^ { 2 } + v _ { 1 } v _ { 2 } ) \\phi ^ { \\prime } ( \\sqrt { B } v _ { 1 } ) : r v \\sim \\mathcal { N } ( 0 , \\mu ^ { * } G ) ] \\right) ^ { 2 } } \\\\ & { = B \\mu ^ { * - 1 } \\left( \\mathbb { E } [ r ^ { - 1 } ( 1 - v _ { 1 } ^ { 2 } + v _ { 1 } v _ { 2 } ) \\phi ^ { \\prime } ( \\sqrt { B } v _ { 1 } ) : r v \\sim \\mathcal { N } ( 0 , G ) ] \\right) ^ { 2 } } \\\\ & { = B \\mu ^ { * - 1 } \\left( \\mathbb { E } [ r ^ { - 1 } ( 1 - ( \\exp ) _ { 1 } ^ { 2 } + ( \\exp ) _ { 1 } ( \\exp ) _ { 2 } ) \\phi ^ { \\prime } ( \\sqrt { B } ( \\exp ) _ { 1 } ) : r w \\sim \\mathcal { N } ( 0 , I _ { B - 1 } ) ] \\right) ^ { 2 } } \\\\ & { = B \\mu ^ { * - 1 } \\biggl ( \\frac { \\Gamma ( ( B - 2 ) / 2 ) } { \\Gamma ( ( B - 3 ) / 2 ) } 2 ^ { - 1 / 2 } \\pi ^ { - 1 } } \\\\ & { \\qquad \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 2 } \\left( 1 - \\zeta ( \\theta _ { 1 } ) ^ { 2 } + \\zeta ( \\theta _ { 1 } ) \\omega ( \\theta _ { 1 } , \\theta _ { 2 } ) \\right) \\phi ^ { \\prime } ( \\sqrt { B } \\zeta ( \\theta _ { 1 } ) ) \\sin ^ { B - 3 } \\theta _ { 1 } \\sin ^ { B - 4 } \\theta _ { 2 } \\biggr ) ^ { 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 191, + 812, + 356 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "where we applied Lemma E.36 with $\\begin{array} { r } { \\zeta ( \\theta ) = - \\sqrt { \\frac { B - 1 } { B } } \\cos \\theta } \\end{array}$ and $\\begin{array} { r } { \\omega ( \\theta _ { 1 } , \\theta _ { 2 } ) = \\frac { 1 } { \\sqrt { B ( B - 1 ) } } \\cos \\theta _ { 1 } - } \\end{array}$ $\\begin{array} { r } { \\sqrt { \\frac { B - 2 } { B - 1 } } \\sin \\theta _ { 1 } \\cos \\theta _ { 2 } . } \\end{array}$ ", + "bbox": [ + 173, + 363, + 826, + 416 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "We can further simplify ", + "bbox": [ + 174, + 420, + 333, + 435 + ], + "page_idx": 88 + }, + { + "type": "equation", + "img_path": "images/38b329465256bb9ae40b9740248fff690a1cfe91703a7aa7a7757ed55215458b.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 2 } ( 1 - \\zeta ( \\theta _ { 1 } ) ^ { 2 } + \\zeta ( \\theta _ { 1 } ) \\omega ( \\theta _ { 1 } , \\theta _ { 2 } ) ) \\phi ^ { \\prime } ( \\sqrt { B } \\zeta ( \\theta _ { 1 } ) ) \\sin ^ { B - 3 } \\theta _ { 1 } \\sin ^ { B - 4 } \\theta _ { 2 } } } \\\\ & { } & { = \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } \\sin ^ { B - 3 } \\theta _ { 1 } \\phi ^ { \\prime } ( \\sqrt { B } \\zeta ( \\theta _ { 1 } ) ) \\Bigg ( \\Bigg ( 1 - \\zeta ( \\theta _ { 1 } ) ^ { 2 } + \\zeta ( \\theta _ { 1 } ) \\frac { 1 } { \\sqrt { B ( B - 1 ) } } \\cos \\theta _ { 1 } \\Bigg ) \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 2 } \\sin ^ { B - 4 } \\theta _ { 2 } } \\\\ & { } & { \\quad - \\sqrt { \\frac { B - 2 } { B - 1 } } \\sin \\theta _ { 1 } \\zeta ( \\theta _ { 1 } ) \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 2 } \\sin ^ { B - 4 } \\theta _ { 2 } \\cos \\theta _ { 2 } \\Bigg ) } \\\\ & { } & { = \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } \\sin ^ { B - 3 } \\theta _ { 1 } \\phi ^ { \\prime } ( \\sqrt { B } \\zeta ( \\theta _ { 1 } ) ) \\Bigg ( \\Bigg ( 1 - \\zeta ( \\theta _ { 1 } ) ^ { 2 } + \\zeta ( \\theta _ { 1 } ) \\frac { 1 } { \\sqrt { B ( B - 1 ) } } \\cos \\theta _ { 1 } \\Bigg ) \\mathrm { B e t a } \\Bigg ( \\frac { B - 3 } { 2 } , \\frac { 1 } { 2 } \\Bigg ) \\Bigg ) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 441, + 839, + 606 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "(by Lemma E.29) ", + "bbox": [ + 730, + 606, + 849, + 621 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "so the cross-batch backward off-diagonal eigenvalue is ", + "bbox": [ + 173, + 696, + 534, + 713 + ], + "page_idx": 88 + }, + { + "type": "equation", + "img_path": "images/48a5a35c25e94cbba504f34052fe290d096e4515f5584e58b220329b9f2eb415.jpg", + "text": "$$\n\\begin{array} { c } { { B \\mu ^ { * - 1 } \\left( \\displaystyle \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } ~ \\mathrm { s i n } ^ { B - 1 } \\theta _ { 1 } \\phi ^ { \\prime } ( - \\sqrt { B - 1 } \\cos \\theta _ { 1 } ) \\right) ^ { 2 } } } \\\\ { { = \\displaystyle \\frac { 1 } { 2 \\pi } B \\mu ^ { * - 1 } \\left( \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta _ { 1 } ~ \\mathrm { s i n } ^ { B - 1 } \\theta _ { 1 } \\phi ^ { \\prime } ( - \\sqrt { B - 1 } \\cos \\theta _ { 1 } ) \\right) ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 299, + 719, + 699, + 796 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "For positive homogeneous functions we can evaluate the eigenvalues explicitly. ", + "bbox": [ + 171, + 837, + 694, + 852 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "Theorem I.3. If $\\phi$ is positive-homogeneous of degree $\\alpha$ with $\\phi ( c ) = a \\rho _ { \\alpha } ( c ) - b \\rho _ { \\alpha } ( - c )$ , then $\\mathcal { T } : \\mathcal { M } _ { B } ^ { G } \\to \\mathcal { M } _ { B } ^ { \\ddot { G } }$ is just multiplication by ", + "bbox": [ + 171, + 854, + 823, + 886 + ], + "page_idx": 88 + }, + { + "type": "equation", + "img_path": "images/e1dbe820c7daf5b28b5a24e271d64e018d51cf1c08842c4a53b3457c771fb453.jpg", + "text": "$$\n\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow } = \\frac { 1 } { 8 \\pi } B ( B - 1 ) ^ { \\alpha - 1 } \\mu ^ { * - 1 } \\alpha ^ { 2 } ( a + b ) ^ { 2 } \\mathrm { B e t a } \\left( \\frac { \\alpha } { 2 } , \\frac { B } { 2 } \\right) ^ { 2 } .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 892, + 691, + 929 + ], + "page_idx": 88 + }, + { + "type": "text", + "text": "Proof. As in the proof of Corollary H.4, ", + "bbox": [ + 174, + 103, + 439, + 118 + ], + "page_idx": 89 + }, + { + "type": "equation", + "img_path": "images/698d42ee03d0bbf612c807c5e5675782f630353f96aaf073551de4278d3d172b.jpg", + "text": "$$\n\\int _ { 0 } ^ { \\pi / 2 } \\mathrm { d } \\theta ( \\cos \\theta ) ^ { \\alpha - 1 } \\sin ^ { B - 1 } \\theta = \\int _ { \\pi / 2 } ^ { \\pi } \\mathrm { d } \\theta ( - \\cos \\theta ) ^ { \\alpha - 1 } \\sin ^ { B - 1 } \\theta = { \\frac { 1 } { 2 } } \\operatorname { B e t a } \\left( { \\frac { \\alpha } { 2 } } , { \\frac { B } { 2 } } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 215, + 122, + 777, + 159 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "So ", + "bbox": [ + 173, + 160, + 196, + 172 + ], + "page_idx": 89 + }, + { + "type": "equation", + "img_path": "images/09e46ff6f663da34f43013e33badca840ab8b2bf21e091ae121a7fce13e81fcb.jpg", + "text": "$$\n\\begin{array} { l } { { { \\frac { 1 } { 2 \\pi } } B \\mu ^ { * - 1 } \\left( \\displaystyle \\int _ { 0 } ^ { \\pi } { \\mathrm { d } \\theta _ { 1 } \\sin ^ { B - 1 } \\theta _ { 1 } \\phi ^ { \\prime } ( - \\sqrt { B - 1 } \\cos \\theta _ { 1 } ) } \\right) ^ { 2 } } } \\\\ { { = { \\frac { 1 } { 2 \\pi } } B \\mu ^ { * - 1 } ( B - 1 ) ^ { \\alpha - 1 } \\alpha ^ { 2 } \\left( a \\displaystyle \\int _ { \\pi / 2 } ^ { \\pi } { \\mathrm { d } \\theta \\left( - \\cos \\theta \\right) ^ { \\alpha - 1 } \\sin ^ { B - 1 } \\theta } + b \\displaystyle \\int _ { 0 } ^ { \\pi / 2 } { \\mathrm { d } \\theta \\left( \\cos \\theta \\right) ^ { \\alpha - 1 } \\sin ^ { B - 1 } \\theta } \\right) ^ { 2 } } } \\\\ { { = { \\frac { 1 } { 2 \\pi } } B \\mu ^ { * - 1 } ( B - 1 ) ^ { \\alpha - 1 } \\alpha ^ { 2 } ( a + b ) ^ { 2 } 2 ^ { - 2 } \\mathrm { B e t a } \\left( { \\frac { \\alpha } { 2 } } , { \\frac { B } { 2 } } \\right) ^ { 2 } } } \\\\ { { = { \\frac { 1 } { 8 \\pi } } B ( B - 1 ) ^ { \\alpha - 1 } \\mu ^ { * - 1 } \\alpha ^ { 2 } ( a + b ) ^ { 2 } \\mathrm { B e t a } \\left( { \\frac { \\alpha } { 2 } } , { \\frac { B } { 2 } } \\right) ^ { 2 } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 174, + 836, + 333 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "It is also straightforward to obtain an expression of the eigenvalue in terms of Gegenbauer coefficients. ", + "bbox": [ + 174, + 359, + 823, + 388 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "Theorem I.4. If φ(√B − 1x) has Gegenbauer expansion P∞l=0 al 1cB−1,l C $\\begin{array} { r } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( x ) } \\end{array}$ , then $\\tau$ $\\mathcal { M } _ { B } ^ { G } \\mathcal { M } _ { B } ^ { G }$ is just multiplication by ", + "bbox": [ + 174, + 390, + 818, + 429 + ], + "page_idx": 89 + }, + { + "type": "equation", + "img_path": "images/65e466a2008b573d92845326e0293edf65871fe7ac4c82d3b71739b668a6a7c0.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow } = \\frac { 1 } { 2 \\pi } B ( B - 1 ) \\mu ^ { \\ast ^ { - 1 } } a _ { 1 } ^ { 2 } \\mathrm { B e t a } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { 2 } } } \\\\ { { \\displaystyle \\quad = \\frac { \\frac { 1 } { 2 \\pi } B ( B - 1 ) a _ { 1 } ^ { 2 } \\mathrm { B e t a } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { 2 } } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } } } \\\\ { { \\displaystyle \\quad = \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } . } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 431, + 684, + 542 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "Proof. It suffices to express the integral in $\\mathrm { T h m } \\ : \\mathrm { I } . 2$ in Gegenbauer coefficients. Changing coordinates $x = - \\cos \\theta$ , so that $\\mathrm { d } x = \\sin \\theta \\mathrm { d } \\theta$ , we get ", + "bbox": [ + 174, + 553, + 823, + 583 + ], + "page_idx": 89 + }, + { + "type": "equation", + "img_path": "images/c10e2acbe0b0c268e750af008b73d0b50e0cc940947a05681577ce7e309462dd.jpg", + "text": "$$\n\\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta \\sin ^ { B - 1 } \\theta \\phi ^ { \\prime } ( - \\sqrt { B - 1 } \\cos \\theta ) = \\int _ { - 1 } ^ { 1 } \\mathrm { d } x ( 1 - x ^ { 2 } ) ^ { \\frac { B - 2 } { 2 } } \\phi ^ { \\prime } ( \\sqrt { B - 1 } x ) .\n$$", + "text_format": "latex", + "bbox": [ + 250, + 584, + 745, + 621 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "By Eq. (35), ", + "bbox": [ + 173, + 621, + 258, + 637 + ], + "page_idx": 89 + }, + { + "type": "equation", + "img_path": "images/b1282918e669da1a87c3ca389c589d5f690cd03646af3c57dee9651d714f0b0c.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\phi ^ { \\prime } ( \\sqrt { B - 1 } x ) = ( B - 1 ) ^ { - 1 / 2 } \\frac { d } { d x } \\phi ( \\sqrt { B - 1 } x ) } } } \\\\ { { \\displaystyle ~ = ( B - 1 ) ^ { - 1 / 2 } \\sum _ { l = 0 } ^ { \\infty } a _ { l } \\frac { 1 } { c _ { B - 1 , l } } C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\prime ( x ) } } \\\\ { { \\displaystyle ~ = ( B - 1 ) ^ { - 1 / 2 } \\sum _ { l = 1 } ^ { \\infty } a _ { l } \\frac { B - 3 } { c _ { B - 1 , l } } C _ { l - 1 } ^ { ( \\frac { B - 1 } { 2 } ) } ( x ) } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 313, + 637, + 683, + 753 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "Then by the orthogonality relations among $\\{ C _ { l - 1 } ^ { ( \\frac { B - 1 } { 2 } ) } \\} _ { l \\ge 1 }$ , ", + "bbox": [ + 173, + 755, + 552, + 776 + ], + "page_idx": 89 + }, + { + "type": "equation", + "img_path": "images/f9f9b550e0888a7cfcdbc6402c23aaf3c38bde3761fa4038448b5f3bdd338245.jpg", + "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\int _ { 0 } ^ { \\pi } \\mathrm { d } \\theta \\sin ^ { B - 1 } \\theta \\phi ^ { \\prime } ( - \\sqrt { B - 1 } \\cos \\theta ) = \\int _ { - 1 } ^ { 1 } \\mathrm { d } x ( 1 - x ^ { 2 } ) ^ { \\frac { B - 2 } { 2 } } \\phi ^ { \\prime } ( \\sqrt { B - 1 } x ) } \\\\ & { \\quad \\quad \\quad \\quad = ( B - 1 ) ^ { - 1 / 2 } a _ { 1 } \\frac { B - 3 } { c _ { B - 1 , 1 } } \\int _ { - 1 } ^ { 1 } \\left[ C _ { 0 } ^ { ( \\frac { B - 1 } { 2 } ) } ( x ) \\right] ^ { 2 } ( 1 - x ^ { 2 } ) ^ { \\frac { B - 2 } { 2 } } d x } \\\\ & { \\quad \\quad \\quad = ( B - 1 ) ^ { - 1 / 2 } a _ { 1 } \\frac { B - 3 } { c _ { B - 1 , 1 } } \\frac { \\sqrt { \\pi } \\Gamma \\left( \\frac { B } { 2 } \\right) } { \\Gamma \\left( \\frac { B + 1 } { 2 } \\right) } } \\\\ & { \\quad \\quad \\quad = a _ { 1 } \\sqrt { B - 1 } \\mathrm { B e t a } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 181, + 779, + 828, + 929 + ], + "page_idx": 89 + }, + { + "type": "text", + "text": "To summarize, ", + "bbox": [ + 174, + 137, + 272, + 151 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Theorem I.5. The backward cross batch dynamics Eq. (59) over $\\mathcal { H } _ { k B } ^ { G }$ is separable over each individual $B \\times B$ block. The diagonal blocks evolve as in Eq. (51), and this linear dynamics has eigenvalues $\\lambda _ { G } ^ { \\downarrow } , \\lambda _ { \\mathbb { L } } ^ { \\downarrow } , \\lambda _ { \\mathbb { M } } ^ { \\downarrow }$ as described by Appendix G. The off-diagonal blocks dynamics is multiplication by $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow } = \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow }$ . ", + "bbox": [ + 174, + 156, + 825, + 222 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Note that ", + "bbox": [ + 173, + 233, + 238, + 247 + ], + "page_idx": 90 + }, + { + "type": "equation", + "img_path": "images/5e13d05b1f7e9b320b292d105de0c2539bc2d4d57f7976ab74081b3603fdd03d.jpg", + "text": "$$\n\\begin{array} { r l } & { \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow } = \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } = \\frac { \\frac { 1 } { 2 \\pi } B ( B - 1 ) a _ { 1 } ^ { 2 } \\mathrm { B e t a } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { 2 } } { \\sum _ { l = 0 } ^ { \\infty } a _ { l } ^ { 2 } \\frac { 1 } { c _ { B - 1 , l } } \\left( C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) - C _ { l } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } } \\\\ & { \\qquad \\leq \\frac { \\frac { 1 } { 2 \\pi } B ( B - 1 ) \\mathrm { B e t a } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { 2 } } { \\frac { 1 } { c _ { B - 1 , 1 } } \\left( C _ { 1 } ^ { ( \\frac { B - 3 } { 2 } ) } ( 1 ) - C _ { 1 } ^ { ( \\frac { B - 3 } { 2 } ) } \\left( \\frac { - 1 } { B - 1 } \\right) \\right) } } \\\\ & { \\qquad = \\frac { B - 1 } { 2 } \\mathrm { P } \\left( \\frac { B } { 2 } , \\frac { 1 } { 2 } \\right) ^ { - 2 } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 292, + 253, + 704, + 391 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "which increases to 1 from below as $B \\infty$ . Thus, ", + "bbox": [ + 174, + 397, + 511, + 411 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Corollary I.6. For any $\\phi$ inducing Eq. (54) to converge to a CBSB1 fixed point, $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow } = \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\uparrow } <$ 1. For any fixed batch size $B$ , $\\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow }$ is maximized by $\\phi \\ : = \\ : \\mathrm { i d } .$ . Furthermore, for $\\phi = \\mathrm { i d } , \\ : \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow } =$ $\\textstyle { \\frac { B - 1 } { 2 } } \\operatorname { P } \\left( { \\frac { B } { 2 } } , { \\frac { 1 } { 2 } } \\right) ^ { - 2 }$ and $\\begin{array} { r } { \\operatorname* { l i m } _ { B \\to \\infty } \\lambda _ { \\widetilde { \\mathbb { M } } } ^ { \\downarrow } = 1 } \\end{array}$ , increasing to the limit from below. ", + "bbox": [ + 173, + 417, + 825, + 478 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "This corollary indicates that stacking batchnorm in a deep neural network will always cause chaotic behavior, in the sense that cross batch correlation, both forward and backward, decreases to 0 exponentially with depth. The $\\phi$ that can maximally ameliorate the exponential loss of information is linear. ", + "bbox": [ + 174, + 488, + 825, + 544 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "J UNCOMMON REGIMES ", + "text_level": 1, + "bbox": [ + 174, + 565, + 392, + 582 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "In the above exposition of the mean field theory of batchnorm, we have assumed $B \\geq 4$ and that $\\phi$ induces BSB1 fixed points in Eq. (8). ", + "bbox": [ + 173, + 598, + 825, + 627 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Small Batch Size What happens when $B \\ < \\ 4 2$ It is clear that for $B = 1$ , batchnorm is not well-defined. For $B = 2$ , ${ \\cal B } _ { \\phi } ( h ) = ( \\pm 1 , \\mp 1 )$ depending on the signs of $h$ . Thus, the gradient of a batchnorm network with $B = 2$ is 0. Therefore, we see an abrupt phase transition from the immediate gradient vanishing of $B = 2$ to the gradient explosion of $B \\geq 4$ . We empirically see that $B = 3$ suffers similar gradient explosion as $B = 4$ and conjecture that a generalization of $\\cdot$ holds for $B = 3$ . ", + "bbox": [ + 173, + 642, + 825, + 727 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Batch Symmetry Breaking What about other nonlinearities? Empirically, we observe that if the fixed point is not BSB1, then it is BSB2, like in Fig. 1, where a submatrix of $\\Sigma$ (the dominant block) is much larger in magnitude than everything else (see Defn K.1). If the initial $\\Sigma ^ { 0 }$ is permutationinvariant, then convergence to this fixed point requires spontaneous symmetry breaking, as the dominant block can appear in any part of $\\Sigma$ along the diagonal. This symmetry breaking is lost when we take the mean field limit, but in real networks, the symmetry is broken by the network weight randomness. Because small fluctuation in the input can also direct the dynamics toward one BSB1 fixed point against others, causing large change in the output, the gradient is intuitively large as well. Additionally, at the BSB2 fixed point, we expect the dominant block goes through a similar dynamics as if it were a BSB1 fixed point for a smaller $B$ , thus suffering from similar gradient explosion. Appendix K discusses several results on our current understanding of BSB2 fixed points. A specific form of BSB2 fixed point with a $1 \\times 1$ dominant block can be analyzed much further, and this is done in Appendix K.1. ", + "bbox": [ + 173, + 742, + 825, + 924 + ], + "page_idx": 90 + }, + { + "type": "text", + "text": "Finite width effect For certain nonlinearities, the favored fixed point can be different between the large it and small width. For example, for $\\phi = C _ { 5 } ^ { ( \\frac { B - 3 } { 2 } ) } ( x / \\sqrt { B - 1 } ) - C _ { 7 } ^ { ( \\frac { B - 3 } { 2 } ) } ( x / \\sqrt { B - 1 } )$ $B = 1 0$ , one can observe that for width 100, Eq. (43) favors BSB2 fixed points, but for width 1000 and more, Eq. (43) favors BSB1 fixed points. ", + "bbox": [ + 173, + 102, + 826, + 166 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "K TOWARD UNDERSTANDING BSB2 FIXED POINTS ", + "text_level": 1, + "bbox": [ + 174, + 184, + 619, + 202 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Definition K.1. For $B _ { 1 } , B _ { 2 } \\geq 2$ , a BSB2 matrix is one of the form ", + "bbox": [ + 173, + 214, + 616, + 232 + ], + "page_idx": 91 + }, + { + "type": "equation", + "img_path": "images/ba08433328272c0623b39648e4622c72ba34510020bf39070fc6146845450896.jpg", + "text": "$$\n\\mathrm { B S B 2 } _ { B _ { 1 } , B _ { 2 } } ( d _ { 1 } , f _ { 1 } , d _ { 2 } , f _ { 2 } , c ) : = \\left( \\begin{array} { c c } { { \\mathrm { B S B 1 } _ { B _ { 1 } } ( d _ { 1 } , f _ { 1 } ) } } & { { c } } \\\\ { { c } } & \\mathrm { B S B 1 } _ { B _ { 2 } } ( d _ { 2 } , f _ { 2 } ) \\right) \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 259, + 236, + 736, + 270 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "up to simultaneous permutation of rows and columns. A $\\widehat { \\mathrm { B S B 2 } }$ matrix is a specific kind of BSB2 matrix $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { B ^ { \\prime } } ( d , f , c , b ) = \\mathrm { B S B 2 } _ { B ^ { \\prime } , B - B ^ { \\prime } } ( d , f , c , b , b )$ , where $B ^ { \\prime } \\in [ 2 , B - 1 ]$ (where for $B ^ { \\prime } =$ $B { - } 1$ the lower right hand block is a scalar $b$ ). We call the upper left hand block of $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { B ^ { \\prime } } ( d , f , c , b )$ its main block. ", + "bbox": [ + 173, + 273, + 826, + 349 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "When $\\phi$ grows quickly, Eq. (43) can converge to BSB2 fixed points of the form $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { B ^ { \\prime } } ( d , f , c , b )$ , up to simultaneous permutation of rows and columns. In this section we present several results on BSB2 fixed points that sheds light on their structure. We leave the study of the forward Eq. (43) and backward dynamics Eq. (51) to future work. ", + "bbox": [ + 173, + 359, + 825, + 422 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "First, we give the eigendecomposition of BSB2 matrices. ", + "bbox": [ + 173, + 428, + 549, + 443 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Theorem K.2. Let $\\Sigma = \\mathrm { B S B 2 } _ { B _ { 1 } , B _ { 2 } } ( d _ { 1 } , f _ { 1 } , d _ { 2 } , f _ { 2 } , c )$ , where $B _ { 1 } , B _ { 2 } \\ \\geq \\ 2$ . Then ${ \\cal G } ^ { \\otimes 2 } \\{ \\Sigma \\} =$ $G \\Sigma G = \\Sigma ^ { G }$ has the following eigenspaces and eigenvalues. ", + "bbox": [ + 174, + 445, + 825, + 477 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "1. $( B _ { 1 } - 1 )$ -dimensional eigenspace $\\mathbb { Z } _ { 1 } : = \\{ ( x _ { 1 } , \\ldots , x _ { B _ { 1 } } , 0 , \\ldots , 0 ) : \\textstyle \\sum _ { i = 1 } ^ { B _ { 1 } } x _ { i } = 0 \\}$ , with eigenvalue $d _ { 1 } - f _ { 1 }$ . \n2. $( B _ { 2 } - 1 )$ -dimensional eigenspace $\\mathbb { Z } _ { 2 } : = \\{ ( 0 , \\ldots , 0 , y _ { 1 } , \\ldots , y _ { B _ { 2 } } ) : \\textstyle \\sum _ { j = 1 } ^ { B _ { 2 } } y _ { j } = 0 \\} _ { }$ , with eigenvalue $d _ { 2 } - f _ { 2 }$ . ", + "bbox": [ + 207, + 487, + 825, + 561 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "3. $^ { l }$ -dimensional $\\mathbb { R } q$ with eigenvalue $\\frac { ( d _ { 2 } - f _ { 2 } ) B _ { 1 } + ( d _ { 1 } - f _ { 1 } ) B _ { 2 } + ( f _ { 1 } + f _ { 2 } - 2 c ) B _ { 1 } B _ { 2 } } { B _ { 1 } + B _ { 2 } }$ , where $\\begin{array} { r l } { \\pmb { q } } & { { } = } \\end{array}$ $( \\overbrace { B _ { 1 } ^ { - 1 } , \\ldots , B _ { 1 } ^ { - 1 } } ^ { B _ { 1 } } , \\overbrace { - B _ { 2 } ^ { - 1 } , \\ldots , - B _ { 2 } ^ { - 1 } } ^ { B _ { 2 } } ) .$ ", + "bbox": [ + 210, + 568, + 825, + 626 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "4. $^ { l }$ -dimensional $\\mathbb { R } \\mathbf { 1 }$ with eigenvalue $O$ . ", + "bbox": [ + 212, + 631, + 475, + 646 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Proof. We can verify all eigenspaces and their eigenvalues in a straightforward manner. These then must be all of them by a dimensionality argument. □ ", + "bbox": [ + 173, + 660, + 826, + 690 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Specializing to $\\widehat { \\mathrm { B S B 2 } }$ matrices, we get \nCorollary K.3. Let $\\Sigma = \\widehat { \\mathrm { B S B 2 } } _ { B } ^ { B ^ { \\prime } } ( d , f , c , b )$ with $B ^ { \\prime } \\geq 2$ . Then $G ^ { \\otimes 2 } \\{ \\Sigma \\} = G \\Sigma G = \\Sigma ^ { G }$ has the \nfollowing eigenspaces and eigenvalues ", + "bbox": [ + 173, + 702, + 825, + 763 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "$( B ^ { \\prime } - 1 )$ -de ional eigenspace. $\\mathbb { Z } _ { 1 } : = \\{ ( x _ { 1 } , \\ldots , x _ { B ^ { \\prime } } , 0 , \\ldots , 0 ) : \\textstyle \\sum _ { i = 1 } ^ { B ^ { \\prime } } x _ { i } = 0 \\} _ { }$ , with $d - f$ ", + "bbox": [ + 215, + 772, + 825, + 806 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "• $^ { l }$ -dimensional $\\mathbb { R } q$ with eigenvalue $\\frac { ( d - f ) ( B - B ^ { \\prime } ) + ( f - c ) B ( B - B ^ { \\prime } ) } { B }$ where q = , $B ^ { \\prime }$ B $( \\overbrace { { B ^ { \\prime } } ^ { - 1 } , \\ldots , { B ^ { \\prime } } ^ { - 1 } } ^ { B } , \\overbrace { - { B _ { 2 } ^ { - 1 } } , \\ldots , - { B _ { 2 } ^ { - 1 } } } ^ { B _ { 2 } } )$ where $B _ { 2 } = B - B ^ { \\prime }$ . ", + "bbox": [ + 217, + 811, + 828, + 871 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "B 0 • (B − B0)-dimensional eigenspace {(z }| {−µ, . . . , −µ, y1, . . . , yB−B0 : µ = 1B−B0 PB−B j=1 with eigenvalue $O$ . ", + "bbox": [ + 215, + 878, + 825, + 925 + ], + "page_idx": 91 + }, + { + "type": "text", + "text": "Note also that when $B ^ { \\prime } = B$ , then $\\mathbb { Z } _ { 1 }$ and $\\mathbb { R } q$ become the usual eigenspaces of $\\operatorname { B S B 1 } ( d , f )$ . ", + "bbox": [ + 171, + 102, + 777, + 119 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "For fixed $B ^ { \\prime } < B$ , specializing the forward dynamics Eq. (43) to $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { B ^ { \\prime } }$ fixed points yields a 2-dimensional dynamics on the eigenvalues for the eigenspaces $\\mathbb { Z } _ { 1 }$ and $\\mathbb { R } q$ . This dynamics in general is not degenerate, so that the fixed point seems difficult to obtain analytically. Moreoever, the Gegenbauer method, essential for proving that gradient explosion is unavoidable when Eq. (43) has a BSB1 fixed point, does not immediately generalize to the BSB2 case, since $\\mathbb { Z } _ { 1 }$ and $\\mathbb { R } q$ in general do not have the same eigenvalues so that we cannot reduce the integrals to that on a sphere. For this reason, at present we do not have any rigorous result on the BSB2 case. ", + "bbox": [ + 173, + 125, + 825, + 231 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "However, we expect that the main block of the $\\widehat { \\mathrm { B S B 2 } }$ fixed point should undergo a similar dynamics to that of a BSB1 fixed point, leading to similar gradient explosion. ", + "bbox": [ + 171, + 237, + 823, + 270 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "K.1 $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { 1 }$ FIXED POINTS ", + "text_level": 1, + "bbox": [ + 174, + 286, + 377, + 306 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "When $\\phi$ grows extremely rapidly, for example $\\phi ( x ) = \\mathrm { r e l u } ( x ) ^ { 3 0 }$ , and the width of the network is not too large, then we sometimes observe $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { 1 }$ fixed points ", + "bbox": [ + 174, + 315, + 825, + 353 + ], + "page_idx": 92 + }, + { + "type": "equation", + "img_path": "images/50a5192b345e0731eb3edeb0e6b1eceaedbb8ac87c254adcb734478865f1d5bd.jpg", + "text": "$$\n\\widehat { \\mathbf { B S B 2 } } _ { B } ^ { 1 } ( d , c , b ) = \\left( \\begin{array} { c c } { d } & { c \\mathbf { 1 } ^ { T } } \\\\ { c \\mathbf { 1 } } & { b \\mathbf { 1 } \\mathbf { 1 } ^ { T } } \\end{array} \\right)\n$$", + "text_format": "latex", + "bbox": [ + 388, + 356, + 609, + 391 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "where $\\mathbf { 1 } \\in \\mathbb { R } ^ { B - 1 }$ is the all 1s column vector. In these situations we can in fact see pathological gradient vanishing, in a way reminiscent of the gradient vanishing for $B = 2$ batchnorm, as we shall see shortly. ", + "bbox": [ + 173, + 395, + 826, + 438 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "We can extend Corollary K.3 to the $B ^ { \\prime } = 1$ case. ", + "bbox": [ + 176, + 444, + 495, + 459 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Theorem K.4. $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { 1 } ( d , c , b ) ^ { G }$ is rank $^ { l }$ and equal to $\\lambda \\hat { \\pmb q } ^ { \\otimes 2 }$ , where $\\begin{array} { r } { \\lambda = \\frac { B - 1 } { B } ( d - 2 c + b ) } \\end{array}$ and $\\begin{array} { r } { \\hat { \\pmb q } = ( 1 , \\frac { - 1 } { B - 1 } , \\ldots , \\frac { - 1 } { B - 1 } ) \\sqrt { \\frac { B - 1 } { B } } } \\end{array}$ . ", + "bbox": [ + 174, + 463, + 826, + 508 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Proof. Straightforward verification. ", + "bbox": [ + 174, + 522, + 410, + 537 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Because $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { 1 }$ under $G$ -projection is rank 1, such a fixed point of Eq. (43) can be determined $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { B ^ { \\prime } }$ \n(unlike the general case). ", + "bbox": [ + 174, + 551, + 826, + 595 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Theorem K.5. There is a unique $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { 1 }$ fixed point of Eq. (43), $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { 1 } ( d ^ { * } , c ^ { * } , b ^ { * } )$ where ", + "bbox": [ + 171, + 598, + 771, + 621 + ], + "page_idx": 92 + }, + { + "type": "equation", + "img_path": "images/71ea061548ba1a9d6b997b958421d365ea9f252fa85051f44bfce2c40dfa13b8.jpg", + "text": "$$\n\\begin{array} { l } { \\displaystyle d ^ { * } = \\frac { 1 } { 2 } \\left( \\phi ( \\sqrt { B - 1 } ) ^ { 2 } + \\phi ( - \\sqrt { B - 1 } ) ^ { 2 } \\right) } \\\\ { \\displaystyle c ^ { * } = \\frac { 1 } { 2 } \\left( \\phi ( \\sqrt { B - 1 } ) \\phi ( \\frac { - 1 } { \\sqrt { B - 1 } } ) + \\phi ( - \\sqrt { B - 1 } ) \\phi ( \\frac { 1 } { \\sqrt { B - 1 } } ) \\right) } \\\\ { \\displaystyle b ^ { * } = \\frac { 1 } { 2 } \\left( \\phi ( \\frac { - 1 } { \\sqrt { B - 1 } } ) ^ { 2 } + \\phi ( \\frac { 1 } { \\sqrt { B - 1 } } ) ^ { 2 } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 282, + 623, + 715, + 724 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Additionally, $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { 1 } ( d ^ { * } , c ^ { * } , b ^ { * } ) ^ { G } = \\lambda ^ { * } \\hat { \\pmb q } ^ { \\otimes 2 }$ , where ", + "bbox": [ + 173, + 728, + 514, + 748 + ], + "page_idx": 92 + }, + { + "type": "equation", + "img_path": "images/b03c028277141f7c3519023615f096b05e4a33ba43adb53ce91f94e977db1ed7.jpg", + "text": "$$\n\\lambda ^ { * } = \\frac { B - 1 } { 2 B } \\left[ \\left( \\phi ( \\sqrt { B - 1 } ) - \\phi ( \\frac { - 1 } { \\sqrt { B - 1 } } ) \\right) ^ { 2 } + \\left( \\phi ( - \\sqrt { B - 1 } ) - \\phi ( \\frac { 1 } { \\sqrt { B - 1 } } ) \\right) ^ { 2 } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 217, + 752, + 777, + 795 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "Proof. If $\\Sigma = { \\widehat { \\mathrm { B S B 2 } } } _ { B } ^ { 1 } ( d , c , b )$ , then ", + "bbox": [ + 174, + 809, + 411, + 830 + ], + "page_idx": 92 + }, + { + "type": "equation", + "img_path": "images/73eca332a86101495e28dfda73db903a67c3448871686f764986f27d43858dae.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\mathrm { V } _ { \\mathcal { B } _ { \\phi } } ( { \\Sigma } ) = \\underset { y \\sim \\mathcal { N } ( 0 , G \\Sigma G ) } { \\mathbb { E } } \\phi ( { \\mathsf { n } } ( y ) ) ^ { \\otimes 2 } } } \\\\ { { \\displaystyle = \\frac { 1 } { 2 } \\left[ \\phi ( \\sqrt { B } \\hat { \\pmb q } ) ^ { \\otimes 2 } + \\phi ( - \\sqrt { B } \\hat { \\pmb q } ) ^ { \\otimes 2 } \\right] } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 351, + 833, + 647, + 892 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "since $G \\Sigma G = \\lambda \\hat { q } ^ { \\otimes 2 }$ for some $\\lambda$ by Thm K.4. The rest then follows from straightforward computation and another application of Thm K.4. □ ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 92 + }, + { + "type": "text", + "text": "The rank-1 nature of the $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { 1 }$ fixed point under $G$ -projection is reminiscent of the case of $B = 2$ batchnorm. As we will see, gradients of a certain form will vanish similarly. ", + "bbox": [ + 173, + 101, + 825, + 136 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Let $\\delta$ be a gradient vector. Then by Proposition G.4, for $\\Sigma ^ { * }$ being the $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { 1 }$ fixed point, one step of backpropagation yields ", + "bbox": [ + 174, + 141, + 820, + 178 + ], + "page_idx": 93 + }, + { + "type": "equation", + "img_path": "images/49f5e91cb3001dbd82393c1997a229dbc6928e8407e6f60cc6d22fadec10298a.jpg", + "text": "$$\n\\frac { \\mathrm { d } B _ { \\phi } } { \\mathrm { d } \\Sigma } \\bigg | _ { \\Sigma = \\Sigma ^ { * } } ^ { \\dagger } \\delta = \\sqrt { B } r ^ { - 1 } G ( I - v v ^ { T } ) D \\delta\n$$", + "text_format": "latex", + "bbox": [ + 366, + 184, + 633, + 223 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "where $D = \\mathrm { D i a g } ( \\phi ^ { \\prime } ( { \\mathsf { n } } ( y ) ) ) , r = \\| y \\| , v = y / \\| y \\|$ , and $y \\sim { \\mathcal { N } } ( 0 , G \\Sigma ^ { * } G )$ . Because $G \\Sigma ^ { * } G =$ $\\lambda ^ { * } \\hat { \\pmb q } ^ { \\otimes 2 }$ by $\\cdot$ , $\\boldsymbol { \\mathsf { n } } ( y ) = \\sqrt { B } \\boldsymbol { v } = \\pm \\sqrt { B } \\boldsymbol { \\hat { q } }$ with equal probability and $r \\sim | \\mathcal { N } ( 0 , \\lambda ^ { * } ) |$ (the absolute value of a Gaussian). Furthermore, $G ( \\bar { I } - \\hat { q } ^ { \\otimes 2 } ) ^ { \\bullet } = \\acute { G } - \\hat { q } ^ { \\otimes 2 }$ because $G \\hat { \\pmb q } = \\hat { \\pmb q }$ . Thus ", + "bbox": [ + 174, + 228, + 825, + 275 + ], + "page_idx": 93 + }, + { + "type": "equation", + "img_path": "images/357206f23568ce53c7fb2b4851490fbf05dfdf18708ea4227eca286aa9c88141.jpg", + "text": "$$\n\\frac { \\mathrm { d } \\mathcal { B } _ { \\phi } } { \\mathrm { d } \\Sigma } \\bigg | _ { \\Sigma = \\Sigma ^ { * } } ^ { \\dagger } \\delta = \\sqrt { B } | r | ^ { - 1 } ( G - \\hat { q } ^ { \\otimes 2 } ) \\big ( \\phi ^ { \\prime } ( \\sqrt { B } \\mathrm { s g n } ( r ) \\hat { q } ) \\odot \\delta ) , r \\sim \\mathcal { N } ( 0 , \\lambda ^ { * } )\n$$", + "text_format": "latex", + "bbox": [ + 253, + 282, + 746, + 321 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Now notice the following ", + "bbox": [ + 173, + 334, + 344, + 349 + ], + "page_idx": 93 + }, + { + "type": "equation", + "img_path": "images/9dc2777fefc5ce02cdb5e11dd95d20955031a08a9a0f9b9a486138e9221155a0.jpg", + "text": "$$\nG _ { B } - \\hat { q } ^ { \\otimes 2 } = \\left( \\begin{array} { c c } { { 0 } } & { { 0 } } \\\\ { { 0 } } & { { G _ { B - 1 } } } \\end{array} \\right) = \\left( \\begin{array} { c c c c c } { { 0 } } & { { 0 } } & { { 0 } } & { { \\cdots } } & { { 0 } } \\\\ { { 0 } } & { { 1 - \\frac { 1 } { B - 1 } } } & { { \\frac { - 1 } { B - 1 } } } & { { \\cdots } } & { { \\frac { - 1 } { B - 1 } } } \\\\ { { 0 } } & { { \\frac { - 1 } { B - 1 } } } & { { 1 - \\frac { 1 } { B - 1 } } } & { { \\cdots } } & { { \\frac { - 1 } { B - 1 } } } \\\\ { { 0 } } & { { \\frac { - 1 } { B - 1 } } } & { { \\frac { - 1 } { B - 1 } } } & { { \\cdots } } & { { 1 - \\frac { 1 } { B - 1 } } } \\end{array} \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 263, + 352, + 782, + 420 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Proof. Straightforward computation. ", + "bbox": [ + 174, + 435, + 416, + 450 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Thus if $\\delta = ( \\delta _ { 1 } , \\delta _ { 2 } ) ^ { T } \\in \\mathbb { R } ^ { 1 } \\times \\mathbb { R } ^ { B - 1 }$ , then ", + "bbox": [ + 174, + 467, + 452, + 484 + ], + "page_idx": 93 + }, + { + "type": "equation", + "img_path": "images/ca4572ad0f391decfe90bb6ce39b9c2a7e58c2d28ab20d2039e0a93dc751cc0e.jpg", + "text": "$$\n\\frac { \\mathrm { d } \\mathcal { B } _ { \\phi } } { \\mathrm { d } \\Sigma } \\bigg | _ { \\Sigma = \\Sigma ^ { * } } ^ { \\dagger } \\delta = \\sqrt { B } \\phi ^ { \\prime } \\left( \\frac { - \\mathrm { s g n } ( r ) } { \\sqrt { B - 1 } } \\right) | r | ^ { - 1 } ( 0 , G _ { B - 1 } \\delta _ { 2 } ) ^ { T } , r \\sim \\mathcal { N } ( 0 , \\lambda ^ { * } )\n$$", + "text_format": "latex", + "bbox": [ + 264, + 491, + 735, + 530 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "If $\\delta _ { 2 } = 0$ , then this is deterministically 0. Otherwise, this random variable does not have finite mean because of the $| r | ^ { - 1 }$ term, and this is solely due to $\\Sigma ^ { * }$ being rank 1. Thus as long as the gradient $\\delta$ is not constant in $\\delta _ { 2 }$ , then we expect severe gradient explosion; otherwise we expect severe gradient vanishing. ", + "bbox": [ + 173, + 536, + 825, + 593 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Remark K.7. We would like to note that, in pytorch, with either Float Tensors or Double Tensors, the gradient empirically vanishes rapidly with generic gradient $\\delta$ , when $\\phi$ is rapidly increasing and leads to a $\\widehat { \\mathrm { B S B 2 } } _ { B } ^ { 1 }$ fixed point, e.g. $\\phi ( x ) = \\mathrm { r e l u } ( x ) ^ { 3 0 }$ . This is due to numerical issues rather than mathematical issues, where the computation of $( \\dot { G } - \\dot { \\pmb q } ^ { \\otimes 2 } ) \\delta$ in backprop does not kill $\\delta _ { 1 }$ completely. ", + "bbox": [ + 173, + 594, + 825, + 657 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "L WEIGHT GRADIENTS", + "text_level": 1, + "bbox": [ + 174, + 678, + 385, + 694 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "Recall weight gradient and the backpropagation equation Eq. (16) of the main text. ", + "bbox": [ + 173, + 709, + 715, + 726 + ], + "page_idx": 93 + }, + { + "type": "equation", + "img_path": "images/0a6084996a0413f4d5720c941a7f38f113d8ef3094f60b03b01d50162a0161fa.jpg", + "text": "$$\n\\frac { \\partial \\mathcal { L } } { \\partial W _ { \\alpha \\beta } ^ { l } } = \\sum _ { i } \\delta _ { \\alpha i } ^ { l } x _ { \\beta i } ^ { l - 1 } , \\qquad \\delta _ { \\alpha i } ^ { l } = \\sum _ { \\beta j } \\frac { \\partial x _ { \\alpha j } ^ { l } } { \\partial h _ { \\alpha i } ^ { l } } W _ { \\beta \\alpha } ^ { l + 1 } \\delta _ { \\beta j } ^ { l + 1 }\n$$", + "text_format": "latex", + "bbox": [ + 312, + 732, + 684, + 775 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "where we have used $\\boldsymbol { x } _ { \\beta i } ^ { l }$ to denote $B _ { \\phi } ( { \\pmb h } _ { \\beta } ^ { l } ) _ { i }$ , $\\delta _ { i } ^ { l } = \\nabla _ { h _ { i } ^ { l } } \\mathcal { L }$ , and subscript $i$ denotes slice over sample $i$ in the batch, and subscript $\\alpha$ denotes slice over neuron $\\alpha$ in the layer. Then ", + "bbox": [ + 173, + 782, + 825, + 815 + ], + "page_idx": 93 + }, + { + "type": "equation", + "img_path": "images/eb18b8d3184802e94923013ce079a1c90e2f0a316a821fa1c151483d2ea6c0f5.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\mathbb { E } [ ( \\frac { \\partial \\mathcal { L } } { \\partial W _ { \\alpha \\beta } ^ { l } } ) ^ { 2 } ] = \\mathbb { E } [ ( \\sum _ { i } \\delta _ { \\alpha i } ^ { l } x _ { \\beta i } ^ { l - 1 } ) ^ { 2 } ] } } \\\\ & { } & { = \\mathbb { E } [ \\sum _ { i , j } \\delta _ { \\alpha i } ^ { l } x _ { \\beta i } ^ { l - 1 } \\delta _ { \\alpha j } ^ { l } x _ { \\beta j } ^ { l - 1 } ] . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 341, + 821, + 656, + 924 + ], + "page_idx": 93 + }, + { + "type": "text", + "text": "By gradient independence assumption (Appendix B), ", + "bbox": [ + 173, + 103, + 522, + 118 + ], + "page_idx": 94 + }, + { + "type": "equation", + "img_path": "images/e7dd7590117124beaa452636580d898140f202466738b0def601ec4badaf1560.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } \\delta _ { \\alpha i } ^ { l } x _ { \\beta i } ^ { l - 1 } \\delta _ { \\alpha j } ^ { l } x _ { \\beta j } ^ { l - 1 } = \\mathbb { E } [ \\delta _ { \\alpha i } ^ { l } \\delta _ { \\alpha j } ^ { l } ] \\mathbb { E } [ x _ { \\beta i } ^ { l - 1 } x _ { \\beta j } ^ { l - 1 } ] } \\\\ & { \\qquad = \\Pi _ { i j } ^ { l } \\Sigma _ { i j } ^ { l - 1 } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 351, + 125, + 645, + 170 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Thus ", + "bbox": [ + 173, + 172, + 210, + 188 + ], + "page_idx": 94 + }, + { + "type": "equation", + "img_path": "images/94d87d5867207e524d7a3a0c8265fddb5e2587f491bf75ef80946027bc1ed7b9.jpg", + "text": "$$\n\\mathbb { E } \\left[ \\left( \\frac { \\partial \\mathcal { L } } { \\partial W _ { \\alpha \\beta } ^ { l } } \\right) ^ { 2 } \\right] = \\sum _ { i , j } \\Pi _ { i j } ^ { l } \\Sigma _ { i j } ^ { l - 1 } = \\langle \\Pi ^ { l } , \\Sigma ^ { l - 1 } \\rangle .\n$$", + "text_format": "latex", + "bbox": [ + 339, + 190, + 658, + 241 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "If $l$ is large and $\\phi$ is well-behaved, then we expect $\\Sigma ^ { l - 1 } \\approx \\Sigma ^ { * }$ , the unique BSB1 fixed point of Eq. (43). Assuming equality, we then have ", + "bbox": [ + 174, + 247, + 825, + 277 + ], + "page_idx": 94 + }, + { + "type": "equation", + "img_path": "images/87bdf827aadcd2ba8536d1f76709cf0f3231a3b7549fdaeda4ff9476c4ab6fbf.jpg", + "text": "$$\n\\begin{array} { r l r } { { \\mathbb { E } [ ( \\frac { \\partial \\mathcal { L } } { \\partial W _ { \\alpha \\beta } ^ { l } } ) ^ { 2 } ] = \\langle \\Pi ^ { l } , \\Sigma ^ { * } \\rangle = \\langle G ^ { \\otimes 2 } \\{ \\Pi ^ { l } \\} , \\Sigma ^ { * } \\rangle } } \\\\ & { } & { = \\langle \\Pi ^ { l } , G ^ { \\otimes 2 } \\{ \\Sigma ^ { * } \\} \\rangle = \\langle \\Pi ^ { l } , \\mu ^ { * } G \\rangle } \\\\ & { } & { = \\mu ^ { * } \\operatorname { t r } \\Pi ^ { l } . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 331, + 284, + 665, + 376 + ], + "page_idx": 94 + }, + { + "type": "text", + "text": "Thus in general, we expect the magnitude of weight gradients to follow the magnitude of hidden unit gradient $\\delta _ { i } ^ { l }$ . 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\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad 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B + 1 : 2 B } = \\mathbb { E } [ \\mathcal { B } _ { \\phi } ( z _ { 1 : B } ) \\otimes \\mathcal { B } _ { \\phi } ( z _ { B + 1 : 2 B } ) : z \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ) ] } & { } \\\\ & { = \\mathbb { E } [ \\phi \\circ \\boldsymbol { \\mathrm { n } } ( y _ { 1 : B } ) \\otimes \\phi \\circ \\boldsymbol { \\mathrm { n } } ( y _ { B + 1 : 2 B } ) : y \\sim \\mathcal { N } ( 0 , \\widetilde { \\Sigma } ^ { G } ) ] } \\\\ { \\widetilde { \\Sigma } ^ { G } } & : = \\begin{array} { r l } { G ^ { \\oplus 2 } \\widetilde { \\Sigma } G ^ { \\oplus 2 } } & { = \\begin{array} { r l } { \\binom { G } { 0 } \\widetilde { \\Sigma } \\left( \\begin{array} { c c } { G } & { 0 } \\\\ { 0 } & { G } \\end{array} \\right) \\widetilde { \\Sigma } \\left( \\begin{array} { c c } { G } & { 0 } \\\\ { 0 } & { G } \\end{array} \\right) } & { = \\begin{array} { r l } { \\binom { G \\Sigma G } { c 4 1 ^ { T } G } } & { G \\Sigma G \\pmb { \\mathrm { \\Delta } } } \\end{array} } 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ABSTRACT + +Recent unsupervised representation learning techniques show remarkable success on many single image tasks by using instance discrimination: learning to differentiate between two augmented versions of the same image and a large batch of unrelated images. Prior work uses artificial data augmentation techniques such as cropping, and color jitter which can only affect the image in superficial ways and are not aligned with how objects actually change e.g. occlusion, deformation, viewpoint change. We argue that videos offer this natural augmentation for free. Videos can provide entirely new views of objects, show deformation, and even connect semantically similar but visually distinct concepts. We propose Video Noise Contrastive Estimation, a method for using unlabeled video to learn strong, transferable, single image representations. We demonstrate improvements over recent unsupervised single image techniques, as well as over fully supervised ImageNet pretraining, across temporal and non-temporal tasks.1 + +# 1 INTRODUCTION + +The world seen through our eyes is constantly changing. As we move through the world, we see much more than a single static image: objects rotate revealing occluded regions, deform, the surroundings change, and we ourselves move. Our internal visual systems are constantly seeing temporally coherent images. Yet many popular computer vision models learn representations which are limited to inference on single images, lacking temporal context. Representations learned from static images are inherently limited to an understanding of the world as many unrelated static snapshots. + +This is especially true of recent unsupervised learning techniques (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Hénaff et al., 2019; Hjelm et al., 2019; Misra & Maaten, 2020; Tian et al., 2019; Wu et al., 2018), all of which train on a set of highly-curated, well-balanced data: ImageNet (Deng et al., 2009). Scaling up these techniques to larger, less-curated datasets like Instagram1B (Mahajan et al., 2018) has not provided large improvements in performance (He et al., 2020). Only so much can be learned from a single image: no amount of artificial augmentation can show a new view of an object or what might happen next in a scene. This dichotomy can be seen in Figure 1. + +In order to move beyond this limitation, we argue that video supplies significantly more semantically meaningful content than a single image. With video, we can see how the world changes, find connections between images, and more directly observe the underlying scene. Prior work using temporal cues has shown success in learning from unlabeled videos (Misra et al., 2016; Wang et al., 2019; Srivastava et al., 2015), but has not been able to surpass supervised pretraining. On the other hand, single image techniques (Gutmann & Hyvärinen, 2010) have shown improvements over stateof-the-art by using Noise Contrastive Estimation (NCE). In this work, we merge the two concepts with Video Noise Contrastive Estimation (VINCE), a method for using unlabeled videos as a basis for learning visual representations. Instead of predicting whether two feature vectors come from the same underlying image, we task our network with predicting whether two images originate from the same video. Not only does this allow our method to learn how a single object might change, it also enables learning which things might be in a scene together, e.g. cats are more likely to be in videos with dogs than with sharks. Additionally, we generalize the NCE technique to operate on multiple positive pairs from a single source. To facilitate this learning, we construct Random Related Video + +![](images/e2b1139403385d66fe60b510873896d5fada3f69ff9c16783797309800e869e7.jpg) +Figure 1: NCE vs VINCE. The standard contrastive setup learns to separate artificial augmentations of the same image. Our method uses novel views and temporal consistency which single images cannot provide. + +Views (R2V2), a set 960,000 frames from 240,000 uncurated videos. Using our learning technique, we achieve across-the-board improvements over the recent Momentum Contrast method (He et al., 2020) as well as over a network pretrained on supervised ImageNet on diverse tasks such as scene classification, activity recognition, and object tracking. + +# 2 RELATED WORK + +# 2.1 NOISE CONTRASTIVE ESTIMATION (NCE) + +The NCE loss (Gutmann & Hyvärinen, 2010) is at the center of many recent representation learning methods (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Hénaff et al., 2019; Hjelm et al., 2019; Misra & Maaten, 2020; Tian et al., 2019; Wu et al., 2018). Similar to the triplet loss (Chechik et al., 2010), the basic principle behind NCE is to maximize the similarity between an anchor data point and a positive data point while minimizing similarity to all other (negative) points. + +A challenge for using NCE in an unsupervised fashion is devising a way to construct positive pairs. Pairs should be different enough that a network learns a non-trivial representation, but structured enough that the learned representation is useful for downstream tasks. A standard approach used by (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020) is to generate the pairs via artificial data augmentation techniques such as color jitter, cropping, and flipping. Contrastive Multiview Coding (Tian et al., 2019) uses multiple “views” of a single source image such as intensity (L), color (ab), depth, or segmentation, training separate encoders for each view. PIRL (Misra & Maaten, 2020) uses the jigsaw technique (Noroozi & Favaro, 2016) to break the image into non-overlapping regions and learns a shared representation for the full image and the shuffled image patches. Similarly, Contrastive Predictive Coding (CPC) (Hénaff et al., 2019) uses crops of an image as “context” and predicts features for the unseen portions of the image. We provide a more natural data augmentation by using multiple frames from a single video. As a video progresses, the objects in the scene, the background, and the camera itself may move, providing new views. Whereas augmentations on an image are constrained by a single snapshot in time, using different frames from a single video gives entirely new information about the scene. Additionally, rather than restricting our method to only use two frames from a video, we generalize the NCE technique to use many images from a single video, resulting in more computational reuse and a better final representation (Bachman et al. (2019) similarly makes multiple comparisons per pair, but each anchor has only one positive). + +# 2.2 UNSUPERVISED LEARNING USING VIDEO CUES + +In contrast with supervised learning which requires hand-labeling, self-supervised and unsupervised learning acquire their labels for free. These techniques create datasets which are orders of magnitude larger than comparable fully-supervised datasets. Whereas self-supervised learning requires extra setup during data generation (Godard et al., 2017; Pinto et al., 2016; Schmidt et al., 2016), unsupervised learning can use existing data without any specific data generation constraints. Unsupervised single image methods such as auto-encoders (Kramer, 1991), colorization (Zhang et al., 2016), GANs (Radford et al., 2015), jigsaw (Noroozi & Favaro, 2016), and NCE (Wu et al., 2018) rely on properties of the images themselves and can be applied to arbitrary datasets. However these datasets cannot represent temporal information, nor can they show novel object views or occlusions. + +Video data automatically provides temporal cohesion which can be used as additional supervisory signal to learn these phenomena. There is a long history of using videos for low level (Li et al., 2019; Meister et al., 2018; Srivastava et al., 2015) and high-level tasks (Misra et al., 2016; Wang et al., 2019). One of the most common unsupervised setups is using the present to predict the future. The Natural Language Processing community has embraced language modeling as an unsupervised task which has resulted in numerous breakthroughs (Devlin et al., 2019; Mikolov et al., 2013; Radford et al., 2019). However, similar systems applied to unlabeled videos have not revolutionized computer vision. These representations still underperform supervised methods due to several issues. Primarily, neighboring video frames do not change nearly as much as neighboring words in a sentence, so a network which learns the identity function would perform well at next frame prediction. Additionally, words are reused and can thus be tokenized in an effective way whereas images never repeat, especially between two disparate video sources. + +To avoid these issues, many have opted for other methods. Anand et al. (2019) use the NCE loss to discriminate between temporally near frames and temporally far frames of ATARI gameplay but do not compare across games. Han et al. (2019) also use the NCE loss and the CPC technique on a 3D-ResNet to learn spatio-temporal features. Similar to our proposal, Tschannen et al. (2020) and Purushwalkam & Gupta (2020) advocates using videos to learn invariances to deformations, color changes, occlusions, and other difficult scenarios which preserve the semantic relationships of the video subjects but drastically change the appearance. These works provide additional evidence that videos offer a richer visual backing as compared with single image methods. + +Aside from the NCE approach, others have proposed alternative video training tasks. Misra et al. (2016) shuffle the frames of a video and train a network to predict whether they are correctly temporally ordered. Wang Wang et al. (2019) and Vondrick et al. (2018) use cycle consistency and color as a form of tracking from one frame onto another. Earlier work from Wang & Gupta (2015) uses hand-crafted features to track patches of a video and learn similarities between the patches. Others have explored learning representations from multi-modal inputs provided by video such as optical flow or audio Korbar et al. (2018); Owens & Efros (2018); Piergiovanni et al. (2020); Zhao et al. (2018). Our approach is inspired by these works but focuses on learning a semantic representation of the entire scene based on a single frame from a video. Many of these prior works require multiple frames for their representation, and thus cannot be used on single-image downstream tasks. However, if a network can consistently represent visually dissimilar images from the same video with similar vectors, then not only has it learned how to recognize what is in each image, but it can also represent what might happen in the past or future of that scene. + +# 3 METHODS + +In order to learn a semantically meaningful representation, we exploit the natural augmentations provided by unlabeled videos. In this section, we first outline the dataset generation process. We then describe the learning algorithm used to train our representation. + +# 3.1 DATASET + +Using ImageNet as a basis for representation learning has shown remarkable success both with supervised pretraining as well as unsupervised learning. However, even without labels, the images of ImageNet have been hand selected and are unnaturally balanced. To improve learned representations using existing techniques may require significantly larger datasets (He et al., 2020), but obtaining + +
DatasetNumber of Images (Train)Number of Videos (Train)Number of CategoriesMean Image Size
ImageNet1K (Deng et al.,2009)1.3 M01000(428,406)
YouTube 8M(Abu-El-Haija et al.,2016)03.7M3862
Kinetics 40o (Kay et al.,2017)0220K400
GOT-10k (Huang et al., 2019)1.4M9K563(1600,912)
Random Related Video Views (Ours)0.96M240K-(467,280)
+ +Table 1: Comparison of various image and video datasets. While we have neither the most images nor the most videos, we provide good diversity between videos which is crucial for learning a strong, generic image representation. + +data with similar properties automatically and at scale is not practical. Instead, we turn to unlabeled videos as a source of additional supervision. + +In order to train on a diverse set of realistic video frames, we collect a new dataset which we call Random Related Video Views (R2V2). We use the following fast and automated procedure to generate the images in our dataset. + +1. Use YouTube Search to find videos for a set of queries, and download the top K videos licensed under the Creative Commons. In practice we use the ImageNet 1K classes. +2. Filter out videos with static images. +3. Pick a random point in the video and extract $T$ images with a gap of $G$ frames between each image. In practice $T = 4$ and $G = 1 5 0$ . + +Using this procedure, we are able to construct R2V2 in under a day on a single machine. Because our dataset is constructed automatically, we can easily gather more data (more frames per video, more videos overall). In this work we limit the scale to roughly that of comparable datasets (see Table 1). More details on data collection are provided in Appendix 1. + +# 3.2 NOISE CONTRASTIVE ESTIMATION (NCE) LEARNING + +Given a dataset of diverse video frames, we learn a representation which takes advantage of the structure of the data. We choose the Noise Contrastive Estimation technique (Gutmann & Hyvärinen, 2010) which has been popular in many recent works (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Hénaff et al., 2019; Hjelm et al., 2019; Misra & Maaten, 2020; Tian et al., 2019; Wu et al., 2018), augmented with temporal supervision. + +The standard NCE implementation (used in (Chen et al., 2020a; Bachman et al., 2019; Hjelm et al., 2019)) uses the following procedure. First, a batch of anchor images $A$ is selected. Second, a batch of positive images $P$ is selected, one for each anchor. Positive matches for one example are reused as negatives for the other samples without the need to recompute the features. The NCE loss for a single batch is shown in Equation 1 where $s i m ( X , Y )$ is any similarity metric between the two inputs. This similarity is in a learned feature space rather than directly in image space, and this learned feature encoding is what we use on downstream tasks. The NCE Loss produces a strong representational encoding for inputs by maximizing the similarity of the encoding of positive pairs while minimizing similarity of negative ones, encouraging the encoder to be discriminative over many possible matches, but general in that it must work on any inputs. As in other works, we use the dot product of the feature embeddings of the data points (as seen in Equation 2) as the similarity metric due to its computational efficiency (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Hénaff et al., 2019; Misra $\&$ Maaten, 2020; Wu et al., 2018). The similarity is rescaled by a temperature scalar $\tau$ to create peaked softmax distributions. + +$$ +\begin{array} { c } { { \displaystyle \mathcal { L } _ { \mathrm { N C E } } = - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \log \frac { e ^ { s i m ( A _ { i } , P _ { i } ) } } { \sum _ { j = 1 } ^ { n } e ^ { s i m ( A _ { i } , P _ { j } ) } } } } \\ { { \displaystyle s i m ( X , Y ) = \tau * \frac { f ( X ) } { \| f ( X ) \| } \cdot \frac { g ( Y ) } { \| g ( Y ) \| } } } \end{array} +$$ + +# 3.3 MULTI-FRAME NCE + +All recent works perform some sort of transformation on a single image to create Anchor-Positive pairs for the NCE loss (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Hénaff et al., 2019; + +![](images/7821717a4ff18a11ce37a18819d0e222869c17f58d61e0f16e4f7dd5362c2233.jpg) +Figure 2: Left: Standard NCE using “Same Frame” where all positive pairs come from the same image. Middle: NCE using “Multi-Frame” where positive pairs come from the same video. Right: Multi-Frame Multi-Pair NCE which uses more than one positive pair per video. The gray boxes indicate the true match pairs. The Memory Bank adds more negatives for each anchor. + +Misra & Maaten, 2020; Wu et al., 2018). We refer to this as “Same Frame.” We differ from these works by using multiple images from a single video to form our pairs. This allows our network to see truly different views, deformations, similar objects, and larger scene changes. Additionally, this encodes temporal consistency as the semantic contents of a video are unlikely to change suddenly. For example in a video of two cats playing, the camera may focus on one cat, or may even pan to a previously unseen dog, but it is unlikely to pan to a shark. Note that in practice, we select frames with replacement, making our potential pairs a strict superset of those in prior works. + +# 3.4 MEMORY BANKS AND MOMENTUM CONTRAST + +NCE-based methods benefit greatly from large pools of negatives because this increases the likelihood of finding at least one hard negative for each positive example. In some works (Misra & Maaten, 2020; Wu et al., 2018), negatives are sampled from a large memory bank which was filled with earlier outputs of the network. The NCE loss can be modified to use negatives from prior batches as shown in equation 3 for a memory bank of negatives $N _ { 1 \dots m }$ . + +$$ +{ \mathcal { L } } _ { \mathrm { N C E } } = - { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \log { \frac { e ^ { s i m ( A _ { i } , P _ { i } ) } } { e ^ { s i m ( A _ { i } , P _ { i } ) } + \sum _ { j = 1 } ^ { m } e ^ { s i m ( A _ { i } , N _ { j } ) } } } +$$ + +As the network trains, its output distribution will shift. A potential issue when using a memory bank is the network learns a simple classifier between the current distribution and an old one. Momentum Contrast (MoCo) (He et al., 2020) alleviates this issue by using a quickly updating primary network $f$ in equation 2) and a slowly updating secondary network $( g )$ . $f$ is updated based on the NCE loss in equation 3 and $g$ is updated using a momentum rule $g \alpha g + ( 1 - \alpha ) f$ . The memory bank is filled with previous outputs from the slowly changing network $g$ , reducing the likelihood that $f$ will be able to learn a simple recent batch/old batch classifier. For more details, see (He et al., 2020). + +# 3.5 MULTI-PAIR NCE + +By using a memory bank, we increase the number of negatives without a large computational cost. Yet standard NCE only uses $n$ positive pairs per batch of size $n$ . We can further increase the number of positives per batch (while holding batch size constant) by selecting $v$ videos and $k$ samples from each video where $\textstyle k = { \frac { n } { v } }$ . By computing the pairwise similarity between each pair, we reuse each positive sample $k$ times, resulting in $\textstyle k ^ { 2 } v = { \frac { n ^ { 2 } } { v } }$ positives per batch. Using a simple block-diagonal mask as shown in Figure 2 we can efficiently compute the similarities and NCE loss both between elements of the batch and across a memory bank, achieving a large number of positive comparisons per batch while retaining a large negative size. In practice, we notice no meaningful computational cost to this approach. We refer to this full method using Multi-Pair on video data and the MultiFrame learning procedure as Video Noise Contrastive Estimation (VINCE). See Appendix 2 for detailed pseudocode. + +# 4 EXPERIMENTS + +We evaluate our method (VINCE) on both single image and temporal tasks by freezing our learned representation and adding a small network (in most cases a single linear layer) for adaptation to new end-tasks. Our learned representation transfers well to a variety of visual tasks, especially tasks which require temporal reasoning. To show this, we compare with multiple strong baselines: + +• MoCo-IN: Network pretrained on ImageNet (Deng et al., 2009) using the MoCo algorithm. • MoCo-R2V2: Network pretrained on R2V2 using the MoCo algorithm. This uses exactly the same data as VINCE but the Same Frame technique described in Sec. 3.3. We also prevent multiple images from the same video being in the Memory Bank at the same time. • Sup-IN: Network pretrained on fully supervised ImageNet. + +To validate the benefits of learning from videos, we additionally compare against an unsupervised image dataset. Since ImageNet itself required time, effort, and money to create, we construct a new static image dataset analogous to our video dataset. Specifically, we search Google Images for the ImageNet synsets and download the top K results for each category. We refer to this as MoCoG). Although other unsupervised video-based methods exist, they have generally underperformed pretrained supervised ImageNet weights or are only applicable to video tasks. + +# 4.1 TARGET TASKS + +We compare each method on several diverse end-tasks using both ResNet18 (He et al., 2016) and ResNet50 backbones. More training implementation details can be found in Appendix 3. Results for these tasks are shown in Table 2. We train all end-task models using Adam (Kingma & Ba, 2014) and a shared learning rate schedule per task. For each dataset, we use standard data augmentation approaches (crop, flip, color jitter except for tracking). One overall trend to note is the relative gain over MoCo-R2V2. If the single-frame algorithm performed as well as our multi-frame method on temporal tasks, it would indicate minimal temporal understanding. However, the relative gain of VINCE over MoCo-R2V2 on Kinetics $( 1 3 . 8 5 \% )$ and tracking ( $1 3 . 3 3 \%$ and $1 5 . 3 8 \%$ ) are higher than those on single-frame tasks, showing that our method can incorporate temporal cues. + +ImageNet: For this task, we use our frozen learned representations, adding a single linear layer after the global average pool. Although none of the methods match the fully supervised performance of ResNet on ImageNet, they do achieve reasonable performance given only single linear layer. It is unsurprising that MoCo pretrained on ImageNet images (MoCo-IN) outperforms our method (0.447 vs 0.400) due to the domain shift between pretrain and end-task. However MoCo pretrained on R2V2 (MoCo-R2V2) suffers nearly a $2 \times$ drop in accuracy $( 8 . 9 \% )$ compared to our method $( 4 . 7 \% )$ , indicating that pretraining on multi-frame matching provides a clear benefit over single frame pairs. Our method, which has never seen an image from ImageNet before, still learns a representation which generalizes well to this new type of data. Even drawing images from a similar class distribution (MoCo-G) does not outperform our method. + +SUN Scenes: SUN Scenes is a classification dataset in which each image is categorized into one of 397 possible scene types such as airplane cabin, bedroom, and coffee shop. Again we train a single linear layer on top of each pretrained network. This data is quite similar to ImageNet in that each image is well-curated and contains single, unambiguous subjects. As such, the ImageNet fully supervised baseline transfers quite well to SUN Scenes. However VINCE outperforms Sup-IN by a small margin. Again we note a large improvement of VINCE over MoCo-R2V2 (0.495 vs. 0.450). This shows that our method learns to recognize not just the main subject of an image but also the surrounding scene, which requires a richer understanding of the world. + +Kinetics 400: This dataset consists of 10 second clips from YouTube videos and action labels for each segment. We first download each video and subsample each clip to one frame per second (10 frames per clip). We train a single layer LSTM (Hochreiter & Schmidhuber, 1997) followed by a single linear layer to predict the action category for each segment. Kinetics acts as a crucial test to evaluate whether our model learns temporal cues. VINCE greatly outperform all other methods, whereas traditional baselines such as fine-tuning supervised ImageNet do not adapt well at all. This shows that contrary to popular belief, representations pretrained on ImageNet many not be a good fit for other visual domains, especially on temporal tasks. + +Table 2: Comparison of representation performance across a variety of end tasks. We show improvements over MoCo trained on the same data on all tasks, and outperform MoCo trained on ImageNet as well as supervised pretraining on ImageNet on all tasks but ImageNet itself (and tracking for ResNet50). Each representation uses the same ResNet convolutional backbone, sharing weights across all tasks. Linear (for Kinetics LSTM Linear) classifiers are the only learned weights for each end task. + +
Image Classification ImageNet(Deng et al.,2009)Scene Classification SUN Scenes(Xiao et al.,2010)Action Recognition Kinetics 400(Kay et al.,2017)Tracking OTB 2015(Wu et al.,2015)
Trained Layer(s) MetricLinear Accuracy (Top 1)Linear Accuracy (Top 1)LSTM→Linear Accuracy (Top 1)1x1 Conv
Sup-INPrecisionSuccess
MoCo-IN0.6960.4910.2070.5570.396
0.4470.4870.3360.5830.429
BeerMoCo-G0.3930.4440.3130.5510.413
MoCo-R2V20.3580.450 0.4950.3180.5550.403
VINCE (Ours)0.4000.3620.6290.465
Relative Gain over MoCo-R2V211.91%9.93%13.85%13.33%15.38%
PssSDSup-IN0.7620.5930.3050.4580.320
MoCo-V2-IN (our impl.)0.6520.6080.4590.3000.260
MoCo-R2V20.5360.5810.4560.3860.299
VINCE (Ours)0.5440.6110.4910.4020.300
Relative Gain over MoCo-R2V21.36%5.29%7.72%4.15%0.33%
+ +Object Tracking Benchmark (OTB) 2015: In this dataset, given an initial bounding box around an arbitrary object in the first image, a model must locate the object in the following frames. Difficult tracking datasets like OTB 2015 implicitly require robustness to occlusion and appearance change of the target object, so improved performance would indicate a representation’s relative robustness to these challenges. We use the SiamFC (Bertinetto et al., 2016) tracking algorithm on top of our learned representation. SiamFC localizes an object of interest by convolving deep features from the initial bounding box with the current image, similar to template matching (Briechle & Hanebeck, 2001) but in deep feature space. For a more complete explanation, see (Bertinetto et al., 2016). OTB is evaluated using two metrics – precision and success. Precision measures the percentage of frames where the (normalized) center error is less than a certain threshold, averaging over many thresholds. Similarly, success measures the percentage of frames where the Intersection Over Union is more than a range of thresholds. + +A representation which works well for SiamFC would have the property that the cross correlation of two images of the same object is high, but the cross correlation of two different objects, or a poorly cropped image of the same object, is low. Pretraining our representations on multiple frames from the same video coincides well with the first objective, however since we use cropped data augmentations, the representations tend to be somewhat invariant to poorly-cropped candidates. Still, the models perform quite well across a variety of difficult tracking instances. Our ResNet18 model transfers significantly better than all other methods indicating a clear benefit to using temporal cues during pretraining. We find, somewhat unexpectedly, that the ResNet18 network fares better than the ResNet50. A likely explanation for this is that the original SiamFC method uses AlexNet (Krizhevsky et al., 2012) with no padding in a fully-convolutional manner. When using ResNet, padding must be applied to keep the outputs the same dimensionality as the inputs. Thus, at training time the network may latch onto zero-padding cues which will not be applicable at test time. This becomes more of an issue the larger the receptive field which is why ResNet50 struggles but ResNet18 is somewhat less affected. + +# 4.2 METHOD ABLATION + +We validate the effectiveness of Multi-Frame (Sec. 3.3) and Multi-Pair (Sec. 3.5) learning by ablating the number of images from each video used in a batch of comparisons shown in Table 3 (due to computational constraints, we only perform ablations on ResNet18). The first row is equivalent to the procedure done in MoCo (He et al., 2020) i.e. the anchor and positive pairs are two data augmentations of the same image. The second row uses the MoCo procedure as well, however the anchors and positives may be from different images from the same video. The third row uses our Multi-Pair NCE method taking 4 positives and 4 anchors from each video, resulting in 16 positive pairs. A pictorial representation can be seen in Figure 2. Note that when selecting images for row 2 and 3, we use sampling with replacement, making our method a strict super-set of MoCo. + +
Images Per VideoTest Task
ImageNetSUN SceneKinetics 400OTB 2015 PrecisionOTB 2015 Success
1: Same Frame0.3580.4500.3180.5550.403
2: Multi-Frame0.3810.4780.3610.6220.464
8:Multi-Frame Multi-Pair0.4000.4950.3620.6290.465
+ +Table 3: Method ablation for VINCE. We compare using one source image with two augmentations (the standard approach), two different images, or a set of different images. Using Multi-Frame results in a large boost across the board. Multi-Frame Multi-Pair further increases the power of the representation. Note that all methods use the entire dataset, but only Multi-Frame methods use multiple images from a video within one batch. + +Table 4: Pretraining data ablation for VINCE. Each method uses the same training setup but different training data. Since R2V2 uses ImageNet search queries, it outperforms the others on ImageNet. Similarly, pretraining on Kinetics increases performance on the Kinetics task. + +
Pretraining DataTest Task
ImageNet SUN SceneKinetics 400OTB 2015 PrecisionOTB 2015 Success
R2V2 IN-Queries0.4000.4950.3620.6290.465
R2V2 YT8MURLs0.3670.4780.3430.6670.492
Kinetics 400 URLs0.3680.4940.3900.6120.456
+ +We observe across-the-board improvements from both modifications to the MoCo approach. The majority of the improvement comes from using two non-identical frames for matching, but we still gain an additional improvement from using Multi-Pair NCE. Our intuition is that using the MultiPair NCE creates gradients that pull each feature towards a global video representation whereas the standard NCE remains more instance-based, only moving a representation in one direction at a time. Thus, we would expect the Multi-Pair NCE features to be more holistically semantic whereas the standard NCE may retain more uniquely identifying features. In fact, we observe a larger performance gap on the more semantic ImageNet and SUN Scene tasks. In contrast, because the Kinetics model uses an LSTM to reason over all input images at once, instance-level features are equally useful as global video features for overall accuracy. + +# 4.3 PRETRAINING DATA ABLATION + +In Table 4 we explore the effect of different pretraining datasets on end-task performance. For each experiment, we use VINCE but use video data from three different sources: our method of searching ImageNet synset queries, using the URLs from YouTube 8M (Abu-El-Haija et al., 2016), and using the URLs from Kinetics 400 (Kay et al., 2017). Again, we only test ResNet18. Our YouTube8M(YT8M) pretraining data uses the same filtering procedure as R2V2 and contains 5.8 million images from 1.4 million videos. As noted in Table 2 MoCo-IN results, using the same dataset for pretraining as the end-task results in a boost in performance on that specific task but does not indicate that the representation will be better on all tasks. We see this trend is true again when pretraining on Kinetics data. Similarly, since R2V2 uses ImageNet synset for search queries, pretraining on it performs better on ImageNet than the other less-aligned datasets. + +In general, this would indicate that given a large enough set of diverse videos, pretraining directly on the unlabeled source data would result in the best performing representation on that data. If this is not possible, then pretraining on a large external source of data may still result in a useful representation. It also indicates that the VINCE method works well on a variety of different pretraining datasets. The increased performance on tracking when using YT8M data could be explained by it simply having access to a larger number of video sources and frames. For generic object tracking, class diversity may be less important than number of samples because the class identity is ignored. + +# 5 CONCLUSIONS + +In this work we introduced Video Noise Contrastive Estimation, a process for using unlabeled videos to learn an unsupervised image representation. By training on multiple images from the same video, we learn from natural changes such as deformation and viewpoint rather than artificial 2D augmentations. To learn from a large variety of diverse videos, we collect Random Related Video Views in a completely automated fashion. We show across-the-board improvements over the recently proposed MoCo (He et al., 2020) technique on a wide variety of tasks, and we believe Video Noise Contrastive Estimation will extend to other unsupervised methods such as SimCLR (Chen et al., 2020a) and PIRL (Misra & Maaten, 2020) as well as other end-tasks. 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In Proceedings of the European Conference on Computer Vision (ECCV), September 2018. + +![](images/8e99c9e1888b1f509194ec70749e16697122f5048d817e2552c11a6d0f660fe6.jpg) +Figure A1: Sample from Random Related Video Views (train set). + +# APPENDIX 1 DATASET COMPARISON + +Random Related Video Views provides nearly 250,000 video links as well as multiple images per video across a broad variety of categories. We show more samples from R2V2 in Figure A1. Videos each have four images 150 frames apart. Each separate video (outlined in blue) lists its corresponding YouTube link. Using ImageNet synsets for search queries provides reasonable visual diversity, but could be substituted with another set of queries. While we acknowledge that using YouTube’s Search feature is not truly random, this procedure resulted in significantly more diverse samples than using existing datasets like YouTube8M (Abu-El-Haija et al., 2016) which is heavily unbalanced with unnatural videos.2 We do no additional data cleaning to ensure that the videos or extracted images actually contain the search term (many do not), nor do we search for “high interest” video segments as in Misra et al. (2016). We also discard the search term itself as a form of supervision. We find that a gap of 5 seconds between each saved image typically results in visually distinct but semantically related images. Too much shorter results in images which are less individually distinct, and too much longer may result in large and unpredictable changes. A sample from each dataset can be seen in the supplemental material. + +![](images/2a73ace7d06bf23a2fb379edcf93ff0973f48b41f0087f5b714f1f928e1f1b5e.jpg) +Figure A2: Random sampling of pairs of images from videos in each dataset. In GOT-10k, sometimes different video clips are segments from the same original video as seen in the first and second sample. Images are square cropped for visualization purposes only. + +We show samples from various existing video datasets in Figure A2. Datasets such as YouTube8M (Abu-El-Haija et al., 2016) and Kinetics400 (Kay et al., 2017). Kinetics contains only videos of humans performing actions. Alternative datasets such as GOT-10k (Huang et al., 2019) provide a comparatively small number of videos but with dense annotations (in GOT-10k’s case for object tracking). + +# APPENDIX 2 ALGORITHM PSEUDOCODE + +The pseudocode below efficiently computes the Multi-Pair NCE for a batch of inputs. Shapes of each input are provided for clarity where: + +• $v =$ The number of videos in a batch. +• $k =$ The number of frames per video. +• $n = ( v * k ) =$ The total number of images in the batch. +• $m =$ The number of elements in the Memory Bank. +• $d =$ The dimensionality of the feature space. + +Algorithm 1 Python-style pseudo code for Multi-Pair NCE. + +
def multi_pair_nce([v，k，d] output of f encoder
f_output,# #
g_output,#output of g encoder
moco_mem, mask,[m，d]
temperature):# #(n + m)] block diagonal bool
f_output = f_output.reshape(v * k，d)[n，d]
g_output = g_output.reshape(v * k，d)#[n，d]
compare = concatenate((g_output，moco_mem)，axis=0)#[(n + m)，d]
similarities = matmul(f_output，compare.T)(n + m)]
similarities /= temperature
pos_similarities = similarities[mask]#[n，k]
neg similarities = similarities[!mask] exp_pos_sim = exp(pos_similarities)#(n + m -k)]
normalizing_constant = broadcast([n， k]
reduce_sum(exp(neg_similarities)，axis=1),[n，k]
shape(numerator))))
score = exp_pos_sim / (exp_pos_sim + normalizing_constant)
loss = -mean(log(score)) return loss
+ +In practice we use the Log-Sum-Exp trick for numerical stability but omit here for clarity. broadcast repeats the input until it matches the provided dimensions. !mask flips the booleans of each point in the mask. + +# APPENDIX 3 VINCE IMPLEMENTATION DETAILS + +For training our ResNet18 (He et al., 2016) representations we use the network up to the global average pooling followed by a fully connected layer $( 5 1 2 \mathrm { x } 5 1 2 )$ , Leaky-ReLU and a final embedding layer $( 5 1 2 \mathrm { ~ x ~ } 6 4 )$ ). The features are L2-normalized and multiplied by $\begin{array} { r } { \dot { \tau } = \frac { 1 } { 0 . 0 7 } } \end{array}$ 10.07 as in MoCo (He et al., 2020). We use 8 GPUs, a training batch size of 256 and a MoCo Memory Bank of size 65536 and a $g$ -network momentum of 0.999. For multi-GPU training we employ the Shuffle-BN technique shuffling both the anchors and the positives to reduce the correlation between batches filled with multiple images from the same video. We use SGD with a learning rate of 0.03, momentum $\scriptstyle 1 = 0 . 9$ , and weight decay $\scriptstyle = 0 . 0 0 0 1$ . All of these hyperparameters are shared with MoCo (He et al., 2020). All methods are trained for approximately $4 5 0 \mathrm { k }$ iterations. When selecting frames from a video, we pick with replacement. In addition to the natural augmentation, we perform standard data augmentation (color jitter, cropping, flipping) on the inputs. This prevents the network from relying too heavily on shared video statistics like mean frame color. After cropping, all images are resized to $2 2 4 \times 2 2 4$ . It is worth noting that both VINCE and our data (R2V2) can be used with other network architectures or learning algorithms such as AMDIM (Bachman et al., 2019), PIRL (Misra & Maaten, 2020), or SimCLR (Chen et al., 2020a). We choose ResNet18 and MoCo due to implementation simplicity and relatively low computational constraints. + +For ResNet50 training, we closely match the hyperparameters in MoCo v2 (Chen et al., 2020b). Specifically, we use the blur augmentation and stronger color augmentation and a cosine annealing learning rate for 286,000 iterations (200 epochs of ImageNet or equivalent on R2V2 regardless of the number of unique positives per batch). The batch size we use is 896 with an initial learning rate of 0.105 and an embedding dimensionality of 128. The temperature parameter used was $\tau \stackrel { } { = } \frac { 1 } { 0 . 2 }$ . We did not perform any hyperparameter searches for VINCE or R2V2, so these results may be suboptimal. + +![](images/98dfdc7828efc0513ffa867293128fcc9da3691d331c460f71eb37125f0855a1.jpg) +Figure A3: Nearest neighbor results for a sampling of query images from R2V2 and ImageNet using various models. VINCE shows a clear understanding of each image and finds highly relevant neighbors. + +# APPENDIX 4 QUALITATIVE RESULTS + +We additionally provide two qualitative analyses to better understand the success and failure cases of VINCE: Nearest Neighbors, and t-SNE. + +# APPENDIX 4.1 NEAREST NEIGHBORS: + +We query ImageNet Val and a set of test videos for nearest neighbor matches, taking at most one neighbor per video. We visualize the top 5 neighbors for VINCE, MoCo-R2V2, and MoCo-IN in Figure A3. We observe that VINCE seems to understand the semantics of an image more than MoCo-R2V2 and MoCo-IN. For instance, although MoCo-R2V2 and MoCo-IN find other control panels and buttons in query 1, they do not make the scene-level connection to car interiors as well as VINCE does. Query 2 shows an interesting quirk case of our method. Rather than matching the semantics of the image, VINCE relies on the news logo as a differentiating feature due to its discriminative nature. Each image in VINCE’s query 2 results is from a separate video, but from the same news source. For the ImageNet queries, despite never seeing ImageNet inputs during pretraining, VINCE is able to find good matches as well as MoCo-IN which was trained using only ImageNet inputs. + +![](images/bc55ce9cb816226d8d030d252ea576e9d6515d16c77c31bdda33ee70452a958a.jpg) +Figure A4: t-SNE embedding of images from R2V2 test set. + +# APPENDIX 4.2 T-SNE: + +Using a set of held-out video frames, we project the 64-D embedding space from VINCE to 2D using t-SNE (Maaten & Hinton, 2008) and visualize the formed clusters in Figure A4. Not only does this assist in verifying the quality of the embedding, it also serves as a visual method for evaluating the diversity of the dataset itself. The largest of the clusters seems to be the face cluster. YouTube is full of videos of people looking and talking directly to a camera, and our random subset reflects this pattern. Other interesting, yet unexpected clusters emerge as well such as cats (YouTube loves cats), hands (demo videos), and food (cooking videos). + +# APPENDIX 5 PRECISION AND SUCCESS PLOTS FOR OTB 2015 + +We provide full breakdowns of the Precision and Success of each method on OTB 2015 (Wu et al., 2015). The values in the legend correspond to the (mean) area under the curve for each method. + +![](images/9d1a5e59c5fd92d5b0acb55809eff39331bf6dce134be0f08d1cfe2660f6b429.jpg) +Figure A5: Precision (a) and Success (b) plots for OTB 2015 for various backbones. \ No newline at end of file diff --git a/parse/train/iktA2PtTRsK/iktA2PtTRsK_content_list.json b/parse/train/iktA2PtTRsK/iktA2PtTRsK_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..a4ce445088f3e4bb0aa787a6b3a9a7eafbae4237 --- /dev/null +++ b/parse/train/iktA2PtTRsK/iktA2PtTRsK_content_list.json @@ -0,0 +1,1653 @@ +[ + { + "type": "text", + "text": "WATCHING THE WORLD GO BY: REPRESENTATION LEARNING FROM UNLABELED VIDEOS ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent unsupervised representation learning techniques show remarkable success on many single image tasks by using instance discrimination: learning to differentiate between two augmented versions of the same image and a large batch of unrelated images. Prior work uses artificial data augmentation techniques such as cropping, and color jitter which can only affect the image in superficial ways and are not aligned with how objects actually change e.g. occlusion, deformation, viewpoint change. We argue that videos offer this natural augmentation for free. Videos can provide entirely new views of objects, show deformation, and even connect semantically similar but visually distinct concepts. We propose Video Noise Contrastive Estimation, a method for using unlabeled video to learn strong, transferable, single image representations. We demonstrate improvements over recent unsupervised single image techniques, as well as over fully supervised ImageNet pretraining, across temporal and non-temporal tasks.1 ", + "bbox": [ + 233, + 267, + 764, + 446 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 178, + 476, + 336, + 492 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The world seen through our eyes is constantly changing. As we move through the world, we see much more than a single static image: objects rotate revealing occluded regions, deform, the surroundings change, and we ourselves move. Our internal visual systems are constantly seeing temporally coherent images. Yet many popular computer vision models learn representations which are limited to inference on single images, lacking temporal context. Representations learned from static images are inherently limited to an understanding of the world as many unrelated static snapshots. ", + "bbox": [ + 174, + 508, + 823, + 592 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This is especially true of recent unsupervised learning techniques (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Hénaff et al., 2019; Hjelm et al., 2019; Misra & Maaten, 2020; Tian et al., 2019; Wu et al., 2018), all of which train on a set of highly-curated, well-balanced data: ImageNet (Deng et al., 2009). Scaling up these techniques to larger, less-curated datasets like Instagram1B (Mahajan et al., 2018) has not provided large improvements in performance (He et al., 2020). Only so much can be learned from a single image: no amount of artificial augmentation can show a new view of an object or what might happen next in a scene. This dichotomy can be seen in Figure 1. ", + "bbox": [ + 174, + 598, + 825, + 696 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In order to move beyond this limitation, we argue that video supplies significantly more semantically meaningful content than a single image. With video, we can see how the world changes, find connections between images, and more directly observe the underlying scene. Prior work using temporal cues has shown success in learning from unlabeled videos (Misra et al., 2016; Wang et al., 2019; Srivastava et al., 2015), but has not been able to surpass supervised pretraining. On the other hand, single image techniques (Gutmann & Hyvärinen, 2010) have shown improvements over stateof-the-art by using Noise Contrastive Estimation (NCE). In this work, we merge the two concepts with Video Noise Contrastive Estimation (VINCE), a method for using unlabeled videos as a basis for learning visual representations. Instead of predicting whether two feature vectors come from the same underlying image, we task our network with predicting whether two images originate from the same video. Not only does this allow our method to learn how a single object might change, it also enables learning which things might be in a scene together, e.g. cats are more likely to be in videos with dogs than with sharks. Additionally, we generalize the NCE technique to operate on multiple positive pairs from a single source. To facilitate this learning, we construct Random Related Video ", + "bbox": [ + 174, + 704, + 825, + 897 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/e2b1139403385d66fe60b510873896d5fada3f69ff9c16783797309800e869e7.jpg", + "image_caption": [ + "Figure 1: NCE vs VINCE. The standard contrastive setup learns to separate artificial augmentations of the same image. Our method uses novel views and temporal consistency which single images cannot provide. " + ], + "image_footnote": [], + "bbox": [ + 243, + 103, + 756, + 392 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Views (R2V2), a set 960,000 frames from 240,000 uncurated videos. Using our learning technique, we achieve across-the-board improvements over the recent Momentum Contrast method (He et al., 2020) as well as over a network pretrained on supervised ImageNet on diverse tasks such as scene classification, activity recognition, and object tracking. ", + "bbox": [ + 174, + 458, + 825, + 515 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 537, + 343, + 554 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 NOISE CONTRASTIVE ESTIMATION (NCE) ", + "text_level": 1, + "bbox": [ + 174, + 570, + 509, + 584 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The NCE loss (Gutmann & Hyvärinen, 2010) is at the center of many recent representation learning methods (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Hénaff et al., 2019; Hjelm et al., 2019; Misra & Maaten, 2020; Tian et al., 2019; Wu et al., 2018). Similar to the triplet loss (Chechik et al., 2010), the basic principle behind NCE is to maximize the similarity between an anchor data point and a positive data point while minimizing similarity to all other (negative) points. ", + "bbox": [ + 174, + 597, + 825, + 667 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A challenge for using NCE in an unsupervised fashion is devising a way to construct positive pairs. Pairs should be different enough that a network learns a non-trivial representation, but structured enough that the learned representation is useful for downstream tasks. A standard approach used by (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020) is to generate the pairs via artificial data augmentation techniques such as color jitter, cropping, and flipping. Contrastive Multiview Coding (Tian et al., 2019) uses multiple “views” of a single source image such as intensity (L), color (ab), depth, or segmentation, training separate encoders for each view. PIRL (Misra & Maaten, 2020) uses the jigsaw technique (Noroozi & Favaro, 2016) to break the image into non-overlapping regions and learns a shared representation for the full image and the shuffled image patches. Similarly, Contrastive Predictive Coding (CPC) (Hénaff et al., 2019) uses crops of an image as “context” and predicts features for the unseen portions of the image. We provide a more natural data augmentation by using multiple frames from a single video. As a video progresses, the objects in the scene, the background, and the camera itself may move, providing new views. Whereas augmentations on an image are constrained by a single snapshot in time, using different frames from a single video gives entirely new information about the scene. Additionally, rather than restricting our method to only use two frames from a video, we generalize the NCE technique to use many images from a single video, resulting in more computational reuse and a better final representation (Bachman et al. (2019) similarly makes multiple comparisons per pair, but each anchor has only one positive). ", + "bbox": [ + 173, + 674, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.2 UNSUPERVISED LEARNING USING VIDEO CUES ", + "text_level": 1, + "bbox": [ + 174, + 103, + 547, + 117 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In contrast with supervised learning which requires hand-labeling, self-supervised and unsupervised learning acquire their labels for free. These techniques create datasets which are orders of magnitude larger than comparable fully-supervised datasets. Whereas self-supervised learning requires extra setup during data generation (Godard et al., 2017; Pinto et al., 2016; Schmidt et al., 2016), unsupervised learning can use existing data without any specific data generation constraints. Unsupervised single image methods such as auto-encoders (Kramer, 1991), colorization (Zhang et al., 2016), GANs (Radford et al., 2015), jigsaw (Noroozi & Favaro, 2016), and NCE (Wu et al., 2018) rely on properties of the images themselves and can be applied to arbitrary datasets. However these datasets cannot represent temporal information, nor can they show novel object views or occlusions. ", + "bbox": [ + 174, + 128, + 825, + 255 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Video data automatically provides temporal cohesion which can be used as additional supervisory signal to learn these phenomena. There is a long history of using videos for low level (Li et al., 2019; Meister et al., 2018; Srivastava et al., 2015) and high-level tasks (Misra et al., 2016; Wang et al., 2019). One of the most common unsupervised setups is using the present to predict the future. The Natural Language Processing community has embraced language modeling as an unsupervised task which has resulted in numerous breakthroughs (Devlin et al., 2019; Mikolov et al., 2013; Radford et al., 2019). However, similar systems applied to unlabeled videos have not revolutionized computer vision. These representations still underperform supervised methods due to several issues. Primarily, neighboring video frames do not change nearly as much as neighboring words in a sentence, so a network which learns the identity function would perform well at next frame prediction. Additionally, words are reused and can thus be tokenized in an effective way whereas images never repeat, especially between two disparate video sources. ", + "bbox": [ + 174, + 262, + 825, + 428 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To avoid these issues, many have opted for other methods. Anand et al. (2019) use the NCE loss to discriminate between temporally near frames and temporally far frames of ATARI gameplay but do not compare across games. Han et al. (2019) also use the NCE loss and the CPC technique on a 3D-ResNet to learn spatio-temporal features. Similar to our proposal, Tschannen et al. (2020) and Purushwalkam & Gupta (2020) advocates using videos to learn invariances to deformations, color changes, occlusions, and other difficult scenarios which preserve the semantic relationships of the video subjects but drastically change the appearance. These works provide additional evidence that videos offer a richer visual backing as compared with single image methods. ", + "bbox": [ + 174, + 435, + 825, + 547 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Aside from the NCE approach, others have proposed alternative video training tasks. Misra et al. (2016) shuffle the frames of a video and train a network to predict whether they are correctly temporally ordered. Wang Wang et al. (2019) and Vondrick et al. (2018) use cycle consistency and color as a form of tracking from one frame onto another. Earlier work from Wang & Gupta (2015) uses hand-crafted features to track patches of a video and learn similarities between the patches. Others have explored learning representations from multi-modal inputs provided by video such as optical flow or audio Korbar et al. (2018); Owens & Efros (2018); Piergiovanni et al. (2020); Zhao et al. (2018). Our approach is inspired by these works but focuses on learning a semantic representation of the entire scene based on a single frame from a video. Many of these prior works require multiple frames for their representation, and thus cannot be used on single-image downstream tasks. However, if a network can consistently represent visually dissimilar images from the same video with similar vectors, then not only has it learned how to recognize what is in each image, but it can also represent what might happen in the past or future of that scene. ", + "bbox": [ + 174, + 554, + 825, + 733 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 METHODS ", + "text_level": 1, + "bbox": [ + 174, + 752, + 290, + 768 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In order to learn a semantically meaningful representation, we exploit the natural augmentations provided by unlabeled videos. In this section, we first outline the dataset generation process. We then describe the learning algorithm used to train our representation. ", + "bbox": [ + 176, + 784, + 823, + 825 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 DATASET ", + "text_level": 1, + "bbox": [ + 174, + 842, + 279, + 856 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Using ImageNet as a basis for representation learning has shown remarkable success both with supervised pretraining as well as unsupervised learning. However, even without labels, the images of ImageNet have been hand selected and are unnaturally balanced. To improve learned representations using existing techniques may require significantly larger datasets (He et al., 2020), but obtaining ", + "bbox": [ + 176, + 867, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/7e0a30b722fb92c72431436230b21a03e95ef9a441f276993fc6562a3f15a757.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
DatasetNumber of Images (Train)Number of Videos (Train)Number of CategoriesMean Image Size
ImageNet1K (Deng et al.,2009)1.3 M01000(428,406)
YouTube 8M(Abu-El-Haija et al.,2016)03.7M3862
Kinetics 40o (Kay et al.,2017)0220K400
GOT-10k (Huang et al., 2019)1.4M9K563(1600,912)
Random Related Video Views (Ours)0.96M240K-(467,280)
", + "bbox": [ + 176, + 101, + 821, + 167 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Table 1: Comparison of various image and video datasets. While we have neither the most images nor the most videos, we provide good diversity between videos which is crucial for learning a strong, generic image representation. ", + "bbox": [ + 173, + 174, + 825, + 215 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "data with similar properties automatically and at scale is not practical. Instead, we turn to unlabeled videos as a source of additional supervision. ", + "bbox": [ + 176, + 222, + 821, + 251 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In order to train on a diverse set of realistic video frames, we collect a new dataset which we call Random Related Video Views (R2V2). We use the following fast and automated procedure to generate the images in our dataset. ", + "bbox": [ + 176, + 257, + 825, + 299 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1. Use YouTube Search to find videos for a set of queries, and download the top K videos licensed under the Creative Commons. In practice we use the ImageNet 1K classes. \n2. Filter out videos with static images. \n3. Pick a random point in the video and extract $T$ images with a gap of $G$ frames between each image. In practice $T = 4$ and $G = 1 5 0$ . ", + "bbox": [ + 210, + 310, + 823, + 386 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Using this procedure, we are able to construct R2V2 in under a day on a single machine. Because our dataset is constructed automatically, we can easily gather more data (more frames per video, more videos overall). In this work we limit the scale to roughly that of comparable datasets (see Table 1). More details on data collection are provided in Appendix 1. ", + "bbox": [ + 174, + 396, + 825, + 452 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 NOISE CONTRASTIVE ESTIMATION (NCE) LEARNING ", + "text_level": 1, + "bbox": [ + 174, + 468, + 589, + 483 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Given a dataset of diverse video frames, we learn a representation which takes advantage of the structure of the data. We choose the Noise Contrastive Estimation technique (Gutmann & Hyvärinen, 2010) which has been popular in many recent works (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Hénaff et al., 2019; Hjelm et al., 2019; Misra & Maaten, 2020; Tian et al., 2019; Wu et al., 2018), augmented with temporal supervision. ", + "bbox": [ + 174, + 493, + 825, + 564 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The standard NCE implementation (used in (Chen et al., 2020a; Bachman et al., 2019; Hjelm et al., 2019)) uses the following procedure. First, a batch of anchor images $A$ is selected. Second, a batch of positive images $P$ is selected, one for each anchor. Positive matches for one example are reused as negatives for the other samples without the need to recompute the features. The NCE loss for a single batch is shown in Equation 1 where $s i m ( X , Y )$ is any similarity metric between the two inputs. This similarity is in a learned feature space rather than directly in image space, and this learned feature encoding is what we use on downstream tasks. The NCE Loss produces a strong representational encoding for inputs by maximizing the similarity of the encoding of positive pairs while minimizing similarity of negative ones, encouraging the encoder to be discriminative over many possible matches, but general in that it must work on any inputs. As in other works, we use the dot product of the feature embeddings of the data points (as seen in Equation 2) as the similarity metric due to its computational efficiency (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Hénaff et al., 2019; Misra $\\&$ Maaten, 2020; Wu et al., 2018). The similarity is rescaled by a temperature scalar $\\tau$ to create peaked softmax distributions. ", + "bbox": [ + 173, + 570, + 825, + 765 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/528d7a7854229bf656e7ccc4b20f92311882db2b3182a46ee64354c1f5a50817.jpg", + "text": "$$\n\\begin{array} { c } { { \\displaystyle \\mathcal { L } _ { \\mathrm { N C E } } = - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\log \\frac { e ^ { s i m ( A _ { i } , P _ { i } ) } } { \\sum _ { j = 1 } ^ { n } e ^ { s i m ( A _ { i } , P _ { j } ) } } } } \\\\ { { \\displaystyle s i m ( X , Y ) = \\tau * \\frac { f ( X ) } { \\| f ( X ) \\| } \\cdot \\frac { g ( Y ) } { \\| g ( Y ) \\| } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 349, + 779, + 647, + 858 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.3 MULTI-FRAME NCE ", + "text_level": 1, + "bbox": [ + 174, + 869, + 357, + 883 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "All recent works perform some sort of transformation on a single image to create Anchor-Positive pairs for the NCE loss (Bachman et al., 2019; Chen et al., 2020a; He et al., 2020; Hénaff et al., 2019; ", + "bbox": [ + 174, + 895, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/7821717a4ff18a11ce37a18819d0e222869c17f58d61e0f16e4f7dd5362c2233.jpg", + "image_caption": [ + "Figure 2: Left: Standard NCE using “Same Frame” where all positive pairs come from the same image. Middle: NCE using “Multi-Frame” where positive pairs come from the same video. Right: Multi-Frame Multi-Pair NCE which uses more than one positive pair per video. The gray boxes indicate the true match pairs. The Memory Bank adds more negatives for each anchor. " + ], + "image_footnote": [], + "bbox": [ + 174, + 99, + 816, + 246 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Misra & Maaten, 2020; Wu et al., 2018). We refer to this as “Same Frame.” We differ from these works by using multiple images from a single video to form our pairs. This allows our network to see truly different views, deformations, similar objects, and larger scene changes. Additionally, this encodes temporal consistency as the semantic contents of a video are unlikely to change suddenly. For example in a video of two cats playing, the camera may focus on one cat, or may even pan to a previously unseen dog, but it is unlikely to pan to a shark. Note that in practice, we select frames with replacement, making our potential pairs a strict superset of those in prior works. ", + "bbox": [ + 173, + 327, + 825, + 426 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.4 MEMORY BANKS AND MOMENTUM CONTRAST ", + "text_level": 1, + "bbox": [ + 174, + 449, + 544, + 464 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "NCE-based methods benefit greatly from large pools of negatives because this increases the likelihood of finding at least one hard negative for each positive example. In some works (Misra & Maaten, 2020; Wu et al., 2018), negatives are sampled from a large memory bank which was filled with earlier outputs of the network. The NCE loss can be modified to use negatives from prior batches as shown in equation 3 for a memory bank of negatives $N _ { 1 \\dots m }$ . ", + "bbox": [ + 174, + 478, + 825, + 549 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/0bd3e60fa2a98ff7c51d944db3eaa9ef4d38594a1e1c610ed7acdbc014000895.jpg", + "text": "$$\n{ \\mathcal { L } } _ { \\mathrm { N C E } } = - { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } \\log { \\frac { e ^ { s i m ( A _ { i } , P _ { i } ) } } { e ^ { s i m ( A _ { i } , P _ { i } ) } + \\sum _ { j = 1 } ^ { m } e ^ { s i m ( A _ { i } , N _ { j } ) } } }\n$$", + "text_format": "latex", + "bbox": [ + 320, + 561, + 676, + 603 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "As the network trains, its output distribution will shift. A potential issue when using a memory bank is the network learns a simple classifier between the current distribution and an old one. Momentum Contrast (MoCo) (He et al., 2020) alleviates this issue by using a quickly updating primary network $f$ in equation 2) and a slowly updating secondary network $( g )$ . $f$ is updated based on the NCE loss in equation 3 and $g$ is updated using a momentum rule $g \\alpha g + ( 1 - \\alpha ) f$ . The memory bank is filled with previous outputs from the slowly changing network $g$ , reducing the likelihood that $f$ will be able to learn a simple recent batch/old batch classifier. For more details, see (He et al., 2020). ", + "bbox": [ + 174, + 614, + 825, + 713 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.5 MULTI-PAIR NCE ", + "text_level": 1, + "bbox": [ + 174, + 737, + 341, + 751 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "By using a memory bank, we increase the number of negatives without a large computational cost. Yet standard NCE only uses $n$ positive pairs per batch of size $n$ . We can further increase the number of positives per batch (while holding batch size constant) by selecting $v$ videos and $k$ samples from each video where $\\textstyle k = { \\frac { n } { v } }$ . By computing the pairwise similarity between each pair, we reuse each positive sample $k$ times, resulting in $\\textstyle k ^ { 2 } v = { \\frac { n ^ { 2 } } { v } }$ positives per batch. Using a simple block-diagonal mask as shown in Figure 2 we can efficiently compute the similarities and NCE loss both between elements of the batch and across a memory bank, achieving a large number of positive comparisons per batch while retaining a large negative size. In practice, we notice no meaningful computational cost to this approach. We refer to this full method using Multi-Pair on video data and the MultiFrame learning procedure as Video Noise Contrastive Estimation (VINCE). See Appendix 2 for detailed pseudocode. ", + "bbox": [ + 173, + 766, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 102, + 326, + 117 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We evaluate our method (VINCE) on both single image and temporal tasks by freezing our learned representation and adding a small network (in most cases a single linear layer) for adaptation to new end-tasks. Our learned representation transfers well to a variety of visual tasks, especially tasks which require temporal reasoning. To show this, we compare with multiple strong baselines: ", + "bbox": [ + 176, + 133, + 825, + 190 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "• MoCo-IN: Network pretrained on ImageNet (Deng et al., 2009) using the MoCo algorithm. • MoCo-R2V2: Network pretrained on R2V2 using the MoCo algorithm. This uses exactly the same data as VINCE but the Same Frame technique described in Sec. 3.3. We also prevent multiple images from the same video being in the Memory Bank at the same time. • Sup-IN: Network pretrained on fully supervised ImageNet. ", + "bbox": [ + 215, + 202, + 823, + 276 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To validate the benefits of learning from videos, we additionally compare against an unsupervised image dataset. Since ImageNet itself required time, effort, and money to create, we construct a new static image dataset analogous to our video dataset. Specifically, we search Google Images for the ImageNet synsets and download the top K results for each category. We refer to this as MoCoG). Although other unsupervised video-based methods exist, they have generally underperformed pretrained supervised ImageNet weights or are only applicable to video tasks. ", + "bbox": [ + 174, + 289, + 825, + 372 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 TARGET TASKS ", + "text_level": 1, + "bbox": [ + 174, + 390, + 321, + 404 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We compare each method on several diverse end-tasks using both ResNet18 (He et al., 2016) and ResNet50 backbones. More training implementation details can be found in Appendix 3. Results for these tasks are shown in Table 2. We train all end-task models using Adam (Kingma & Ba, 2014) and a shared learning rate schedule per task. For each dataset, we use standard data augmentation approaches (crop, flip, color jitter except for tracking). One overall trend to note is the relative gain over MoCo-R2V2. If the single-frame algorithm performed as well as our multi-frame method on temporal tasks, it would indicate minimal temporal understanding. However, the relative gain of VINCE over MoCo-R2V2 on Kinetics $( 1 3 . 8 5 \\% )$ and tracking ( $1 3 . 3 3 \\%$ and $1 5 . 3 8 \\%$ ) are higher than those on single-frame tasks, showing that our method can incorporate temporal cues. ", + "bbox": [ + 174, + 416, + 825, + 541 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "ImageNet: For this task, we use our frozen learned representations, adding a single linear layer after the global average pool. Although none of the methods match the fully supervised performance of ResNet on ImageNet, they do achieve reasonable performance given only single linear layer. It is unsurprising that MoCo pretrained on ImageNet images (MoCo-IN) outperforms our method (0.447 vs 0.400) due to the domain shift between pretrain and end-task. However MoCo pretrained on R2V2 (MoCo-R2V2) suffers nearly a $2 \\times$ drop in accuracy $( 8 . 9 \\% )$ compared to our method $( 4 . 7 \\% )$ , indicating that pretraining on multi-frame matching provides a clear benefit over single frame pairs. Our method, which has never seen an image from ImageNet before, still learns a representation which generalizes well to this new type of data. Even drawing images from a similar class distribution (MoCo-G) does not outperform our method. ", + "bbox": [ + 174, + 547, + 825, + 688 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "SUN Scenes: SUN Scenes is a classification dataset in which each image is categorized into one of 397 possible scene types such as airplane cabin, bedroom, and coffee shop. Again we train a single linear layer on top of each pretrained network. This data is quite similar to ImageNet in that each image is well-curated and contains single, unambiguous subjects. As such, the ImageNet fully supervised baseline transfers quite well to SUN Scenes. However VINCE outperforms Sup-IN by a small margin. Again we note a large improvement of VINCE over MoCo-R2V2 (0.495 vs. 0.450). This shows that our method learns to recognize not just the main subject of an image but also the surrounding scene, which requires a richer understanding of the world. ", + "bbox": [ + 174, + 694, + 825, + 805 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Kinetics 400: This dataset consists of 10 second clips from YouTube videos and action labels for each segment. We first download each video and subsample each clip to one frame per second (10 frames per clip). We train a single layer LSTM (Hochreiter & Schmidhuber, 1997) followed by a single linear layer to predict the action category for each segment. Kinetics acts as a crucial test to evaluate whether our model learns temporal cues. VINCE greatly outperform all other methods, whereas traditional baselines such as fine-tuning supervised ImageNet do not adapt well at all. This shows that contrary to popular belief, representations pretrained on ImageNet many not be a good fit for other visual domains, especially on temporal tasks. ", + "bbox": [ + 174, + 811, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/89ec095f968fc680ad2aa736f80c7a12b24b7fcf386b487ba2830828798e1d63.jpg", + "table_caption": [ + "Table 2: Comparison of representation performance across a variety of end tasks. We show improvements over MoCo trained on the same data on all tasks, and outperform MoCo trained on ImageNet as well as supervised pretraining on ImageNet on all tasks but ImageNet itself (and tracking for ResNet50). Each representation uses the same ResNet convolutional backbone, sharing weights across all tasks. Linear (for Kinetics LSTM Linear) classifiers are the only learned weights for each end task. " + ], + "table_footnote": [], + "table_body": "
Image Classification ImageNet(Deng et al.,2009)Scene Classification SUN Scenes(Xiao et al.,2010)Action Recognition Kinetics 400(Kay et al.,2017)Tracking OTB 2015(Wu et al.,2015)
Trained Layer(s) MetricLinear Accuracy (Top 1)Linear Accuracy (Top 1)LSTM→Linear Accuracy (Top 1)1x1 Conv
Sup-INPrecisionSuccess
MoCo-IN0.6960.4910.2070.5570.396
0.4470.4870.3360.5830.429
BeerMoCo-G0.3930.4440.3130.5510.413
MoCo-R2V20.3580.450 0.4950.3180.5550.403
VINCE (Ours)0.4000.3620.6290.465
Relative Gain over MoCo-R2V211.91%9.93%13.85%13.33%15.38%
PssSDSup-IN0.7620.5930.3050.4580.320
MoCo-V2-IN (our impl.)0.6520.6080.4590.3000.260
MoCo-R2V20.5360.5810.4560.3860.299
VINCE (Ours)0.5440.6110.4910.4020.300
Relative Gain over MoCo-R2V21.36%5.29%7.72%4.15%0.33%
", + "bbox": [ + 176, + 102, + 821, + 270 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Object Tracking Benchmark (OTB) 2015: In this dataset, given an initial bounding box around an arbitrary object in the first image, a model must locate the object in the following frames. Difficult tracking datasets like OTB 2015 implicitly require robustness to occlusion and appearance change of the target object, so improved performance would indicate a representation’s relative robustness to these challenges. We use the SiamFC (Bertinetto et al., 2016) tracking algorithm on top of our learned representation. SiamFC localizes an object of interest by convolving deep features from the initial bounding box with the current image, similar to template matching (Briechle & Hanebeck, 2001) but in deep feature space. For a more complete explanation, see (Bertinetto et al., 2016). OTB is evaluated using two metrics – precision and success. Precision measures the percentage of frames where the (normalized) center error is less than a certain threshold, averaging over many thresholds. Similarly, success measures the percentage of frames where the Intersection Over Union is more than a range of thresholds. ", + "bbox": [ + 174, + 378, + 825, + 544 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "A representation which works well for SiamFC would have the property that the cross correlation of two images of the same object is high, but the cross correlation of two different objects, or a poorly cropped image of the same object, is low. Pretraining our representations on multiple frames from the same video coincides well with the first objective, however since we use cropped data augmentations, the representations tend to be somewhat invariant to poorly-cropped candidates. Still, the models perform quite well across a variety of difficult tracking instances. Our ResNet18 model transfers significantly better than all other methods indicating a clear benefit to using temporal cues during pretraining. We find, somewhat unexpectedly, that the ResNet18 network fares better than the ResNet50. A likely explanation for this is that the original SiamFC method uses AlexNet (Krizhevsky et al., 2012) with no padding in a fully-convolutional manner. When using ResNet, padding must be applied to keep the outputs the same dimensionality as the inputs. Thus, at training time the network may latch onto zero-padding cues which will not be applicable at test time. This becomes more of an issue the larger the receptive field which is why ResNet50 struggles but ResNet18 is somewhat less affected. ", + "bbox": [ + 174, + 551, + 825, + 746 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 METHOD ABLATION ", + "text_level": 1, + "bbox": [ + 176, + 770, + 356, + 784 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We validate the effectiveness of Multi-Frame (Sec. 3.3) and Multi-Pair (Sec. 3.5) learning by ablating the number of images from each video used in a batch of comparisons shown in Table 3 (due to computational constraints, we only perform ablations on ResNet18). The first row is equivalent to the procedure done in MoCo (He et al., 2020) i.e. the anchor and positive pairs are two data augmentations of the same image. The second row uses the MoCo procedure as well, however the anchors and positives may be from different images from the same video. The third row uses our Multi-Pair NCE method taking 4 positives and 4 anchors from each video, resulting in 16 positive pairs. A pictorial representation can be seen in Figure 2. Note that when selecting images for row 2 and 3, we use sampling with replacement, making our method a strict super-set of MoCo. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/5ea59c0567bba694dc1c048d6b1198a34416a159856acafde6976be958668754.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Images Per VideoTest Task
ImageNetSUN SceneKinetics 400OTB 2015 PrecisionOTB 2015 Success
1: Same Frame0.3580.4500.3180.5550.403
2: Multi-Frame0.3810.4780.3610.6220.464
8:Multi-Frame Multi-Pair0.4000.4950.3620.6290.465
", + "bbox": [ + 176, + 102, + 820, + 174 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3: Method ablation for VINCE. We compare using one source image with two augmentations (the standard approach), two different images, or a set of different images. Using Multi-Frame results in a large boost across the board. Multi-Frame Multi-Pair further increases the power of the representation. Note that all methods use the entire dataset, but only Multi-Frame methods use multiple images from a video within one batch. ", + "bbox": [ + 173, + 178, + 825, + 247 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/236be6c741a0e5de7ce64d1270d4725fcde13154089fd2b5d22a1557b9a4fc63.jpg", + "table_caption": [ + "Table 4: Pretraining data ablation for VINCE. Each method uses the same training setup but different training data. Since R2V2 uses ImageNet search queries, it outperforms the others on ImageNet. Similarly, pretraining on Kinetics increases performance on the Kinetics task. " + ], + "table_footnote": [], + "table_body": "
Pretraining DataTest Task
ImageNet SUN SceneKinetics 400OTB 2015 PrecisionOTB 2015 Success
R2V2 IN-Queries0.4000.4950.3620.6290.465
R2V2 YT8MURLs0.3670.4780.3430.6670.492
Kinetics 400 URLs0.3680.4940.3900.6120.456
", + "bbox": [ + 176, + 263, + 820, + 339 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We observe across-the-board improvements from both modifications to the MoCo approach. The majority of the improvement comes from using two non-identical frames for matching, but we still gain an additional improvement from using Multi-Pair NCE. Our intuition is that using the MultiPair NCE creates gradients that pull each feature towards a global video representation whereas the standard NCE remains more instance-based, only moving a representation in one direction at a time. Thus, we would expect the Multi-Pair NCE features to be more holistically semantic whereas the standard NCE may retain more uniquely identifying features. In fact, we observe a larger performance gap on the more semantic ImageNet and SUN Scene tasks. In contrast, because the Kinetics model uses an LSTM to reason over all input images at once, instance-level features are equally useful as global video features for overall accuracy. ", + "bbox": [ + 174, + 401, + 825, + 540 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.3 PRETRAINING DATA ABLATION ", + "text_level": 1, + "bbox": [ + 176, + 558, + 431, + 571 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In Table 4 we explore the effect of different pretraining datasets on end-task performance. For each experiment, we use VINCE but use video data from three different sources: our method of searching ImageNet synset queries, using the URLs from YouTube 8M (Abu-El-Haija et al., 2016), and using the URLs from Kinetics 400 (Kay et al., 2017). Again, we only test ResNet18. Our YouTube8M(YT8M) pretraining data uses the same filtering procedure as R2V2 and contains 5.8 million images from 1.4 million videos. As noted in Table 2 MoCo-IN results, using the same dataset for pretraining as the end-task results in a boost in performance on that specific task but does not indicate that the representation will be better on all tasks. We see this trend is true again when pretraining on Kinetics data. Similarly, since R2V2 uses ImageNet synset for search queries, pretraining on it performs better on ImageNet than the other less-aligned datasets. ", + "bbox": [ + 174, + 584, + 825, + 723 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In general, this would indicate that given a large enough set of diverse videos, pretraining directly on the unlabeled source data would result in the best performing representation on that data. If this is not possible, then pretraining on a large external source of data may still result in a useful representation. It also indicates that the VINCE method works well on a variety of different pretraining datasets. The increased performance on tracking when using YT8M data could be explained by it simply having access to a larger number of video sources and frames. For generic object tracking, class diversity may be less important than number of samples because the class identity is ignored. ", + "bbox": [ + 174, + 731, + 825, + 828 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 849, + 328, + 864 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In this work we introduced Video Noise Contrastive Estimation, a process for using unlabeled videos to learn an unsupervised image representation. By training on multiple images from the same video, we learn from natural changes such as deformation and viewpoint rather than artificial 2D augmentations. To learn from a large variety of diverse videos, we collect Random Related Video Views in a completely automated fashion. We show across-the-board improvements over the recently proposed MoCo (He et al., 2020) technique on a wide variety of tasks, and we believe Video Noise Contrastive Estimation will extend to other unsupervised methods such as SimCLR (Chen et al., 2020a) and PIRL (Misra & Maaten, 2020) as well as other end-tasks. As representation learning techniques improve, we believe that videos – rather than images – will prove an invaluable resource for pushing the state-of-the-art forward. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 200 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 222, + 285, + 237 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Sami Abu-El-Haija, Nisarg Kothari, Joonseok Lee, Paul Natsev, George Toderici, Balakrishnan Varadarajan, and Sudheendra Vijayanarasimhan. 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Self-supervised visual descriptor learning for dense correspondence. IEEE Robotics and Automation Letters, 2(2):420–427, 2016. ", + "bbox": [ + 174, + 319, + 821, + 349 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Nitish Srivastava, Elman Mansimov, and Ruslan Salakhudinov. Unsupervised learning of video representations using lstms. In International conference on machine learning, pp. 843–852, 2015. ", + "bbox": [ + 176, + 357, + 821, + 387 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. arXiv preprint arXiv:1906.05849, 2019. ", + "bbox": [ + 174, + 395, + 821, + 424 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Michael Tschannen, Josip Djolonga, Marvin Ritter, Aravindh Mahendran, Neil Houlsby, Sylvain Gelly, and Mario Lucic. Self-supervised learning of video-induced visual invariances. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 13806– 13815, 2020. ", + "bbox": [ + 174, + 433, + 825, + 488 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Carl Vondrick, Abhinav Shrivastava, Alireza Fathi, Sergio Guadarrama, and Kevin Murphy. Tracking emerges by colorizing videos. In European Conference on Computer Vision, pp. 391–408, 2018. ", + "bbox": [ + 174, + 497, + 825, + 540 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Xiaolong Wang and Abhinav Gupta. Unsupervised learning of visual representations using videos. In IEEE International Conference on Computer Vision, pp. 2794–2802, 2015. ", + "bbox": [ + 173, + 549, + 821, + 579 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Xiaolong Wang, Allan Jabri, and Alexei A Efros. Learning correspondence from the cycleconsistency of time. In IEEE Conference on Computer Vision and Pattern Recognition, pp. 2566–2576, 2019. ", + "bbox": [ + 173, + 587, + 821, + 630 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Yi Wu, Jongwoo Lim, and Ming-Hsuan Yang. Object tracking benchmark. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(9):1834–1848, 2015. ", + "bbox": [ + 169, + 638, + 823, + 667 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via nonparametric instance discrimination. In IEEE Conference on Computer Vision and Pattern Recognition, pp. 3733–3742, 2018. ", + "bbox": [ + 174, + 676, + 821, + 719 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Jianxiong Xiao, James Hays, Krista A Ehinger, Aude Oliva, and Antonio Torralba. Sun database: Large-scale scene recognition from abbey to zoo. In IEEE Conference on Computer Vision and Pattern Recognition, pp. 3485–3492. IEEE, 2010. ", + "bbox": [ + 174, + 728, + 823, + 772 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In European conference on computer vision, pp. 649–666. Springer, 2016. ", + "bbox": [ + 171, + 780, + 823, + 809 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Hang Zhao, Chuang Gan, Andrew Rouditchenko, Carl Vondrick, Josh McDermott, and Antonio Torralba. The sound of pixels. In Proceedings of the European Conference on Computer Vision (ECCV), September 2018. ", + "bbox": [ + 174, + 818, + 825, + 861 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/8e99c9e1888b1f509194ec70749e16697122f5048d817e2552c11a6d0f660fe6.jpg", + "image_caption": [ + "Figure A1: Sample from Random Related Video Views (train set). " + ], + "image_footnote": [], + "bbox": [ + 238, + 101, + 758, + 313 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "APPENDIX 1 DATASET COMPARISON ", + "text_level": 1, + "bbox": [ + 174, + 349, + 495, + 366 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Random Related Video Views provides nearly 250,000 video links as well as multiple images per video across a broad variety of categories. We show more samples from R2V2 in Figure A1. Videos each have four images 150 frames apart. Each separate video (outlined in blue) lists its corresponding YouTube link. Using ImageNet synsets for search queries provides reasonable visual diversity, but could be substituted with another set of queries. While we acknowledge that using YouTube’s Search feature is not truly random, this procedure resulted in significantly more diverse samples than using existing datasets like YouTube8M (Abu-El-Haija et al., 2016) which is heavily unbalanced with unnatural videos.2 We do no additional data cleaning to ensure that the videos or extracted images actually contain the search term (many do not), nor do we search for “high interest” video segments as in Misra et al. (2016). We also discard the search term itself as a form of supervision. We find that a gap of 5 seconds between each saved image typically results in visually distinct but semantically related images. Too much shorter results in images which are less individually distinct, and too much longer may result in large and unpredictable changes. A sample from each dataset can be seen in the supplemental material. ", + "bbox": [ + 174, + 383, + 825, + 577 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/2a73ace7d06bf23a2fb379edcf93ff0973f48b41f0087f5b714f1f928e1f1b5e.jpg", + "image_caption": [ + "Figure A2: Random sampling of pairs of images from videos in each dataset. In GOT-10k, sometimes different video clips are segments from the same original video as seen in the first and second sample. Images are square cropped for visualization purposes only. " + ], + "image_footnote": [], + "bbox": [ + 174, + 661, + 820, + 858 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "We show samples from various existing video datasets in Figure A2. Datasets such as YouTube8M (Abu-El-Haija et al., 2016) and Kinetics400 (Kay et al., 2017). Kinetics contains only videos of humans performing actions. Alternative datasets such as GOT-10k (Huang et al., 2019) provide a comparatively small number of videos but with dense annotations (in GOT-10k’s case for object tracking). ", + "bbox": [ + 174, + 103, + 825, + 174 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX 2 ALGORITHM PSEUDOCODE ", + "text_level": 1, + "bbox": [ + 176, + 193, + 524, + 209 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "The pseudocode below efficiently computes the Multi-Pair NCE for a batch of inputs. Shapes of each input are provided for clarity where: ", + "bbox": [ + 174, + 224, + 825, + 252 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "• $v =$ The number of videos in a batch. \n• $k =$ The number of frames per video. \n• $n = ( v * k ) =$ The total number of images in the batch. \n• $m =$ The number of elements in the Memory Bank. \n• $d =$ The dimensionality of the feature space. ", + "bbox": [ + 215, + 263, + 598, + 353 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Algorithm 1 Python-style pseudo code for Multi-Pair NCE. ", + "bbox": [ + 174, + 372, + 566, + 387 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/9c51c3ac4b7b427f39ed5d21ad5c6148000bf6d77d25d092e07def9616521416.jpg", + "table_caption": [], + "table_footnote": [ + "In practice we use the Log-Sum-Exp trick for numerical stability but omit here for clarity. broadcast repeats the input until it matches the provided dimensions. !mask flips the booleans of each point in the mask. " + ], + "table_body": "
def multi_pair_nce([v，k，d] output of f encoder
f_output,# #
g_output,#output of g encoder
moco_mem, mask,[m，d]
temperature):# #(n + m)] block diagonal bool
f_output = f_output.reshape(v * k，d)[n，d]
g_output = g_output.reshape(v * k，d)#[n，d]
compare = concatenate((g_output，moco_mem)，axis=0)#[(n + m)，d]
similarities = matmul(f_output，compare.T)(n + m)]
similarities /= temperature
pos_similarities = similarities[mask]#[n，k]
neg similarities = similarities[!mask] exp_pos_sim = exp(pos_similarities)#(n + m -k)]
normalizing_constant = broadcast([n， k]
reduce_sum(exp(neg_similarities)，axis=1),[n，k]
shape(numerator))))
score = exp_pos_sim / (exp_pos_sim + normalizing_constant)
loss = -mean(log(score)) return loss
", + "bbox": [ + 174, + 397, + 769, + 577 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX 3 VINCE IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 174, + 635, + 598, + 651 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For training our ResNet18 (He et al., 2016) representations we use the network up to the global average pooling followed by a fully connected layer $( 5 1 2 \\mathrm { x } 5 1 2 )$ , Leaky-ReLU and a final embedding layer $( 5 1 2 \\mathrm { ~ x ~ } 6 4 )$ ). The features are L2-normalized and multiplied by $\\begin{array} { r } { \\dot { \\tau } = \\frac { 1 } { 0 . 0 7 } } \\end{array}$ 10.07 as in MoCo (He et al., 2020). We use 8 GPUs, a training batch size of 256 and a MoCo Memory Bank of size 65536 and a $g$ -network momentum of 0.999. For multi-GPU training we employ the Shuffle-BN technique shuffling both the anchors and the positives to reduce the correlation between batches filled with multiple images from the same video. We use SGD with a learning rate of 0.03, momentum $\\scriptstyle 1 = 0 . 9$ , and weight decay $\\scriptstyle = 0 . 0 0 0 1$ . All of these hyperparameters are shared with MoCo (He et al., 2020). All methods are trained for approximately $4 5 0 \\mathrm { k }$ iterations. When selecting frames from a video, we pick with replacement. In addition to the natural augmentation, we perform standard data augmentation (color jitter, cropping, flipping) on the inputs. This prevents the network from relying too heavily on shared video statistics like mean frame color. After cropping, all images are resized to $2 2 4 \\times 2 2 4$ . It is worth noting that both VINCE and our data (R2V2) can be used with other network architectures or learning algorithms such as AMDIM (Bachman et al., 2019), PIRL (Misra & Maaten, 2020), or SimCLR (Chen et al., 2020a). We choose ResNet18 and MoCo due to implementation simplicity and relatively low computational constraints. ", + "bbox": [ + 174, + 666, + 825, + 888 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "For ResNet50 training, we closely match the hyperparameters in MoCo v2 (Chen et al., 2020b). Specifically, we use the blur augmentation and stronger color augmentation and a cosine annealing learning rate for 286,000 iterations (200 epochs of ImageNet or equivalent on R2V2 regardless of the number of unique positives per batch). The batch size we use is 896 with an initial learning rate of 0.105 and an embedding dimensionality of 128. The temperature parameter used was $\\tau \\stackrel { } { = } \\frac { 1 } { 0 . 2 }$ . We did not perform any hyperparameter searches for VINCE or R2V2, so these results may be suboptimal. ", + "bbox": [ + 174, + 895, + 821, + 924 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/98dfdc7828efc0513ffa867293128fcc9da3691d331c460f71eb37125f0855a1.jpg", + "image_caption": [ + "Figure A3: Nearest neighbor results for a sampling of query images from R2V2 and ImageNet using various models. VINCE shows a clear understanding of each image and finds highly relevant neighbors. " + ], + "image_footnote": [], + "bbox": [ + 192, + 99, + 800, + 333 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 428, + 825, + 497 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "APPENDIX 4 QUALITATIVE RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 525, + 493, + 541 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We additionally provide two qualitative analyses to better understand the success and failure cases of VINCE: Nearest Neighbors, and t-SNE. ", + "bbox": [ + 176, + 560, + 823, + 589 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "APPENDIX 4.1 NEAREST NEIGHBORS: ", + "text_level": 1, + "bbox": [ + 176, + 613, + 450, + 627 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We query ImageNet Val and a set of test videos for nearest neighbor matches, taking at most one neighbor per video. We visualize the top 5 neighbors for VINCE, MoCo-R2V2, and MoCo-IN in Figure A3. We observe that VINCE seems to understand the semantics of an image more than MoCo-R2V2 and MoCo-IN. For instance, although MoCo-R2V2 and MoCo-IN find other control panels and buttons in query 1, they do not make the scene-level connection to car interiors as well as VINCE does. Query 2 shows an interesting quirk case of our method. Rather than matching the semantics of the image, VINCE relies on the news logo as a differentiating feature due to its discriminative nature. Each image in VINCE’s query 2 results is from a separate video, but from the same news source. For the ImageNet queries, despite never seeing ImageNet inputs during pretraining, VINCE is able to find good matches as well as MoCo-IN which was trained using only ImageNet inputs. ", + "bbox": [ + 174, + 642, + 825, + 795 + ], + "page_idx": 13 + }, + { + "type": "image", + "img_path": "images/bc55ce9cb816226d8d030d252ea576e9d6515d16c77c31bdda33ee70452a958a.jpg", + "image_caption": [ + "Figure A4: t-SNE embedding of images from R2V2 test set. " + ], + "image_footnote": [], + "bbox": [ + 245, + 103, + 753, + 416 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "APPENDIX 4.2 T-SNE: ", + "text_level": 1, + "bbox": [ + 176, + 478, + 343, + 491 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Using a set of held-out video frames, we project the 64-D embedding space from VINCE to 2D using t-SNE (Maaten & Hinton, 2008) and visualize the formed clusters in Figure A4. Not only does this assist in verifying the quality of the embedding, it also serves as a visual method for evaluating the diversity of the dataset itself. The largest of the clusters seems to be the face cluster. YouTube is full of videos of people looking and talking directly to a camera, and our random subset reflects this pattern. Other interesting, yet unexpected clusters emerge as well such as cats (YouTube loves cats), hands (demo videos), and food (cooking videos). ", + "bbox": [ + 173, + 503, + 825, + 602 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "APPENDIX 5 PRECISION AND SUCCESS PLOTS FOR OTB 2015 ", + "text_level": 1, + "bbox": [ + 176, + 621, + 705, + 637 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We provide full breakdowns of the Precision and Success of each method on OTB 2015 (Wu et al., 2015). The values in the legend correspond to the (mean) area under the curve for each method. ", + "bbox": [ + 173, + 652, + 823, + 681 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/9d1a5e59c5fd92d5b0acb55809eff39331bf6dce134be0f08d1cfe2660f6b429.jpg", + "image_caption": [ + "Figure A5: Precision (a) and Success (b) plots for OTB 2015 for various backbones. " + ], + "image_footnote": [], + "bbox": [ + 240, + 715, + 759, + 887 + ], + "page_idx": 14 + } +] \ No newline at end of file diff --git a/parse/train/iktA2PtTRsK/iktA2PtTRsK_middle.json b/parse/train/iktA2PtTRsK/iktA2PtTRsK_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..6adc632fb41bc6ff2260734c4f895913bd3f3983 --- /dev/null +++ b/parse/train/iktA2PtTRsK/iktA2PtTRsK_middle.json @@ -0,0 +1,34446 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 503, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 77, + 505, + 98 + ], + "spans": [ + { + "bbox": [ + 106, + 77, + 505, + 98 + ], + "score": 1.0, + "content": "WATCHING THE WORLD GO BY: REPRESENTATION", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 98, + 394, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 394, + 118 + ], + "score": 1.0, + "content": "LEARNING FROM UNLABELED VIDEOS", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 135, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 135, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "spans": [ + { + "bbox": [ + 111, + 146, + 245, + 158 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 185, + 336, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 212, + 468, + 354 + ], + "lines": [ + { + "bbox": [ + 141, + 212, + 470, + 225 + ], + "spans": [ + { + "bbox": [ + 141, + 212, + 470, + 225 + ], + "score": 1.0, + "content": "Recent unsupervised representation learning techniques show remarkable success", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 223, + 469, + 235 + ], + "spans": [ + { + "bbox": [ + 141, + 223, + 469, + 235 + ], + "score": 1.0, + "content": "on many single image tasks by using instance discrimination: learning to differ-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 234, + 470, + 246 + ], + "spans": [ + { + "bbox": [ + 141, + 234, + 470, + 246 + ], + "score": 1.0, + "content": "entiate between two augmented versions of the same image and a large batch of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 245, + 470, + 257 + ], + "spans": [ + { + "bbox": [ + 141, + 245, + 470, + 257 + ], + "score": 1.0, + "content": "unrelated images. 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Yet many popular computer vision models learn representations which are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "limited to inference on single images, lacking temporal context. Representations learned from static", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 459, + 500, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 500, + 470 + ], + "score": 1.0, + "content": "images are inherently limited to an understanding of the world as many unrelated static snapshots.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "This is especially true of recent unsupervised learning techniques (Bachman et al., 2019; Chen", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 486, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 497 + ], + "score": 1.0, + "content": "et al., 2020a; He et al., 2020; Hénaff et al., 2019; Hjelm et al., 2019; Misra & Maaten, 2020; Tian", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "et al., 2019; Wu et al., 2018), all of which train on a set of highly-curated, well-balanced data: Ima-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "geNet (Deng et al., 2009). Scaling up these techniques to larger, less-curated datasets like Instagram-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "1B (Mahajan et al., 2018) has not provided large improvements in performance (He et al., 2020).", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 530, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 506, + 542 + ], + "score": 1.0, + "content": "Only so much can be learned from a single image: no amount of artificial augmentation can show a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "new view of an object or what might happen next in a scene. This dichotomy can be seen in Figure 1.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "In order to move beyond this limitation, we argue that video supplies significantly more semanti-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "cally meaningful content than a single image. With video, we can see how the world changes, find", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 578, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 506, + 594 + ], + "score": 1.0, + "content": "connections between images, and more directly observe the underlying scene. Prior work using", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 589, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 506, + 604 + ], + "score": 1.0, + "content": "temporal cues has shown success in learning from unlabeled videos (Misra et al., 2016; Wang et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "2019; Srivastava et al., 2015), but has not been able to surpass supervised pretraining. 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In this work, we merge the two concepts", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "with Video Noise Contrastive Estimation (VINCE), a method for using unlabeled videos as a basis", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 646, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 646, + 505, + 657 + ], + "score": 1.0, + "content": "for learning visual representations. Instead of predicting whether two feature vectors come from the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 657, + 505, + 669 + ], + "score": 1.0, + "content": "same underlying image, we task our network with predicting whether two images originate from the", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 680 + ], + "score": 1.0, + "content": "same video. Not only does this allow our method to learn how a single object might change, it also", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 678, + 506, + 692 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 692 + ], + "score": 1.0, + "content": "enables learning which things might be in a scene together, e.g. cats are more likely to be in videos", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 505, + 702 + ], + "score": 1.0, + "content": "with dogs than with sharks. Additionally, we generalize the NCE technique to operate on multiple", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "positive pairs from a single source. 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Prior work uses artificial data augmentation techniques such", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 256, + 470, + 270 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 470, + 270 + ], + "score": 1.0, + "content": "as cropping, and color jitter which can only affect the image in superficial ways", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 266, + 469, + 280 + ], + "spans": [ + { + "bbox": [ + 141, + 266, + 469, + 280 + ], + "score": 1.0, + "content": "and are not aligned with how objects actually change e.g. occlusion, deformation,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 279, + 469, + 290 + ], + "spans": [ + { + "bbox": [ + 142, + 279, + 469, + 290 + ], + "score": 1.0, + "content": "viewpoint change. We argue that videos offer this natural augmentation for free.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 301 + ], + "score": 1.0, + "content": "Videos can provide entirely new views of objects, show deformation, and even", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 469, + 312 + ], + "score": 1.0, + "content": "connect semantically similar but visually distinct concepts. We propose Video", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 311, + 469, + 323 + ], + "spans": [ + { + "bbox": [ + 141, + 311, + 469, + 323 + ], + "score": 1.0, + "content": "Noise Contrastive Estimation, a method for using unlabeled video to learn", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 322, + 469, + 334 + ], + "score": 1.0, + "content": "strong, transferable, single image representations. We demonstrate improve-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 469, + 344 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 469, + 344 + ], + "score": 1.0, + "content": "ments over recent unsupervised single image techniques, as well as over fully", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 344, + 453, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 344, + 453, + 356 + ], + "score": 1.0, + "content": "supervised ImageNet pretraining, across temporal and non-temporal tasks.1", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 141, + 212, + 470, + 356 + ] + }, + { + "type": "title", + "bbox": [ + 109, + 377, + 206, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 376, + 208, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 208, + 393 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 403, + 504, + 469 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "The world seen through our eyes is constantly changing. As we move through the world, we see", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "much more than a single static image: objects rotate revealing occluded regions, deform, the sur-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "roundings change, and we ourselves move. Our internal visual systems are constantly seeing tem-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 436, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 436, + 506, + 449 + ], + "score": 1.0, + "content": "porally coherent images. Yet many popular computer vision models learn representations which are", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 460 + ], + "score": 1.0, + "content": "limited to inference on single images, lacking temporal context. Representations learned from static", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 459, + 500, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 500, + 470 + ], + "score": 1.0, + "content": "images are inherently limited to an understanding of the world as many unrelated static snapshots.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 402, + 506, + 470 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 474, + 505, + 552 + ], + "lines": [ + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "This is especially true of recent unsupervised learning techniques (Bachman et al., 2019; Chen", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 486, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 505, + 497 + ], + "score": 1.0, + "content": "et al., 2020a; He et al., 2020; Hénaff et al., 2019; Hjelm et al., 2019; Misra & Maaten, 2020; Tian", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 505, + 509 + ], + "score": 1.0, + "content": "et al., 2019; Wu et al., 2018), all of which train on a set of highly-curated, well-balanced data: Ima-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 505, + 520 + ], + "score": 1.0, + "content": "geNet (Deng et al., 2009). Scaling up these techniques to larger, less-curated datasets like Instagram-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 106, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "1B (Mahajan et al., 2018) has not provided large improvements in performance (He et al., 2020).", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 530, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 530, + 506, + 542 + ], + "score": 1.0, + "content": "Only so much can be learned from a single image: no amount of artificial augmentation can show a", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 505, + 553 + ], + "score": 1.0, + "content": "new view of an object or what might happen next in a scene. This dichotomy can be seen in Figure 1.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 474, + 506, + 553 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 558, + 505, + 711 + ], + "lines": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 557, + 505, + 570 + ], + "score": 1.0, + "content": "In order to move beyond this limitation, we argue that video supplies significantly more semanti-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "cally meaningful content than a single image. 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Simi-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "larly, Contrastive Predictive Coding (CPC) (Hénaff et al., 2019) uses crops of an image as “context”", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "and predicts features for the unseen portions of the image. We provide a more natural data augmen-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "tation by using multiple frames from a single video. As a video progresses, the objects in the scene,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "the background, and the camera itself may move, providing new views. Whereas augmentations on", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "an image are constrained by a single snapshot in time, using different frames from a single video", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "gives entirely new information about the scene. Additionally, rather than restricting our method to", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 712 + ], + "score": 1.0, + "content": "only use two frames from a video, we generalize the NCE technique to use many images from a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "single video, resulting in more computational reuse and a better final representation (Bachman et al.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 720, + 483, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 483, + 733 + ], + "score": 1.0, + "content": "(2019) similarly makes multiple comparisons per pair, but each anchor has only one positive).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 534, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 335, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 336, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 336, + 95 + ], + "score": 1.0, + "content": "2.2 UNSUPERVISED LEARNING USING VIDEO CUES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 202 + ], + "lines": [ + { + "bbox": [ + 106, + 103, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 505, + 115 + ], + "score": 1.0, + "content": "In contrast with supervised learning which requires hand-labeling, self-supervised and unsupervised", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "learning acquire their labels for free. These techniques create datasets which are orders of magni-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "score": 1.0, + "content": "tude larger than comparable fully-supervised datasets. Whereas self-supervised learning requires", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "score": 1.0, + "content": "extra setup during data generation (Godard et al., 2017; Pinto et al., 2016; Schmidt et al., 2016),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "unsupervised learning can use existing data without any specific data generation constraints. Un-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "supervised single image methods such as auto-encoders (Kramer, 1991), colorization (Zhang et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 182 + ], + "score": 1.0, + "content": "2016), GANs (Radford et al., 2015), jigsaw (Noroozi & Favaro, 2016), and NCE (Wu et al., 2018)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "score": 1.0, + "content": "rely on properties of the images themselves and can be applied to arbitrary datasets. However these", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "datasets cannot represent temporal information, nor can they show novel object views or occlusions.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 221 + ], + "score": 1.0, + "content": "Video data automatically provides temporal cohesion which can be used as additional supervisory", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "signal to learn these phenomena. There is a long history of using videos for low level (Li et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "score": 1.0, + "content": "2019; Meister et al., 2018; Srivastava et al., 2015) and high-level tasks (Misra et al., 2016; Wang", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "score": 1.0, + "content": "et al., 2019). One of the most common unsupervised setups is using the present to predict the future.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "The Natural Language Processing community has embraced language modeling as an unsupervised", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "task which has resulted in numerous breakthroughs (Devlin et al., 2019; Mikolov et al., 2013; Rad-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "ford et al., 2019). However, similar systems applied to unlabeled videos have not revolutionized", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "computer vision. These representations still underperform supervised methods due to several issues.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "Primarily, neighboring video frames do not change nearly as much as neighboring words in a sen-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 306, + 504, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 504, + 318 + ], + "score": 1.0, + "content": "tence, so a network which learns the identity function would perform well at next frame prediction.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "Additionally, words are reused and can thus be tokenized in an effective way whereas images never", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 329, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 329, + 341 + ], + "score": 1.0, + "content": "repeat, especially between two disparate video sources.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "To avoid these issues, many have opted for other methods. Anand et al. (2019) use the NCE loss", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "to discriminate between temporally near frames and temporally far frames of ATARI gameplay but", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "do not compare across games. Han et al. (2019) also use the NCE loss and the CPC technique on a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "3D-ResNet to learn spatio-temporal features. Similar to our proposal, Tschannen et al. (2020) and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "Purushwalkam & Gupta (2020) advocates using videos to learn invariances to deformations, color", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "changes, occlusions, and other difficult scenarios which preserve the semantic relationships of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "video subjects but drastically change the appearance. These works provide additional evidence that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 422, + 414, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 414, + 434 + ], + "score": 1.0, + "content": "videos offer a richer visual backing as compared with single image methods.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 504, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 504, + 451 + ], + "score": 1.0, + "content": "Aside from the NCE approach, others have proposed alternative video training tasks. Misra et al.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 448, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 463 + ], + "score": 1.0, + "content": "(2016) shuffle the frames of a video and train a network to predict whether they are correctly tempo-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "rally ordered. Wang Wang et al. (2019) and Vondrick et al. (2018) use cycle consistency and color", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 470, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 485 + ], + "score": 1.0, + "content": "as a form of tracking from one frame onto another. Earlier work from Wang & Gupta (2015) uses", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "hand-crafted features to track patches of a video and learn similarities between the patches. Others", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "have explored learning representations from multi-modal inputs provided by video such as optical", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "flow or audio Korbar et al. (2018); Owens & Efros (2018); Piergiovanni et al. (2020); Zhao et al.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "(2018). Our approach is inspired by these works but focuses on learning a semantic representation", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "of the entire scene based on a single frame from a video. Many of these prior works require multiple", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "frames for their representation, and thus cannot be used on single-image downstream tasks. How-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "ever, if a network can consistently represent visually dissimilar images from the same video with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "similar vectors, then not only has it learned how to recognize what is in each image, but it can also", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 571, + 360, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 360, + 583 + ], + "score": 1.0, + "content": "represent what might happen in the past or future of that scene.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 596, + 178, + 609 + ], + "lines": [ + { + "bbox": [ + 104, + 595, + 181, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 181, + 612 + ], + "score": 1.0, + "content": "3 METHODS", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 108, + 621, + 504, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "In order to learn a semantically meaningful representation, we exploit the natural augmentations", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "provided by unlabeled videos. In this section, we first outline the dataset generation process. We", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 642, + 381, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 381, + 656 + ], + "score": 1.0, + "content": "then describe the learning algorithm used to train our representation.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45 + }, + { + "type": "title", + "bbox": [ + 107, + 667, + 171, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 173, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 173, + 680 + ], + "score": 1.0, + "content": "3.1 DATASET", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "Using ImageNet as a basis for representation learning has shown remarkable success both with", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "supervised pretraining as well as unsupervised learning. However, even without labels, the images of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 107, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 107, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "ImageNet have been hand selected and are unnaturally balanced. To improve learned representations", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "using existing techniques may require significantly larger datasets (He et al., 2020), but obtaining", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 335, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 336, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 336, + 95 + ], + "score": 1.0, + "content": "2.2 UNSUPERVISED LEARNING USING VIDEO CUES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 102, + 505, + 202 + ], + "lines": [ + { + "bbox": [ + 106, + 103, + 505, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 103, + 505, + 115 + ], + "score": 1.0, + "content": "In contrast with supervised learning which requires hand-labeling, self-supervised and unsupervised", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 504, + 127 + ], + "score": 1.0, + "content": "learning acquire their labels for free. These techniques create datasets which are orders of magni-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 124, + 506, + 138 + ], + "score": 1.0, + "content": "tude larger than comparable fully-supervised datasets. Whereas self-supervised learning requires", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 135, + 505, + 149 + ], + "score": 1.0, + "content": "extra setup during data generation (Godard et al., 2017; Pinto et al., 2016; Schmidt et al., 2016),", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "spans": [ + { + "bbox": [ + 106, + 147, + 505, + 159 + ], + "score": 1.0, + "content": "unsupervised learning can use existing data without any specific data generation constraints. Un-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 171 + ], + "score": 1.0, + "content": "supervised single image methods such as auto-encoders (Kramer, 1991), colorization (Zhang et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 168, + 505, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 168, + 505, + 182 + ], + "score": 1.0, + "content": "2016), GANs (Radford et al., 2015), jigsaw (Noroozi & Favaro, 2016), and NCE (Wu et al., 2018)", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 505, + 193 + ], + "score": 1.0, + "content": "rely on properties of the images themselves and can be applied to arbitrary datasets. However these", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "spans": [ + { + "bbox": [ + 105, + 191, + 505, + 204 + ], + "score": 1.0, + "content": "datasets cannot represent temporal information, nor can they show novel object views or occlusions.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 103, + 506, + 204 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 208, + 505, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 207, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 207, + 505, + 221 + ], + "score": 1.0, + "content": "Video data automatically provides temporal cohesion which can be used as additional supervisory", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 505, + 231 + ], + "score": 1.0, + "content": "signal to learn these phenomena. There is a long history of using videos for low level (Li et al.,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "score": 1.0, + "content": "2019; Meister et al., 2018; Srivastava et al., 2015) and high-level tasks (Misra et al., 2016; Wang", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 505, + 254 + ], + "score": 1.0, + "content": "et al., 2019). One of the most common unsupervised setups is using the present to predict the future.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 506, + 264 + ], + "score": 1.0, + "content": "The Natural Language Processing community has embraced language modeling as an unsupervised", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 276 + ], + "score": 1.0, + "content": "task which has resulted in numerous breakthroughs (Devlin et al., 2019; Mikolov et al., 2013; Rad-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 505, + 285 + ], + "score": 1.0, + "content": "ford et al., 2019). However, similar systems applied to unlabeled videos have not revolutionized", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 505, + 297 + ], + "score": 1.0, + "content": "computer vision. These representations still underperform supervised methods due to several issues.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 505, + 308 + ], + "score": 1.0, + "content": "Primarily, neighboring video frames do not change nearly as much as neighboring words in a sen-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 306, + 504, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 504, + 318 + ], + "score": 1.0, + "content": "tence, so a network which learns the identity function would perform well at next frame prediction.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 105, + 316, + 505, + 330 + ], + "score": 1.0, + "content": "Additionally, words are reused and can thus be tokenized in an effective way whereas images never", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 329, + 329, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 329, + 341 + ], + "score": 1.0, + "content": "repeat, especially between two disparate video sources.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 207, + 506, + 341 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 345, + 505, + 434 + ], + "lines": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 505, + 357 + ], + "score": 1.0, + "content": "To avoid these issues, many have opted for other methods. Anand et al. (2019) use the NCE loss", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 505, + 369 + ], + "score": 1.0, + "content": "to discriminate between temporally near frames and temporally far frames of ATARI gameplay but", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "do not compare across games. Han et al. (2019) also use the NCE loss and the CPC technique on a", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "3D-ResNet to learn spatio-temporal features. Similar to our proposal, Tschannen et al. (2020) and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 505, + 401 + ], + "score": 1.0, + "content": "Purushwalkam & Gupta (2020) advocates using videos to learn invariances to deformations, color", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 400, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 505, + 412 + ], + "score": 1.0, + "content": "changes, occlusions, and other difficult scenarios which preserve the semantic relationships of the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 506, + 424 + ], + "score": 1.0, + "content": "video subjects but drastically change the appearance. These works provide additional evidence that", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 422, + 414, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 414, + 434 + ], + "score": 1.0, + "content": "videos offer a richer visual backing as compared with single image methods.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 25.5, + "bbox_fs": [ + 105, + 345, + 506, + 434 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 439, + 505, + 581 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 504, + 451 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 504, + 451 + ], + "score": 1.0, + "content": "Aside from the NCE approach, others have proposed alternative video training tasks. Misra et al.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 448, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 505, + 463 + ], + "score": 1.0, + "content": "(2016) shuffle the frames of a video and train a network to predict whether they are correctly tempo-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 474 + ], + "score": 1.0, + "content": "rally ordered. Wang Wang et al. (2019) and Vondrick et al. (2018) use cycle consistency and color", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 470, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 105, + 470, + 505, + 485 + ], + "score": 1.0, + "content": "as a form of tracking from one frame onto another. Earlier work from Wang & Gupta (2015) uses", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 505, + 495 + ], + "score": 1.0, + "content": "hand-crafted features to track patches of a video and learn similarities between the patches. Others", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 505, + 506 + ], + "score": 1.0, + "content": "have explored learning representations from multi-modal inputs provided by video such as optical", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "flow or audio Korbar et al. (2018); Owens & Efros (2018); Piergiovanni et al. (2020); Zhao et al.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "(2018). Our approach is inspired by these works but focuses on learning a semantic representation", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 505, + 539 + ], + "score": 1.0, + "content": "of the entire scene based on a single frame from a video. Many of these prior works require multiple", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "frames for their representation, and thus cannot be used on single-image downstream tasks. How-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 548, + 505, + 561 + ], + "score": 1.0, + "content": "ever, if a network can consistently represent visually dissimilar images from the same video with", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "similar vectors, then not only has it learned how to recognize what is in each image, but it can also", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 571, + 360, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 571, + 360, + 583 + ], + "score": 1.0, + "content": "represent what might happen in the past or future of that scene.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 36, + "bbox_fs": [ + 105, + 439, + 506, + 583 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 596, + 178, + 609 + ], + "lines": [ + { + "bbox": [ + 104, + 595, + 181, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 595, + 181, + 612 + ], + "score": 1.0, + "content": "3 METHODS", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 43 + }, + { + "type": "text", + "bbox": [ + 108, + 621, + 504, + 654 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 634 + ], + "score": 1.0, + "content": "In order to learn a semantically meaningful representation, we exploit the natural augmentations", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "provided by unlabeled videos. In this section, we first outline the dataset generation process. We", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 642, + 381, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 642, + 381, + 656 + ], + "score": 1.0, + "content": "then describe the learning algorithm used to train our representation.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 621, + 506, + 656 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 667, + 171, + 678 + ], + "lines": [ + { + "bbox": [ + 105, + 666, + 173, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 173, + 680 + ], + "score": 1.0, + "content": "3.1 DATASET", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 108, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "Using ImageNet as a basis for representation learning has shown remarkable success both with", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "supervised pretraining as well as unsupervised learning. However, even without labels, the images of", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 107, + 711, + 505, + 721 + ], + "spans": [ + { + "bbox": [ + 107, + 711, + 505, + 721 + ], + "score": 1.0, + "content": "ImageNet have been hand selected and are unnaturally balanced. To improve learned representations", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "using existing techniques may require significantly larger datasets (He et al., 2020), but obtaining", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 49.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 80, + 503, + 133 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 80, + 503, + 133 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 80, + 503, + 133 + ], + "spans": [ + { + "bbox": [ + 108, + 80, + 503, + 133 + ], + "score": 0.974, + "html": "
DatasetNumber of Images (Train)Number of Videos (Train)Number of CategoriesMean Image Size
ImageNet1K (Deng et al.,2009)1.3 M01000(428,406)
YouTube 8M(Abu-El-Haija et al.,2016)03.7M3862
Kinetics 40o (Kay et al.,2017)0220K400
GOT-10k (Huang et al., 2019)1.4M9K563(1600,912)
Random Related Video Views (Ours)0.96M240K-(467,280)
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While we have neither the most", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "score": 1.0, + "content": "images nor the most videos, we provide good diversity between videos which is crucial for learning", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 263, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 263, + 172 + ], + "score": 1.0, + "content": "a strong, generic image representation.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 108, + 176, + 503, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "data with similar properties automatically and at scale is not practical. Instead, we turn to unlabeled", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 187, + 285, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 187, + 285, + 199 + ], + "score": 1.0, + "content": "videos as a source of additional supervision.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 108, + 204, + 505, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 106, + 204, + 505, + 216 + ], + "score": 1.0, + "content": "In order to train on a diverse set of realistic video frames, we collect a new dataset which we call", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 505, + 228 + ], + "score": 1.0, + "content": "Random Related Video Views (R2V2). We use the following fast and automated procedure to gen-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 226, + 232, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 226, + 232, + 239 + ], + "score": 1.0, + "content": "erate the images in our dataset.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 129, + 246, + 504, + 306 + ], + "lines": [ + { + "bbox": [ + 129, + 246, + 504, + 259 + ], + "spans": [ + { + "bbox": [ + 129, + 246, + 504, + 259 + ], + "score": 1.0, + "content": "1. Use YouTube Search to find videos for a set of queries, and download the top K videos", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 257, + 477, + 269 + ], + "spans": [ + { + "bbox": [ + 141, + 257, + 477, + 269 + ], + "score": 1.0, + "content": "licensed under the Creative Commons. In practice we use the ImageNet 1K classes.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 129, + 270, + 287, + 282 + ], + "spans": [ + { + "bbox": [ + 129, + 270, + 287, + 282 + ], + "score": 1.0, + "content": "2. Filter out videos with static images.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 129, + 282, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 129, + 282, + 328, + 296 + ], + "score": 1.0, + "content": "3. Pick a random point in the video and extract", + "type": "text" + }, + { + "bbox": [ + 328, + 283, + 337, + 293 + ], + "score": 0.81, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 282, + 428, + 296 + ], + "score": 1.0, + "content": "images with a gap of", + "type": "text" + }, + { + "bbox": [ + 428, + 283, + 437, + 293 + ], + "score": 0.82, + "content": "G", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 282, + 505, + 296 + ], + "score": 1.0, + "content": "frames between", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 294, + 324, + 306 + ], + "spans": [ + { + "bbox": [ + 141, + 294, + 237, + 306 + ], + "score": 1.0, + "content": "each image. 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Because", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "our dataset is constructed automatically, we can easily gather more data (more frames per video,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 336, + 505, + 348 + ], + "score": 1.0, + "content": "more videos overall). In this work we limit the scale to roughly that of comparable datasets (see", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 347, + 388, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 388, + 359 + ], + "score": 1.0, + "content": "Table 1). 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DatasetNumber of Images (Train)Number of Videos (Train)Number of CategoriesMean Image Size
ImageNet1K (Deng et al.,2009)1.3 M01000(428,406)
YouTube 8M(Abu-El-Haija et al.,2016)03.7M3862
Kinetics 40o (Kay et al.,2017)0220K400
GOT-10k (Huang et al., 2019)1.4M9K563(1600,912)
Random Related Video Views (Ours)0.96M240K-(467,280)
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While we have neither the most", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "score": 1.0, + "content": "images nor the most videos, we provide good diversity between videos which is crucial for learning", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 160, + 263, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 263, + 172 + ], + "score": 1.0, + "content": "a strong, generic image representation.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 138, + 505, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 176, + 503, + 199 + ], + "lines": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "data with similar properties automatically and at scale is not practical. 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More details on data collection are provided in Appendix 1.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 313, + 505, + 359 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 371, + 361, + 383 + ], + "lines": [ + { + "bbox": [ + 104, + 369, + 361, + 386 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 361, + 386 + ], + "score": 1.0, + "content": "3.2 NOISE CONTRASTIVE ESTIMATION (NCE) LEARNING", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 447 + ], + "lines": [ + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "Given a dataset of diverse video frames, we learn a representation which takes advantage of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 505, + 415 + ], + "score": 1.0, + "content": "structure of the data. We choose the Noise Contrastive Estimation technique (Gutmann & Hyväri-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "nen, 2010) which has been popular in many recent works (Bachman et al., 2019; Chen et al., 2020a;", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "He et al., 2020; Hénaff et al., 2019; Hjelm et al., 2019; Misra & Maaten, 2020; Tian et al., 2019; Wu", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 435, + 314, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 314, + 448 + ], + "score": 1.0, + "content": "et al., 2018), augmented with temporal supervision.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 392, + 505, + 448 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 452, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 106, + 453, + 504, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 504, + 464 + ], + "score": 1.0, + "content": "The standard NCE implementation (used in (Chen et al., 2020a; Bachman et al., 2019; Hjelm et al.,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 463, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 105, + 463, + 382, + 476 + ], + "score": 1.0, + "content": "2019)) uses the following procedure. First, a batch of anchor images", + "type": "text" + }, + { + "bbox": [ + 383, + 464, + 391, + 474 + ], + "score": 0.68, + "content": "A", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 463, + 505, + 476 + ], + "score": 1.0, + "content": "is selected. Second, a batch", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 474, + 506, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 182, + 487 + ], + "score": 1.0, + "content": "of positive images", + "type": "text" + }, + { + "bbox": [ + 182, + 475, + 191, + 484 + ], + "score": 0.81, + "content": "P", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 474, + 506, + 487 + ], + "score": 1.0, + "content": "is selected, one for each anchor. Positive matches for one example are reused", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 485, + 506, + 498 + ], + "score": 1.0, + "content": "as negatives for the other samples without the need to recompute the features. The NCE loss for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 290, + 509 + ], + "score": 1.0, + "content": "a single batch is shown in Equation 1 where", + "type": "text" + }, + { + "bbox": [ + 290, + 496, + 336, + 509 + ], + "score": 0.92, + "content": "s i m ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "is any similarity metric between the two", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 506, + 520 + ], + "score": 1.0, + "content": "inputs. This similarity is in a learned feature space rather than directly in image space, and this", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 517, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 104, + 517, + 506, + 532 + ], + "score": 1.0, + "content": "learned feature encoding is what we use on downstream tasks. The NCE Loss produces a strong", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 529, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 104, + 529, + 506, + 542 + ], + "score": 1.0, + "content": "representational encoding for inputs by maximizing the similarity of the encoding of positive pairs", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 506, + 554 + ], + "score": 1.0, + "content": "while minimizing similarity of negative ones, encouraging the encoder to be discriminative over", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "many possible matches, but general in that it must work on any inputs. As in other works, we use", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 562, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 106, + 562, + 505, + 575 + ], + "score": 1.0, + "content": "the dot product of the feature embeddings of the data points (as seen in Equation 2) as the similarity", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 586 + ], + "score": 1.0, + "content": "metric due to its computational efficiency (Bachman et al., 2019; Chen et al., 2020a; He et al.,", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 583, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 239, + 597 + ], + "score": 1.0, + "content": "2020; Hénaff et al., 2019; Misra", + "type": "text" + }, + { + "bbox": [ + 239, + 585, + 248, + 594 + ], + "score": 0.27, + "content": "\\&", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 583, + 506, + 597 + ], + "score": 1.0, + "content": "Maaten, 2020; Wu et al., 2018). 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Middle: NCE using “Multi-Frame” where positive pairs come from the same video. Right:", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "Multi-Frame Multi-Pair NCE which uses more than one positive pair per video. The gray boxes", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 236, + 452, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 452, + 248 + ], + "score": 1.0, + "content": "indicate the true match pairs. The Memory Bank adds more negatives for each anchor.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 106, + 259, + 505, + 338 + ], + "lines": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 505, + 272 + ], + "score": 1.0, + "content": "Misra & Maaten, 2020; Wu et al., 2018). We refer to this as “Same Frame.” We differ from these", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 272, + 505, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 272, + 505, + 283 + ], + "score": 1.0, + "content": "works by using multiple images from a single video to form our pairs. This allows our network to", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 106, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "see truly different views, deformations, similar objects, and larger scene changes. Additionally, this", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 293, + 504, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 504, + 306 + ], + "score": 1.0, + "content": "encodes temporal consistency as the semantic contents of a video are unlikely to change suddenly.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 505, + 317 + ], + "score": 1.0, + "content": "For example in a video of two cats playing, the camera may focus on one cat, or may even pan to", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "a previously unseen dog, but it is unlikely to pan to a shark. Note that in practice, we select frames", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 326, + 448, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 448, + 339 + ], + "score": 1.0, + "content": "with replacement, making our potential pairs a strict superset of those in prior works.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 356, + 333, + 368 + ], + "lines": [ + { + "bbox": [ + 106, + 356, + 334, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 334, + 369 + ], + "score": 1.0, + "content": "3.4 MEMORY BANKS AND MOMENTUM CONTRAST", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 435 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 392 + ], + "score": 1.0, + "content": "NCE-based methods benefit greatly from large pools of negatives because this increases the like-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 403 + ], + "score": 1.0, + "content": "lihood of finding at least one hard negative for each positive example. 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We can further increase the number", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 629, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 106, + 629, + 387, + 642 + ], + "score": 1.0, + "content": "of positives per batch (while holding batch size constant) by selecting", + "type": "text" + }, + { + "bbox": [ + 387, + 631, + 394, + 640 + ], + "score": 0.63, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 629, + 440, + 642 + ], + "score": 1.0, + "content": "videos and", + "type": "text" + }, + { + "bbox": [ + 441, + 630, + 447, + 639 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 629, + 505, + 642 + ], + "score": 1.0, + "content": "samples from", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 103, + 637, + 508, + 658 + ], + "spans": [ + { + "bbox": [ + 103, + 637, + 180, + 658 + ], + "score": 1.0, + "content": "each video where", + "type": "text" + }, + { + "bbox": [ + 180, + 640, + 209, + 653 + ], + "score": 0.93, + "content": "\\textstyle k = { \\frac { n } { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 637, + 508, + 658 + ], + "score": 1.0, + "content": ". By computing the pairwise similarity between each pair, we reuse each", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 103, + 651, + 508, + 673 + ], + "spans": [ + { + "bbox": [ + 103, + 651, + 171, + 673 + ], + "score": 1.0, + "content": "positive sample", + "type": "text" + }, + { + "bbox": [ + 171, + 655, + 178, + 665 + ], + "score": 0.8, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 651, + 254, + 673 + ], + "score": 1.0, + "content": "times, resulting in", + "type": "text" + }, + { + "bbox": [ + 255, + 652, + 297, + 668 + ], + "score": 0.93, + "content": "\\textstyle k ^ { 2 } v = { \\frac { n ^ { 2 } } { v } }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 651, + 508, + 673 + ], + "score": 1.0, + "content": "positives per batch. Using a simple block-diagonal", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "mask as shown in Figure 2 we can efficiently compute the similarities and NCE loss both between", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "elements of the batch and across a memory bank, achieving a large number of positive comparisons", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "per batch while retaining a large negative size. In practice, we notice no meaningful computational", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "cost to this approach. We refer to this full method using Multi-Pair on video data and the Multi-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Frame learning procedure as Video Noise Contrastive Estimation (VINCE). See Appendix 2 for", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 720, + 192, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 192, + 733 + ], + "score": 1.0, + "content": "detailed pseudocode.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 35, + "bbox_fs": [ + 103, + 608, + 508, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 201, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 201, + 96 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 106, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "We evaluate our method (VINCE) on both single image and temporal tasks by freezing our learned", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "representation and adding a small network (in most cases a single linear layer) for adaptation to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 505, + 140 + ], + "score": 1.0, + "content": "new end-tasks. Our learned representation transfers well to a variety of visual tasks, especially tasks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 140, + 477, + 151 + ], + "spans": [ + { + "bbox": [ + 107, + 140, + 477, + 151 + ], + "score": 1.0, + "content": "which require temporal reasoning. To show this, we compare with multiple strong baselines:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 132, + 160, + 504, + 219 + ], + "lines": [ + { + "bbox": [ + 131, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 131, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "• MoCo-IN: Network pretrained on ImageNet (Deng et al., 2009) using the MoCo algorithm.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 131, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 131, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "• MoCo-R2V2: Network pretrained on R2V2 using the MoCo algorithm. This uses exactly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 184, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 142, + 184, + 506, + 197 + ], + "score": 1.0, + "content": "the same data as VINCE but the Same Frame technique described in Sec. 3.3. We also", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 141, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "prevent multiple images from the same video being in the Memory Bank at the same time.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 131, + 208, + 382, + 221 + ], + "spans": [ + { + "bbox": [ + 131, + 208, + 382, + 221 + ], + "score": 1.0, + "content": "• Sup-IN: Network pretrained on fully supervised ImageNet.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 229, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "score": 1.0, + "content": "To validate the benefits of learning from videos, we additionally compare against an unsupervised", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 241, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 252 + ], + "score": 1.0, + "content": "image dataset. Since ImageNet itself required time, effort, and money to create, we construct a new", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "static image dataset analogous to our video dataset. Specifically, we search Google Images for the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "ImageNet synsets and download the top K results for each category. We refer to this as MoCo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "G). Although other unsupervised video-based methods exist, they have generally underperformed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 419, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 419, + 297 + ], + "score": 1.0, + "content": "pretrained supervised ImageNet weights or are only applicable to video tasks.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 309, + 197, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 198, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 198, + 322 + ], + "score": 1.0, + "content": "4.1 TARGET TASKS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "We compare each method on several diverse end-tasks using both ResNet18 (He et al., 2016) and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "ResNet50 backbones. More training implementation details can be found in Appendix 3. Results", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "for these tasks are shown in Table 2. We train all end-task models using Adam (Kingma & Ba, 2014)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "and a shared learning rate schedule per task. For each dataset, we use standard data augmentation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "approaches (crop, flip, color jitter except for tracking). One overall trend to note is the relative gain", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "over MoCo-R2V2. If the single-frame algorithm performed as well as our multi-frame method on", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "temporal tasks, it would indicate minimal temporal understanding. However, the relative gain of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 262, + 420 + ], + "score": 1.0, + "content": "VINCE over MoCo-R2V2 on Kinetics", + "type": "text" + }, + { + "bbox": [ + 263, + 407, + 300, + 418 + ], + "score": 0.85, + "content": "( 1 3 . 8 5 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 406, + 357, + 420 + ], + "score": 1.0, + "content": "and tracking (", + "type": "text" + }, + { + "bbox": [ + 357, + 407, + 389, + 418 + ], + "score": 0.82, + "content": "1 3 . 3 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 406, + 407, + 420 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 407, + 407, + 439, + 418 + ], + "score": 0.84, + "content": "1 5 . 3 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 406, + 505, + 420 + ], + "score": 1.0, + "content": ") are higher than", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 418, + 447, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 447, + 431 + ], + "score": 1.0, + "content": "those on single-frame tasks, showing that our method can incorporate temporal cues.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "score": 1.0, + "content": "ImageNet: For this task, we use our frozen learned representations, adding a single linear layer after", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 446, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 506, + 458 + ], + "score": 1.0, + "content": "the global average pool. Although none of the methods match the fully supervised performance of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 456, + 507, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 507, + 470 + ], + "score": 1.0, + "content": "ResNet on ImageNet, they do achieve reasonable performance given only single linear layer. It", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "is unsurprising that MoCo pretrained on ImageNet images (MoCo-IN) outperforms our method", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "(0.447 vs 0.400) due to the domain shift between pretrain and end-task. However MoCo pretrained", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 279, + 501 + ], + "score": 1.0, + "content": "on R2V2 (MoCo-R2V2) suffers nearly a", + "type": "text" + }, + { + "bbox": [ + 279, + 490, + 294, + 500 + ], + "score": 0.9, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 489, + 369, + 501 + ], + "score": 1.0, + "content": "drop in accuracy", + "type": "text" + }, + { + "bbox": [ + 370, + 489, + 398, + 501 + ], + "score": 0.86, + "content": "( 8 . 9 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "compared to our method", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 107, + 501, + 135, + 512 + ], + "score": 0.86, + "content": "( 4 . 7 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 500, + 506, + 514 + ], + "score": 1.0, + "content": ", indicating that pretraining on multi-frame matching provides a clear benefit over single", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "frame pairs. Our method, which has never seen an image from ImageNet before, still learns a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "score": 1.0, + "content": "representation which generalizes well to this new type of data. Even drawing images from a similar", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 533, + 356, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 356, + 545 + ], + "score": 1.0, + "content": "class distribution (MoCo-G) does not outperform our method.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "SUN Scenes: SUN Scenes is a classification dataset in which each image is categorized into one", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 560, + 507, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 507, + 574 + ], + "score": 1.0, + "content": "of 397 possible scene types such as airplane cabin, bedroom, and coffee shop. Again we train a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "single linear layer on top of each pretrained network. This data is quite similar to ImageNet in that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "each image is well-curated and contains single, unambiguous subjects. As such, the ImageNet fully", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "supervised baseline transfers quite well to SUN Scenes. However VINCE outperforms Sup-IN by a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "small margin. Again we note a large improvement of VINCE over MoCo-R2V2 (0.495 vs. 0.450).", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "This shows that our method learns to recognize not just the main subject of an image but also the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 627, + 390, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 390, + 639 + ], + "score": 1.0, + "content": "surrounding scene, which requires a richer understanding of the world.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5 + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "Kinetics 400: This dataset consists of 10 second clips from YouTube videos and action labels for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "each segment. We first download each video and subsample each clip to one frame per second (10", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "frames per clip). We train a single layer LSTM (Hochreiter & Schmidhuber, 1997) followed by a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "single linear layer to predict the action category for each segment. Kinetics acts as a crucial test", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "to evaluate whether our model learns temporal cues. VINCE greatly outperform all other methods,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "whereas traditional baselines such as fine-tuning supervised ImageNet do not adapt well at all. This", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "shows that contrary to popular belief, representations pretrained on ImageNet many not be a good", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 337, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 337, + 733 + ], + "score": 1.0, + "content": "fit for other visual domains, especially on temporal tasks.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 200, + 93 + ], + "lines": [ + { + "bbox": [ + 105, + 80, + 201, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 80, + 201, + 96 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 106, + 505, + 151 + ], + "lines": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 505, + 119 + ], + "score": 1.0, + "content": "We evaluate our method (VINCE) on both single image and temporal tasks by freezing our learned", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 506, + 130 + ], + "score": 1.0, + "content": "representation and adding a small network (in most cases a single linear layer) for adaptation to", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 129, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 129, + 505, + 140 + ], + "score": 1.0, + "content": "new end-tasks. Our learned representation transfers well to a variety of visual tasks, especially tasks", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 107, + 140, + 477, + 151 + ], + "spans": [ + { + "bbox": [ + 107, + 140, + 477, + 151 + ], + "score": 1.0, + "content": "which require temporal reasoning. To show this, we compare with multiple strong baselines:", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 105, + 506, + 151 + ] + }, + { + "type": "text", + "bbox": [ + 132, + 160, + 504, + 219 + ], + "lines": [ + { + "bbox": [ + 131, + 160, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 131, + 160, + 505, + 173 + ], + "score": 1.0, + "content": "• MoCo-IN: Network pretrained on ImageNet (Deng et al., 2009) using the MoCo algorithm.", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 131, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 131, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "• MoCo-R2V2: Network pretrained on R2V2 using the MoCo algorithm. This uses exactly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 142, + 184, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 142, + 184, + 506, + 197 + ], + "score": 1.0, + "content": "the same data as VINCE but the Same Frame technique described in Sec. 3.3. We also", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 196, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 141, + 196, + 505, + 208 + ], + "score": 1.0, + "content": "prevent multiple images from the same video being in the Memory Bank at the same time.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 131, + 208, + 382, + 221 + ], + "spans": [ + { + "bbox": [ + 131, + 208, + 382, + 221 + ], + "score": 1.0, + "content": "• Sup-IN: Network pretrained on fully supervised ImageNet.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 131, + 160, + 506, + 221 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 229, + 505, + 295 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 242 + ], + "score": 1.0, + "content": "To validate the benefits of learning from videos, we additionally compare against an unsupervised", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 241, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 106, + 241, + 505, + 252 + ], + "score": 1.0, + "content": "image dataset. Since ImageNet itself required time, effort, and money to create, we construct a new", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 505, + 264 + ], + "score": 1.0, + "content": "static image dataset analogous to our video dataset. Specifically, we search Google Images for the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 105, + 262, + 505, + 275 + ], + "score": 1.0, + "content": "ImageNet synsets and download the top K results for each category. We refer to this as MoCo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "G). Although other unsupervised video-based methods exist, they have generally underperformed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 284, + 419, + 297 + ], + "spans": [ + { + "bbox": [ + 105, + 284, + 419, + 297 + ], + "score": 1.0, + "content": "pretrained supervised ImageNet weights or are only applicable to video tasks.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 228, + 505, + 297 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 309, + 197, + 320 + ], + "lines": [ + { + "bbox": [ + 106, + 309, + 198, + 322 + ], + "spans": [ + { + "bbox": [ + 106, + 309, + 198, + 322 + ], + "score": 1.0, + "content": "4.1 TARGET TASKS", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 330, + 505, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 505, + 342 + ], + "score": 1.0, + "content": "We compare each method on several diverse end-tasks using both ResNet18 (He et al., 2016) and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 353 + ], + "score": 1.0, + "content": "ResNet50 backbones. More training implementation details can be found in Appendix 3. Results", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 505, + 365 + ], + "score": 1.0, + "content": "for these tasks are shown in Table 2. We train all end-task models using Adam (Kingma & Ba, 2014)", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 376 + ], + "score": 1.0, + "content": "and a shared learning rate schedule per task. For each dataset, we use standard data augmentation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 506, + 387 + ], + "score": 1.0, + "content": "approaches (crop, flip, color jitter except for tracking). One overall trend to note is the relative gain", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "over MoCo-R2V2. If the single-frame algorithm performed as well as our multi-frame method on", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 408 + ], + "score": 1.0, + "content": "temporal tasks, it would indicate minimal temporal understanding. However, the relative gain of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 406, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 262, + 420 + ], + "score": 1.0, + "content": "VINCE over MoCo-R2V2 on Kinetics", + "type": "text" + }, + { + "bbox": [ + 263, + 407, + 300, + 418 + ], + "score": 0.85, + "content": "( 1 3 . 8 5 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 300, + 406, + 357, + 420 + ], + "score": 1.0, + "content": "and tracking (", + "type": "text" + }, + { + "bbox": [ + 357, + 407, + 389, + 418 + ], + "score": 0.82, + "content": "1 3 . 3 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 406, + 407, + 420 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 407, + 407, + 439, + 418 + ], + "score": 0.84, + "content": "1 5 . 3 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 406, + 505, + 420 + ], + "score": 1.0, + "content": ") are higher than", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 418, + 447, + 431 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 447, + 431 + ], + "score": 1.0, + "content": "those on single-frame tasks, showing that our method can incorporate temporal cues.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 330, + 506, + 431 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 434, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 106, + 435, + 505, + 446 + ], + "score": 1.0, + "content": "ImageNet: For this task, we use our frozen learned representations, adding a single linear layer after", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 446, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 446, + 506, + 458 + ], + "score": 1.0, + "content": "the global average pool. Although none of the methods match the fully supervised performance of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 456, + 507, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 507, + 470 + ], + "score": 1.0, + "content": "ResNet on ImageNet, they do achieve reasonable performance given only single linear layer. It", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "is unsurprising that MoCo pretrained on ImageNet images (MoCo-IN) outperforms our method", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "(0.447 vs 0.400) due to the domain shift between pretrain and end-task. However MoCo pretrained", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 489, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 279, + 501 + ], + "score": 1.0, + "content": "on R2V2 (MoCo-R2V2) suffers nearly a", + "type": "text" + }, + { + "bbox": [ + 279, + 490, + 294, + 500 + ], + "score": 0.9, + "content": "2 \\times", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 489, + 369, + 501 + ], + "score": 1.0, + "content": "drop in accuracy", + "type": "text" + }, + { + "bbox": [ + 370, + 489, + 398, + 501 + ], + "score": 0.86, + "content": "( 8 . 9 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 489, + 506, + 501 + ], + "score": 1.0, + "content": "compared to our method", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 500, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 107, + 501, + 135, + 512 + ], + "score": 0.86, + "content": "( 4 . 7 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 135, + 500, + 506, + 514 + ], + "score": 1.0, + "content": ", indicating that pretraining on multi-frame matching provides a clear benefit over single", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "frame pairs. Our method, which has never seen an image from ImageNet before, still learns a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 506, + 535 + ], + "score": 1.0, + "content": "representation which generalizes well to this new type of data. Even drawing images from a similar", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 533, + 356, + 545 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 356, + 545 + ], + "score": 1.0, + "content": "class distribution (MoCo-G) does not outperform our method.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5, + "bbox_fs": [ + 105, + 435, + 507, + 545 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 638 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 506, + 563 + ], + "score": 1.0, + "content": "SUN Scenes: SUN Scenes is a classification dataset in which each image is categorized into one", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 560, + 507, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 507, + 574 + ], + "score": 1.0, + "content": "of 397 possible scene types such as airplane cabin, bedroom, and coffee shop. Again we train a", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 506, + 585 + ], + "score": 1.0, + "content": "single linear layer on top of each pretrained network. This data is quite similar to ImageNet in that", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "each image is well-curated and contains single, unambiguous subjects. As such, the ImageNet fully", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "supervised baseline transfers quite well to SUN Scenes. However VINCE outperforms Sup-IN by a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "small margin. Again we note a large improvement of VINCE over MoCo-R2V2 (0.495 vs. 0.450).", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "This shows that our method learns to recognize not just the main subject of an image but also the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 627, + 390, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 390, + 639 + ], + "score": 1.0, + "content": "surrounding scene, which requires a richer understanding of the world.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 39.5, + "bbox_fs": [ + 105, + 549, + 507, + 639 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 643, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 655 + ], + "score": 1.0, + "content": "Kinetics 400: This dataset consists of 10 second clips from YouTube videos and action labels for", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 666 + ], + "score": 1.0, + "content": "each segment. We first download each video and subsample each clip to one frame per second (10", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 506, + 679 + ], + "score": 1.0, + "content": "frames per clip). We train a single layer LSTM (Hochreiter & Schmidhuber, 1997) followed by a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "single linear layer to predict the action category for each segment. Kinetics acts as a crucial test", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "to evaluate whether our model learns temporal cues. VINCE greatly outperform all other methods,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "whereas traditional baselines such as fine-tuning supervised ImageNet do not adapt well at all. This", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "shows that contrary to popular belief, representations pretrained on ImageNet many not be a good", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 720, + 337, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 337, + 733 + ], + "score": 1.0, + "content": "fit for other visual domains, especially on temporal tasks.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 47.5, + "bbox_fs": [ + 105, + 644, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 81, + 503, + 214 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 81, + 503, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 81, + 503, + 214 + ], + "spans": [ + { + "bbox": [ + 108, + 81, + 503, + 214 + ], + "score": 0.982, + "html": "
Image Classification ImageNet(Deng et al.,2009)Scene Classification SUN Scenes(Xiao et al.,2010)Action Recognition Kinetics 400(Kay et al.,2017)Tracking OTB 2015(Wu et al.,2015)
Trained Layer(s) MetricLinear Accuracy (Top 1)Linear Accuracy (Top 1)LSTM→Linear Accuracy (Top 1)1x1 Conv
Sup-INPrecisionSuccess
MoCo-IN0.6960.4910.2070.5570.396
0.4470.4870.3360.5830.429
BeerMoCo-G0.3930.4440.3130.5510.413
MoCo-R2V20.3580.450 0.4950.3180.5550.403
VINCE (Ours)0.4000.3620.6290.465
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MoCo-V2-IN (our impl.)0.6520.6080.4590.3000.260
MoCo-R2V20.5360.5810.4560.3860.299
VINCE (Ours)0.5440.6110.4910.4020.300
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We show", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "score": 1.0, + "content": "improvements over MoCo trained on the same data on all tasks, and outperform MoCo trained", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 239, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 251 + ], + "score": 1.0, + "content": "on ImageNet as well as supervised pretraining on ImageNet on all tasks but ImageNet itself (and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 248, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 248, + 506, + 263 + ], + "score": 1.0, + "content": "tracking for ResNet50). Each representation uses the same ResNet convolutional backbone, sharing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 328, + 273 + ], + "score": 1.0, + "content": "weights across all tasks. 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Difficult", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 505, + 335 + ], + "score": 1.0, + "content": "tracking datasets like OTB 2015 implicitly require robustness to occlusion and appearance change", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 332, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 105, + 332, + 506, + 346 + ], + "score": 1.0, + "content": "of the target object, so improved performance would indicate a representation’s relative robustness", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 344, + 505, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 505, + 356 + ], + "score": 1.0, + "content": "to these challenges. 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OTB", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "is evaluated using two metrics – precision and success. Precision measures the percentage of frames", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "score": 1.0, + "content": "where the (normalized) center error is less than a certain threshold, averaging over many thresholds.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "Similarly, success measures the percentage of frames where the Intersection Over Union is more", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 420, + 214, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 214, + 433 + ], + "score": 1.0, + "content": "than a range of thresholds.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "A representation which works well for SiamFC would have the property that the cross correla-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "score": 1.0, + "content": "tion of two images of the same object is high, but the cross correlation of two different objects,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "or a poorly cropped image of the same object, is low. Pretraining our representations on multiple", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "frames from the same video coincides well with the first objective, however since we use cropped", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 494 + ], + "score": 1.0, + "content": "data augmentations, the representations tend to be somewhat invariant to poorly-cropped candidates.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "Still, the models perform quite well across a variety of difficult tracking instances. 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The first row is equivalent", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "to the procedure done in MoCo (He et al., 2020) i.e. the anchor and positive pairs are two data", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "augmentations of the same image. The second row uses the MoCo procedure as well, however the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "anchors and positives may be from different images from the same video. 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Note that when selecting images for row 2", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 465, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 465, + 733 + ], + "score": 1.0, + "content": "and 3, we use sampling with replacement, making our method a strict super-set of MoCo.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 81, + 503, + 214 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 81, + 503, + 214 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 81, + 503, + 214 + ], + "spans": [ + { + "bbox": [ + 108, + 81, + 503, + 214 + ], + "score": 0.982, + "html": "
Image Classification ImageNet(Deng et al.,2009)Scene Classification SUN Scenes(Xiao et al.,2010)Action Recognition Kinetics 400(Kay et al.,2017)Tracking OTB 2015(Wu et al.,2015)
Trained Layer(s) MetricLinear Accuracy (Top 1)Linear Accuracy (Top 1)LSTM→Linear Accuracy (Top 1)1x1 Conv
Sup-INPrecisionSuccess
MoCo-IN0.6960.4910.2070.5570.396
0.4470.4870.3360.5830.429
BeerMoCo-G0.3930.4440.3130.5510.413
MoCo-R2V20.3580.450 0.4950.3180.5550.403
VINCE (Ours)0.4000.3620.6290.465
Relative Gain over MoCo-R2V211.91%9.93%13.85%13.33%15.38%
PssSDSup-IN0.7620.5930.3050.4580.320
MoCo-V2-IN (our impl.)0.6520.6080.4590.3000.260
MoCo-R2V20.5360.5810.4560.3860.299
VINCE (Ours)0.5440.6110.4910.4020.300
Relative Gain over MoCo-R2V21.36%5.29%7.72%4.15%0.33%
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We show", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 506, + 240 + ], + "score": 1.0, + "content": "improvements over MoCo trained on the same data on all tasks, and outperform MoCo trained", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 239, + 506, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 506, + 251 + ], + "score": 1.0, + "content": "on ImageNet as well as supervised pretraining on ImageNet on all tasks but ImageNet itself (and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 248, + 506, + 263 + ], + "spans": [ + { + "bbox": [ + 104, + 248, + 506, + 263 + ], + "score": 1.0, + "content": "tracking for ResNet50). 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We use the SiamFC (Bertinetto et al., 2016) tracking algorithm on top of our", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 356, + 504, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 504, + 366 + ], + "score": 1.0, + "content": "learned representation. SiamFC localizes an object of interest by convolving deep features from the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 105, + 365, + 505, + 379 + ], + "score": 1.0, + "content": "initial bounding box with the current image, similar to template matching (Briechle & Hanebeck,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 505, + 389 + ], + "score": 1.0, + "content": "2001) but in deep feature space. For a more complete explanation, see (Bertinetto et al., 2016). OTB", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 400 + ], + "score": 1.0, + "content": "is evaluated using two metrics – precision and success. Precision measures the percentage of frames", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 412 + ], + "score": 1.0, + "content": "where the (normalized) center error is less than a certain threshold, averaging over many thresholds.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 506, + 423 + ], + "score": 1.0, + "content": "Similarly, success measures the percentage of frames where the Intersection Over Union is more", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 420, + 214, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 214, + 433 + ], + "score": 1.0, + "content": "than a range of thresholds.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5, + "bbox_fs": [ + 105, + 300, + 506, + 433 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 437, + 505, + 591 + ], + "lines": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "A representation which works well for SiamFC would have the property that the cross correla-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 449, + 505, + 460 + ], + "score": 1.0, + "content": "tion of two images of the same object is high, but the cross correlation of two different objects,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 505, + 471 + ], + "score": 1.0, + "content": "or a poorly cropped image of the same object, is low. Pretraining our representations on multiple", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 469, + 505, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 469, + 505, + 483 + ], + "score": 1.0, + "content": "frames from the same video coincides well with the first objective, however since we use cropped", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 480, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 105, + 480, + 505, + 494 + ], + "score": 1.0, + "content": "data augmentations, the representations tend to be somewhat invariant to poorly-cropped candidates.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 506, + 505 + ], + "score": 1.0, + "content": "Still, the models perform quite well across a variety of difficult tracking instances. Our ResNet18", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 503, + 505, + 516 + ], + "score": 1.0, + "content": "model transfers significantly better than all other methods indicating a clear benefit to using tem-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 505, + 526 + ], + "score": 1.0, + "content": "poral cues during pretraining. We find, somewhat unexpectedly, that the ResNet18 network fares", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 505, + 538 + ], + "score": 1.0, + "content": "better than the ResNet50. A likely explanation for this is that the original SiamFC method uses", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 534, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 506, + 550 + ], + "score": 1.0, + "content": "AlexNet (Krizhevsky et al., 2012) with no padding in a fully-convolutional manner. When using", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "score": 1.0, + "content": "ResNet, padding must be applied to keep the outputs the same dimensionality as the inputs. Thus,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 570 + ], + "score": 1.0, + "content": "at training time the network may latch onto zero-padding cues which will not be applicable at test", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "time. This becomes more of an issue the larger the receptive field which is why ResNet50 struggles", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 580, + 268, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 580, + 268, + 591 + ], + "score": 1.0, + "content": "but ResNet18 is somewhat less affected.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 27.5, + "bbox_fs": [ + 105, + 437, + 506, + 591 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 610, + 218, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 609, + 219, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 219, + 623 + ], + "score": 1.0, + "content": "4.2 METHOD ABLATION", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 646 + ], + "score": 1.0, + "content": "We validate the effectiveness of Multi-Frame (Sec. 3.3) and Multi-Pair (Sec. 3.5) learning by ablating", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "the number of images from each video used in a batch of comparisons shown in Table 3 (due to", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "computational constraints, we only perform ablations on ResNet18). The first row is equivalent", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "to the procedure done in MoCo (He et al., 2020) i.e. the anchor and positive pairs are two data", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "augmentations of the same image. The second row uses the MoCo procedure as well, however the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "anchors and positives may be from different images from the same video. The third row uses our", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "Multi-Pair NCE method taking 4 positives and 4 anchors from each video, resulting in 16 positive", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "pairs. A pictorial representation can be seen in Figure 2. Note that when selecting images for row 2", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 721, + 465, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 465, + 733 + ], + "score": 1.0, + "content": "and 3, we use sampling with replacement, making our method a strict super-set of MoCo.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 632, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 108, + 81, + 502, + 138 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 108, + 81, + 502, + 138 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 108, + 81, + 502, + 138 + ], + "spans": [ + { + "bbox": [ + 108, + 81, + 502, + 138 + ], + "score": 0.971, + "html": "
Images Per VideoTest Task
ImageNetSUN SceneKinetics 400OTB 2015 PrecisionOTB 2015 Success
1: Same Frame0.3580.4500.3180.5550.403
2: Multi-Frame0.3810.4780.3610.6220.464
8:Multi-Frame Multi-Pair0.4000.4950.3620.6290.465
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Pretraining DataTest Task
ImageNet SUN SceneKinetics 400OTB 2015 PrecisionOTB 2015 Success
R2V2 IN-Queries0.4000.4950.3620.6290.465
R2V2 YT8MURLs0.3670.4780.3430.6670.492
Kinetics 400 URLs0.3680.4940.3900.6120.456
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Since R2V2 uses ImageNet search queries, it outperforms the others on", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 294, + 462, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 462, + 306 + ], + "score": 1.0, + "content": "ImageNet. Similarly, pretraining on Kinetics increases performance on the Kinetics task.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 318, + 505, + 428 + ], + "lines": [ + { + "bbox": [ + 106, + 318, + 504, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 504, + 330 + ], + "score": 1.0, + "content": "We observe across-the-board improvements from both modifications to the MoCo approach. The", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 330, + 504, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 504, + 341 + ], + "score": 1.0, + "content": "majority of the improvement comes from using two non-identical frames for matching, but we still", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 505, + 353 + ], + "score": 1.0, + "content": "gain an additional improvement from using Multi-Pair NCE. Our intuition is that using the Multi-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 352, + 504, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 352, + 504, + 363 + ], + "score": 1.0, + "content": "Pair NCE creates gradients that pull each feature towards a global video representation whereas the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 505, + 375 + ], + "score": 1.0, + "content": "standard NCE remains more instance-based, only moving a representation in one direction at a time.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 385 + ], + "score": 1.0, + "content": "Thus, we would expect the Multi-Pair NCE features to be more holistically semantic whereas the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "standard NCE may retain more uniquely identifying features. In fact, we observe a larger perfor-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "mance gap on the more semantic ImageNet and SUN Scene tasks. In contrast, because the Kinetics", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 405, + 505, + 419 + ], + "score": 1.0, + "content": "model uses an LSTM to reason over all input images at once, instance-level features are equally", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 416, + 314, + 430 + ], + "spans": [ + { + "bbox": [ + 106, + 416, + 314, + 430 + ], + "score": 1.0, + "content": "useful as global video features for overall accuracy.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 442, + 264, + 453 + ], + "lines": [ + { + "bbox": [ + 106, + 442, + 266, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 266, + 455 + ], + "score": 1.0, + "content": "4.3 PRETRAINING DATA ABLATION", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 463, + 505, + 573 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 464, + 505, + 475 + ], + "score": 1.0, + "content": "In Table 4 we explore the effect of different pretraining datasets on end-task performance. For", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "each experiment, we use VINCE but use video data from three different sources: our method of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "searching ImageNet synset queries, using the URLs from YouTube 8M (Abu-El-Haija et al., 2016),", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "and using the URLs from Kinetics 400 (Kay et al., 2017). Again, we only test ResNet18. Our", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "YouTube8M(YT8M) pretraining data uses the same filtering procedure as R2V2 and contains 5.8", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "million images from 1.4 million videos. As noted in Table 2 MoCo-IN results, using the same", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "score": 1.0, + "content": "dataset for pretraining as the end-task results in a boost in performance on that specific task but", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "does not indicate that the representation will be better on all tasks. We see this trend is true again", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "when pretraining on Kinetics data. Similarly, since R2V2 uses ImageNet synset for search queries,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 562, + 435, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 435, + 574 + ], + "score": 1.0, + "content": "pretraining on it performs better on ImageNet than the other less-aligned datasets.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 579, + 505, + 656 + ], + "lines": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 505, + 591 + ], + "score": 1.0, + "content": "In general, this would indicate that given a large enough set of diverse videos, pretraining directly on", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 506, + 603 + ], + "score": 1.0, + "content": "the unlabeled source data would result in the best performing representation on that data. If this is not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "possible, then pretraining on a large external source of data may still result in a useful representation.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "It also indicates that the VINCE method works well on a variety of different pretraining datasets.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "score": 1.0, + "content": "The increased performance on tracking when using YT8M data could be explained by it simply", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "having access to a larger number of video sources and frames. 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Images Per VideoTest Task
ImageNetSUN SceneKinetics 400OTB 2015 PrecisionOTB 2015 Success
1: Same Frame0.3580.4500.3180.5550.403
2: Multi-Frame0.3810.4780.3610.6220.464
8:Multi-Frame Multi-Pair0.4000.4950.3620.6290.465
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Pretraining DataTest Task
ImageNet SUN SceneKinetics 400OTB 2015 PrecisionOTB 2015 Success
R2V2 IN-Queries0.4000.4950.3620.6290.465
R2V2 YT8MURLs0.3670.4780.3430.6670.492
Kinetics 400 URLs0.3680.4940.3900.6120.456
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For", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 506, + 486 + ], + "score": 1.0, + "content": "each experiment, we use VINCE but use video data from three different sources: our method of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 498 + ], + "score": 1.0, + "content": "searching ImageNet synset queries, using the URLs from YouTube 8M (Abu-El-Haija et al., 2016),", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 506, + 509 + ], + "score": 1.0, + "content": "and using the URLs from Kinetics 400 (Kay et al., 2017). Again, we only test ResNet18. Our", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 520 + ], + "score": 1.0, + "content": "YouTube8M(YT8M) pretraining data uses the same filtering procedure as R2V2 and contains 5.8", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 531 + ], + "score": 1.0, + "content": "million images from 1.4 million videos. As noted in Table 2 MoCo-IN results, using the same", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 506, + 542 + ], + "score": 1.0, + "content": "dataset for pretraining as the end-task results in a boost in performance on that specific task but", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "does not indicate that the representation will be better on all tasks. We see this trend is true again", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 106, + 551, + 505, + 564 + ], + "score": 1.0, + "content": "when pretraining on Kinetics data. 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If this is not", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "possible, then pretraining on a large external source of data may still result in a useful representation.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 624 + ], + "score": 1.0, + "content": "It also indicates that the VINCE method works well on a variety of different pretraining datasets.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 636 + ], + "score": 1.0, + "content": "The increased performance on tracking when using YT8M data could be explained by it simply", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 647 + ], + "score": 1.0, + "content": "having access to a larger number of video sources and frames. For generic object tracking, class", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 645, + 479, + 657 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 479, + 657 + ], + "score": 1.0, + "content": "diversity may be less important than number of samples because the class identity is ignored.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 579, + 506, + 657 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 673, + 201, + 685 + ], + "lines": [ + { + "bbox": [ + 104, + 671, + 203, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 671, + 203, + 689 + ], + "score": 1.0, + "content": "5 CONCLUSIONS", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "In this work we introduced Video Noise Contrastive Estimation, a process for using unlabeled videos", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "to learn an unsupervised image representation. 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In Proceedings of the European Conference on Computer Vision", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 116, + 670, + 223, + 682 + ], + "spans": [ + { + "bbox": [ + 116, + 670, + 223, + 682 + ], + "score": 1.0, + "content": "(ECCV), September 2018.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 648, + 505, + 682 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 146, + 80, + 464, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 146, + 80, + 464, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 80, + 464, + 248 + ], + "spans": [ + { + "bbox": [ + 146, + 80, + 464, + 248 + ], + "score": 0.976, + "type": "image", + "image_path": "8e99c9e1888b1f509194ec70749e16697122f5048d817e2552c11a6d0f660fe6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 146, + 80, + 464, + 136.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 146, + 136.0, + 464, + 192.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 146, + 192.0, + 464, + 248.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 165, + 257, + 445, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 255, + 446, + 270 + ], + "spans": [ + { + "bbox": [ + 165, + 255, + 446, + 270 + ], + "score": 1.0, + "content": "Figure A1: Sample from Random Related Video Views (train set).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "title", + "bbox": [ + 107, + 277, + 303, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 304, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 304, + 292 + ], + "score": 1.0, + "content": "APPENDIX 1 DATASET COMPARISON", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "Random Related Video Views provides nearly 250,000 video links as well as multiple images per", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 315, + 504, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 504, + 327 + ], + "score": 1.0, + "content": "video across a broad variety of categories. We show more samples from R2V2 in Figure A1. Videos", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "each have four images 150 frames apart. Each separate video (outlined in blue) lists its correspond-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "ing YouTube link. Using ImageNet synsets for search queries provides reasonable visual diversity,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "but could be substituted with another set of queries. While we acknowledge that using YouTube’s", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "Search feature is not truly random, this procedure resulted in significantly more diverse samples than", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 370, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 382 + ], + "score": 1.0, + "content": "using existing datasets like YouTube8M (Abu-El-Haija et al., 2016) which is heavily unbalanced", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "with unnatural videos.2 We do no additional data cleaning to ensure that the videos or extracted", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "images actually contain the search term (many do not), nor do we search for “high interest” video", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "segments as in Misra et al. (2016). We also discard the search term itself as a form of supervision.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "We find that a gap of 5 seconds between each saved image typically results in visually distinct but", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "semantically related images. Too much shorter results in images which are less individually distinct,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "and too much longer may result in large and unpredictable changes. A sample from each dataset can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 447, + 257, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 257, + 458 + ], + "score": 1.0, + "content": "be seen in the supplemental material.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5 + }, + { + "type": "image", + "bbox": [ + 107, + 524, + 502, + 680 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 524, + 502, + 680 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 524, + 502, + 680 + ], + "spans": [ + { + "bbox": [ + 107, + 524, + 502, + 680 + ], + "score": 0.975, + "type": "image", + "image_path": "2a73ace7d06bf23a2fb379edcf93ff0973f48b41f0087f5b714f1f928e1f1b5e.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 107, + 524, + 502, + 576.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 107, + 576.0, + 502, + 628.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 107, + 628.0, + 502, + 680.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 690, + 505, + 724 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "Figure A2: Random sampling of pairs of images from videos in each dataset. In GOT-10k,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "sometimes different video clips are segments from the same original video as seen in the first and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 713, + 406, + 725 + ], + "spans": [ + { + "bbox": [ + 106, + 713, + 406, + 725 + ], + "score": 1.0, + "content": "second sample. Images are square cropped for visualization purposes only.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + } + ], + "index": 21.5 + } + ], + "page_idx": 11, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 470, + 504, + 501 + ], + "lines": [ + { + "bbox": [ + 117, + 468, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 117, + 468, + 506, + 484 + ], + "score": 1.0, + "content": "2The second most common category in YouTube8M, comprising 540k of the 3.7 million training videos is", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 506, + 493 + ], + "score": 1.0, + "content": "“Video Game,” and the fifth is “Cartoon” with 240k. The category “Minecraft” itself has over 57,000 videos in", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 490, + 300, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 490, + 300, + 502 + ], + "score": 1.0, + "content": "the dataset whereas the category “Pear” has only 138.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "12", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 146, + 80, + 464, + 248 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 146, + 80, + 464, + 248 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 146, + 80, + 464, + 248 + ], + "spans": [ + { + "bbox": [ + 146, + 80, + 464, + 248 + ], + "score": 0.976, + "type": "image", + "image_path": "8e99c9e1888b1f509194ec70749e16697122f5048d817e2552c11a6d0f660fe6.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 146, + 80, + 464, + 136.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 146, + 136.0, + 464, + 192.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 146, + 192.0, + 464, + 248.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 165, + 257, + 445, + 268 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 255, + 446, + 270 + ], + "spans": [ + { + "bbox": [ + 165, + 255, + 446, + 270 + ], + "score": 1.0, + "content": "Figure A1: Sample from Random Related Video Views (train set).", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "title", + "bbox": [ + 107, + 277, + 303, + 290 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 304, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 304, + 292 + ], + "score": 1.0, + "content": "APPENDIX 1 DATASET COMPARISON", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 304, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 506, + 317 + ], + "score": 1.0, + "content": "Random Related Video Views provides nearly 250,000 video links as well as multiple images per", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 315, + 504, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 504, + 327 + ], + "score": 1.0, + "content": "video across a broad variety of categories. We show more samples from R2V2 in Figure A1. Videos", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "spans": [ + { + "bbox": [ + 106, + 326, + 505, + 339 + ], + "score": 1.0, + "content": "each have four images 150 frames apart. Each separate video (outlined in blue) lists its correspond-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 505, + 349 + ], + "score": 1.0, + "content": "ing YouTube link. Using ImageNet synsets for search queries provides reasonable visual diversity,", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "but could be substituted with another set of queries. While we acknowledge that using YouTube’s", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "Search feature is not truly random, this procedure resulted in significantly more diverse samples than", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 370, + 506, + 382 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 506, + 382 + ], + "score": 1.0, + "content": "using existing datasets like YouTube8M (Abu-El-Haija et al., 2016) which is heavily unbalanced", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "with unnatural videos.2 We do no additional data cleaning to ensure that the videos or extracted", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 505, + 404 + ], + "score": 1.0, + "content": "images actually contain the search term (many do not), nor do we search for “high interest” video", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 416 + ], + "score": 1.0, + "content": "segments as in Misra et al. (2016). We also discard the search term itself as a form of supervision.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 505, + 426 + ], + "score": 1.0, + "content": "We find that a gap of 5 seconds between each saved image typically results in visually distinct but", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "semantically related images. Too much shorter results in images which are less individually distinct,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 505, + 448 + ], + "score": 1.0, + "content": "and too much longer may result in large and unpredictable changes. A sample from each dataset can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 447, + 257, + 458 + ], + "spans": [ + { + "bbox": [ + 106, + 447, + 257, + 458 + ], + "score": 1.0, + "content": "be seen in the supplemental material.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 303, + 506, + 458 + ] + }, + { + "type": "image", + "bbox": [ + 107, + 524, + 502, + 680 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 107, + 524, + 502, + 680 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 107, + 524, + 502, + 680 + ], + "spans": [ + { + "bbox": [ + 107, + 524, + 502, + 680 + ], + "score": 0.975, + "type": "image", + "image_path": "2a73ace7d06bf23a2fb379edcf93ff0973f48b41f0087f5b714f1f928e1f1b5e.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 107, + 524, + 502, + 576.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 107, + 576.0, + 502, + 628.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 107, + 628.0, + 502, + 680.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 690, + 505, + 724 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "Figure A2: Random sampling of pairs of images from videos in each dataset. In GOT-10k,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 701, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 106, + 701, + 505, + 713 + ], + "score": 1.0, + "content": "sometimes different video clips are segments from the same original video as seen in the first and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 713, + 406, + 725 + ], + "spans": [ + { + "bbox": [ + 106, + 713, + 406, + 725 + ], + "score": 1.0, + "content": "second sample. Images are square cropped for visualization purposes only.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + } + ], + "index": 21.5 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 107, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "We show samples from various existing video datasets in Figure A2. Datasets such as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "YouTube8M (Abu-El-Haija et al., 2016) and Kinetics400 (Kay et al., 2017). Kinetics contains only", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "videos of humans performing actions. Alternative datasets such as GOT-10k (Huang et al., 2019)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "provide a comparatively small number of videos but with dense annotations (in GOT-10k’s case for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 174, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 174, + 140 + ], + "score": 1.0, + "content": "object tracking).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 108, + 153, + 321, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 152, + 323, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 323, + 169 + ], + "score": 1.0, + "content": "APPENDIX 2 ALGORITHM PSEUDOCODE", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 178, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "score": 1.0, + "content": "The pseudocode below efficiently computes the Multi-Pair NCE for a batch of inputs. Shapes of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 274, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 274, + 201 + ], + "score": 1.0, + "content": "each input are provided for clarity where:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 132, + 209, + 366, + 280 + ], + "lines": [ + { + "bbox": [ + 132, + 210, + 294, + 221 + ], + "spans": [ + { + "bbox": [ + 132, + 210, + 142, + 221 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 210, + 161, + 220 + ], + "score": 0.67, + "content": "v =", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 210, + 294, + 221 + ], + "score": 1.0, + "content": "The number of videos in a batch.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 132, + 223, + 294, + 236 + ], + "spans": [ + { + "bbox": [ + 132, + 223, + 142, + 236 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 225, + 161, + 235 + ], + "score": 0.86, + "content": "k =", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 223, + 294, + 236 + ], + "score": 1.0, + "content": "The number of frames per video.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 132, + 239, + 366, + 252 + ], + "spans": [ + { + "bbox": [ + 132, + 239, + 142, + 252 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 239, + 203, + 251 + ], + "score": 0.86, + "content": "n = ( v * k ) =", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 239, + 366, + 252 + ], + "score": 1.0, + "content": "The total number of images in the batch.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 132, + 254, + 350, + 266 + ], + "spans": [ + { + "bbox": [ + 132, + 254, + 142, + 266 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 255, + 164, + 264 + ], + "score": 0.83, + "content": "m =", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 254, + 350, + 266 + ], + "score": 1.0, + "content": "The number of elements in the Memory Bank.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 132, + 268, + 323, + 281 + ], + "spans": [ + { + "bbox": [ + 132, + 268, + 142, + 281 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 269, + 161, + 279 + ], + "score": 0.85, + "content": "d =", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 268, + 323, + 281 + ], + "score": 1.0, + "content": "The dimensionality of the feature space.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 347, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 294, + 348, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 348, + 308 + ], + "score": 1.0, + "content": "Algorithm 1 Python-style pseudo code for Multi-Pair NCE.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "table", + "bbox": [ + 107, + 315, + 471, + 457 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 315, + 471, + 457 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 315, + 471, + 457 + ], + "spans": [ + { + "bbox": [ + 107, + 315, + 471, + 457 + ], + "score": 0.696, + "html": "
def multi_pair_nce([v，k，d] output of f encoder
f_output,# #
g_output,#output of g encoder
moco_mem, mask,[m，d]
temperature):# #(n + m)] block diagonal bool
f_output = f_output.reshape(v * k，d)[n，d]
g_output = g_output.reshape(v * k，d)#[n，d]
compare = concatenate((g_output，moco_mem)，axis=0)#[(n + m)，d]
similarities = matmul(f_output，compare.T)(n + m)]
similarities /= temperature
pos_similarities = similarities[mask]#[n，k]
neg similarities = similarities[!mask] exp_pos_sim = exp(pos_similarities)#(n + m -k)]
normalizing_constant = broadcast([n， k]
reduce_sum(exp(neg_similarities)，axis=1),[n，k]
shape(numerator))))
score = exp_pos_sim / (exp_pos_sim + normalizing_constant)
loss = -mean(log(score)) return loss
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The features are L2-normalized and multiplied by", + "type": "text" + }, + { + "bbox": [ + 376, + 550, + 413, + 563 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\dot { \\tau } = \\frac { 1 } { 0 . 0 7 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "10.07 as in MoCo (He et al.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "2020). We use 8 GPUs, a training batch size of 256 and a MoCo Memory Bank of size 65536 and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 114, + 585 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 574, + 121, + 583 + ], + "score": 0.77, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "-network momentum of 0.999. For multi-GPU training we employ the Shuffle-BN technique", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "shuffling both the anchors and the positives to reduce the correlation between batches filled with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 481, + 606 + ], + "score": 1.0, + "content": "multiple images from the same video. We use SGD with a learning rate of 0.03, momentum", + "type": "text" + }, + { + "bbox": [ + 482, + 594, + 501, + 605 + ], + "score": 0.69, + "content": "\\scriptstyle 1 = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 594, + 505, + 606 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 175, + 617 + ], + "score": 1.0, + "content": "and weight decay", + "type": "text" + }, + { + "bbox": [ + 176, + 605, + 209, + 615 + ], + "score": 0.65, + "content": "\\scriptstyle = 0 . 0 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 605, + 505, + 617 + ], + "score": 1.0, + "content": ". All of these hyperparameters are shared with MoCo (He et al., 2020). All", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 259, + 629 + ], + "score": 1.0, + "content": "methods are trained for approximately", + "type": "text" + }, + { + "bbox": [ + 259, + 616, + 281, + 627 + ], + "score": 0.48, + "content": "4 5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "iterations. When selecting frames from a video, we pick", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "with replacement. In addition to the natural augmentation, we perform standard data augmentation", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 639, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 506, + 651 + ], + "score": 1.0, + "content": "(color jitter, cropping, flipping) on the inputs. This prevents the network from relying too heavily on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 649, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 450, + 660 + ], + "score": 1.0, + "content": "shared video statistics like mean frame color. After cropping, all images are resized to", + "type": "text" + }, + { + "bbox": [ + 450, + 649, + 492, + 659 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 649, + 506, + 660 + ], + "score": 1.0, + "content": ". It", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "is worth noting that both VINCE and our data (R2V2) can be used with other network architectures", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "or learning algorithms such as AMDIM (Bachman et al., 2019), PIRL (Misra & Maaten, 2020), or", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 680, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 505, + 695 + ], + "score": 1.0, + "content": "SimCLR (Chen et al., 2020a). We choose ResNet18 and MoCo due to implementation simplicity", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 694, + 288, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 288, + 705 + ], + "score": 1.0, + "content": "and relatively low computational constraints.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 27.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "For ResNet50 training, we closely match the hyperparameters in MoCo v2 (Chen et al., 2020b).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "Specifically, we use the blur augmentation and stronger color augmentation and a cosine annealing", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 36.5 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "spans": [ + { + "bbox": [ + 298, + 750, + 312, + 763 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 13, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 107, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 107, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "We show samples from various existing video datasets in Figure A2. Datasets such as", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "YouTube8M (Abu-El-Haija et al., 2016) and Kinetics400 (Kay et al., 2017). Kinetics contains only", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "videos of humans performing actions. Alternative datasets such as GOT-10k (Huang et al., 2019)", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "provide a comparatively small number of videos but with dense annotations (in GOT-10k’s case for", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 174, + 140 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 174, + 140 + ], + "score": 1.0, + "content": "object tracking).", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2, + "bbox_fs": [ + 105, + 83, + 506, + 140 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 153, + 321, + 166 + ], + "lines": [ + { + "bbox": [ + 106, + 152, + 323, + 169 + ], + "spans": [ + { + "bbox": [ + 106, + 152, + 323, + 169 + ], + "score": 1.0, + "content": "APPENDIX 2 ALGORITHM PSEUDOCODE", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 178, + 505, + 200 + ], + "lines": [ + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "spans": [ + { + "bbox": [ + 105, + 177, + 506, + 191 + ], + "score": 1.0, + "content": "The pseudocode below efficiently computes the Multi-Pair NCE for a batch of inputs. Shapes of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 274, + 201 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 274, + 201 + ], + "score": 1.0, + "content": "each input are provided for clarity where:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6.5, + "bbox_fs": [ + 105, + 177, + 506, + 201 + ] + }, + { + "type": "list", + "bbox": [ + 132, + 209, + 366, + 280 + ], + "lines": [ + { + "bbox": [ + 132, + 210, + 294, + 221 + ], + "spans": [ + { + "bbox": [ + 132, + 210, + 142, + 221 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 210, + 161, + 220 + ], + "score": 0.67, + "content": "v =", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 210, + 294, + 221 + ], + "score": 1.0, + "content": "The number of videos in a batch.", + "type": "text" + } + ], + "index": 8, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 223, + 294, + 236 + ], + "spans": [ + { + "bbox": [ + 132, + 223, + 142, + 236 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 225, + 161, + 235 + ], + "score": 0.86, + "content": "k =", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 223, + 294, + 236 + ], + "score": 1.0, + "content": "The number of frames per video.", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 239, + 366, + 252 + ], + "spans": [ + { + "bbox": [ + 132, + 239, + 142, + 252 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 239, + 203, + 251 + ], + "score": 0.86, + "content": "n = ( v * k ) =", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 239, + 366, + 252 + ], + "score": 1.0, + "content": "The total number of images in the batch.", + "type": "text" + } + ], + "index": 10, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 254, + 350, + 266 + ], + "spans": [ + { + "bbox": [ + 132, + 254, + 142, + 266 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 255, + 164, + 264 + ], + "score": 0.83, + "content": "m =", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 254, + 350, + 266 + ], + "score": 1.0, + "content": "The number of elements in the Memory Bank.", + "type": "text" + } + ], + "index": 11, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 132, + 268, + 323, + 281 + ], + "spans": [ + { + "bbox": [ + 132, + 268, + 142, + 281 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 142, + 269, + 161, + 279 + ], + "score": 0.85, + "content": "d =", + "type": "inline_equation" + }, + { + "bbox": [ + 161, + 268, + 323, + 281 + ], + "score": 1.0, + "content": "The dimensionality of the feature space.", + "type": "text" + } + ], + "index": 12, + "is_list_start_line": true, + "is_list_end_line": true + } + ], + "index": 10, + "bbox_fs": [ + 132, + 210, + 366, + 281 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 347, + 307 + ], + "lines": [ + { + "bbox": [ + 106, + 294, + 348, + 308 + ], + "spans": [ + { + "bbox": [ + 106, + 294, + 348, + 308 + ], + "score": 1.0, + "content": "Algorithm 1 Python-style pseudo code for Multi-Pair NCE.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 294, + 348, + 308 + ] + }, + { + "type": "table", + "bbox": [ + 107, + 315, + 471, + 457 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 107, + 315, + 471, + 457 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 107, + 315, + 471, + 457 + ], + "spans": [ + { + "bbox": [ + 107, + 315, + 471, + 457 + ], + "score": 0.696, + "html": "
def multi_pair_nce([v，k，d] output of f encoder
f_output,# #
g_output,#output of g encoder
moco_mem, mask,[m，d]
temperature):# #(n + m)] block diagonal bool
f_output = f_output.reshape(v * k，d)[n，d]
g_output = g_output.reshape(v * k，d)#[n，d]
compare = concatenate((g_output，moco_mem)，axis=0)#[(n + m)，d]
similarities = matmul(f_output，compare.T)(n + m)]
similarities /= temperature
pos_similarities = similarities[mask]#[n，k]
neg similarities = similarities[!mask] exp_pos_sim = exp(pos_similarities)#(n + m -k)]
normalizing_constant = broadcast([n， k]
reduce_sum(exp(neg_similarities)，axis=1),[n，k]
shape(numerator))))
score = exp_pos_sim / (exp_pos_sim + normalizing_constant)
loss = -mean(log(score)) return loss
", + "type": "table", + "image_path": "9c51c3ac4b7b427f39ed5d21ad5c6148000bf6d77d25d092e07def9616521416.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 107, + 315, + 471, + 362.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 107, + 362.3333333333333, + 471, + 409.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 107, + 409.66666666666663, + 471, + 456.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "table_footnote", + "bbox": [ + 104, + 462, + 505, + 483 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "In practice we use the Log-Sum-Exp trick for numerical stability but omit here for clarity. broadcast repeats", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 472, + 480, + 483 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 480, + 483 + ], + "score": 1.0, + "content": "the input until it matches the provided dimensions. !mask flips the booleans of each point in the mask.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17.5 + } + ], + "index": 16.25 + }, + { + "type": "title", + "bbox": [ + 107, + 503, + 366, + 516 + ], + "lines": [ + { + "bbox": [ + 106, + 502, + 368, + 519 + ], + "spans": [ + { + "bbox": [ + 106, + 502, + 368, + 519 + ], + "score": 1.0, + "content": "APPENDIX 3 VINCE IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 528, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "For training our ResNet18 (He et al., 2016) representations we use the network up to the global", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 538, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 104, + 538, + 313, + 552 + ], + "score": 1.0, + "content": "average pooling followed by a fully connected layer", + "type": "text" + }, + { + "bbox": [ + 314, + 540, + 357, + 550 + ], + "score": 0.81, + "content": "( 5 1 2 \\mathrm { x } 5 1 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 538, + 505, + 552 + ], + "score": 1.0, + "content": ", Leaky-ReLU and a final embedding", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 549, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 131, + 564 + ], + "score": 1.0, + "content": "layer", + "type": "text" + }, + { + "bbox": [ + 131, + 550, + 170, + 561 + ], + "score": 0.76, + "content": "( 5 1 2 \\mathrm { ~ x ~ } 6 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 549, + 376, + 564 + ], + "score": 1.0, + "content": "). The features are L2-normalized and multiplied by", + "type": "text" + }, + { + "bbox": [ + 376, + 550, + 413, + 563 + ], + "score": 0.93, + "content": "\\begin{array} { r } { \\dot { \\tau } = \\frac { 1 } { 0 . 0 7 } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 549, + 506, + 564 + ], + "score": 1.0, + "content": "10.07 as in MoCo (He et al.,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 561, + 505, + 573 + ], + "score": 1.0, + "content": "2020). We use 8 GPUs, a training batch size of 256 and a MoCo Memory Bank of size 65536 and", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 572, + 114, + 585 + ], + "score": 1.0, + "content": "a", + "type": "text" + }, + { + "bbox": [ + 114, + 574, + 121, + 583 + ], + "score": 0.77, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 121, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "-network momentum of 0.999. For multi-GPU training we employ the Shuffle-BN technique", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 505, + 595 + ], + "score": 1.0, + "content": "shuffling both the anchors and the positives to reduce the correlation between batches filled with", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 481, + 606 + ], + "score": 1.0, + "content": "multiple images from the same video. We use SGD with a learning rate of 0.03, momentum", + "type": "text" + }, + { + "bbox": [ + 482, + 594, + 501, + 605 + ], + "score": 0.69, + "content": "\\scriptstyle 1 = 0 . 9", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 594, + 505, + 606 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 175, + 617 + ], + "score": 1.0, + "content": "and weight decay", + "type": "text" + }, + { + "bbox": [ + 176, + 605, + 209, + 615 + ], + "score": 0.65, + "content": "\\scriptstyle = 0 . 0 0 0 1", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 605, + 505, + 617 + ], + "score": 1.0, + "content": ". All of these hyperparameters are shared with MoCo (He et al., 2020). All", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 259, + 629 + ], + "score": 1.0, + "content": "methods are trained for approximately", + "type": "text" + }, + { + "bbox": [ + 259, + 616, + 281, + 627 + ], + "score": 0.48, + "content": "4 5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "iterations. When selecting frames from a video, we pick", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 505, + 639 + ], + "score": 1.0, + "content": "with replacement. In addition to the natural augmentation, we perform standard data augmentation", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 639, + 506, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 506, + 651 + ], + "score": 1.0, + "content": "(color jitter, cropping, flipping) on the inputs. This prevents the network from relying too heavily on", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 649, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 450, + 660 + ], + "score": 1.0, + "content": "shared video statistics like mean frame color. After cropping, all images are resized to", + "type": "text" + }, + { + "bbox": [ + 450, + 649, + 492, + 659 + ], + "score": 0.9, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 649, + 506, + 660 + ], + "score": 1.0, + "content": ". It", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 506, + 672 + ], + "score": 1.0, + "content": "is worth noting that both VINCE and our data (R2V2) can be used with other network architectures", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "or learning algorithms such as AMDIM (Bachman et al., 2019), PIRL (Misra & Maaten, 2020), or", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 680, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 680, + 505, + 695 + ], + "score": 1.0, + "content": "SimCLR (Chen et al., 2020a). We choose ResNet18 and MoCo due to implementation simplicity", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 694, + 288, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 694, + 288, + 705 + ], + "score": 1.0, + "content": "and relatively low computational constraints.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 27.5, + "bbox_fs": [ + 104, + 528, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 503, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "For ResNet50 training, we closely match the hyperparameters in MoCo v2 (Chen et al., 2020b).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 505, + 733 + ], + "score": 1.0, + "content": "Specifically, we use the blur augmentation and stronger color augmentation and a cosine annealing", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "learning rate for 286,000 iterations (200 epochs of ImageNet or equivalent on R2V2 regardless of", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "the number of unique positives per batch). The batch size we use is 896 with an initial learning rate", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 359, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 466, + 376 + ], + "score": 1.0, + "content": "of 0.105 and an embedding dimensionality of 128. The temperature parameter used was", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 467, + 360, + 501, + 374 + ], + "score": 0.91, + "content": "\\tau \\stackrel { } { = } \\frac { 1 } { 0 . 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 501, + 359, + 505, + 376 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 371, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 385 + ], + "score": 1.0, + "content": "We did not perform any hyperparameter searches for VINCE or R2V2, so these results may be", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 382, + 156, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 156, + 396 + ], + "score": 1.0, + "content": "suboptimal.", + "type": "text", + "cross_page": true + } + ], + "index": 10 + } + ], + "index": 36.5, + "bbox_fs": [ + 106, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 118, + 79, + 490, + 264 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 118, + 79, + 490, + 264 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 118, + 79, + 490, + 264 + ], + "spans": [ + { + "bbox": [ + 118, + 79, + 490, + 264 + ], + "score": 0.977, + "type": "image", + "image_path": "98dfdc7828efc0513ffa867293128fcc9da3691d331c460f71eb37125f0855a1.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 118, + 79, + 490, + 140.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 118, + 140.66666666666666, + 490, + 202.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 118, + 202.33333333333331, + 490, + 264.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 274, + 505, + 307 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 273, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 506, + 288 + ], + "score": 1.0, + "content": "Figure A3: Nearest neighbor results for a sampling of query images from R2V2 and ImageNet", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 285, + 504, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 285, + 504, + 297 + ], + "score": 1.0, + "content": "using various models. VINCE shows a clear understanding of each image and finds highly relevant", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 296, + 151, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 296, + 151, + 309 + ], + "score": 1.0, + "content": "neighbors.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 339, + 505, + 394 + ], + "lines": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 339, + 505, + 351 + ], + "score": 1.0, + "content": "learning rate for 286,000 iterations (200 epochs of ImageNet or equivalent on R2V2 regardless of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 350, + 505, + 362 + ], + "score": 1.0, + "content": "the number of unique positives per batch). The batch size we use is 896 with an initial learning rate", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 359, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 359, + 466, + 376 + ], + "score": 1.0, + "content": "of 0.105 and an embedding dimensionality of 128. The temperature parameter used was", + "type": "text" + }, + { + "bbox": [ + 467, + 360, + 501, + 374 + ], + "score": 0.91, + "content": "\\tau \\stackrel { } { = } \\frac { 1 } { 0 . 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 359, + 505, + 376 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 371, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 385 + ], + "score": 1.0, + "content": "We did not perform any hyperparameter searches for VINCE or R2V2, so these results may be", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 382, + 156, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 382, + 156, + 396 + ], + "score": 1.0, + "content": "suboptimal.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 108, + 416, + 302, + 429 + ], + "lines": [ + { + "bbox": [ + 106, + 414, + 304, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 304, + 432 + ], + "score": 1.0, + "content": "APPENDIX 4 QUALITATIVE RESULTS", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 108, + 444, + 504, + 467 + ], + "lines": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 505, + 457 + ], + "score": 1.0, + "content": "We additionally provide two qualitative analyses to better understand the success and failure cases", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 456, + 279, + 468 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 279, + 468 + ], + "score": 1.0, + "content": "of VINCE: Nearest Neighbors, and t-SNE.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 486, + 276, + 497 + ], + "lines": [ + { + "bbox": [ + 106, + 486, + 278, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 486, + 278, + 499 + ], + "score": 1.0, + "content": "APPENDIX 4.1 NEAREST NEIGHBORS:", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 107, + 509, + 505, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "We query ImageNet Val and a set of test videos for nearest neighbor matches, taking at most one", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "spans": [ + { + "bbox": [ + 105, + 520, + 505, + 533 + ], + "score": 1.0, + "content": "neighbor per video. We visualize the top 5 neighbors for VINCE, MoCo-R2V2, and MoCo-IN in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "Figure A3. We observe that VINCE seems to understand the semantics of an image more than", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "MoCo-R2V2 and MoCo-IN. For instance, although MoCo-R2V2 and MoCo-IN find other control", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 105, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "panels and buttons in query 1, they do not make the scene-level connection to car interiors as well", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 577 + ], + "score": 1.0, + "content": "as VINCE does. Query 2 shows an interesting quirk case of our method. Rather than matching", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 105, + 575, + 505, + 587 + ], + "score": 1.0, + "content": "the semantics of the image, VINCE relies on the news logo as a differentiating feature due to its", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "discriminative nature. Each image in VINCE’s query 2 results is from a separate video, but from", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 505, + 610 + ], + "score": 1.0, + "content": "the same news source. 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We visualize the top 5 neighbors for VINCE, MoCo-R2V2, and MoCo-IN in", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 531, + 505, + 543 + ], + "score": 1.0, + "content": "Figure A3. We observe that VINCE seems to understand the semantics of an image more than", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 554 + ], + "score": 1.0, + "content": "MoCo-R2V2 and MoCo-IN. 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ImageNet1K (Deng et al.,2009)1.3 M01000(428,406)
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Kinetics 40o (Kay et al.,2017)0220K400
GOT-10k (Huang et al., 2019)1.4M9K563(1600,912)
Random Related Video Views (Ours)0.96M240K-(467,280)
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Image Classification ImageNet(Deng et al.,2009)Scene Classification SUN Scenes(Xiao et al.,2010)Action Recognition Kinetics 400(Kay et al.,2017)Tracking OTB 2015(Wu et al.,2015)
Trained Layer(s) MetricLinear Accuracy (Top 1)Linear Accuracy (Top 1)LSTM→Linear Accuracy (Top 1)1x1 Conv
Sup-INPrecisionSuccess
MoCo-IN0.6960.4910.2070.5570.396
0.4470.4870.3360.5830.429
BeerMoCo-G0.3930.4440.3130.5510.413
MoCo-R2V20.3580.450 0.4950.3180.5550.403
VINCE (Ours)0.4000.3620.6290.465
Relative Gain over MoCo-R2V211.91%9.93%13.85%13.33%15.38%
PssSDSup-IN0.7620.5930.3050.4580.320
MoCo-V2-IN (our impl.)0.6520.6080.4590.3000.260
MoCo-R2V20.5360.5810.4560.3860.299
VINCE (Ours)0.5440.6110.4910.4020.300
Relative Gain over MoCo-R2V21.36%5.29%7.72%4.15%0.33%
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Pretraining DataTest Task
ImageNet SUN SceneKinetics 400OTB 2015 PrecisionOTB 2015 Success
R2V2 IN-Queries0.4000.4950.3620.6290.465
R2V2 YT8MURLs0.3670.4780.3430.6670.492
Kinetics 400 URLs0.3680.4940.3900.6120.456
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Images Per VideoTest Task
ImageNetSUN SceneKinetics 400OTB 2015 PrecisionOTB 2015 Success
1: Same Frame0.3580.4500.3180.5550.403
2: Multi-Frame0.3810.4780.3610.6220.464
8:Multi-Frame Multi-Pair0.4000.4950.3620.6290.465
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def multi_pair_nce([v，k，d] output of f encoder
f_output,# #
g_output,#output of g encoder
moco_mem, mask,[m，d]
temperature):# #(n + m)] block diagonal bool
f_output = f_output.reshape(v * k，d)[n，d]
g_output = g_output.reshape(v * k，d)#[n，d]
compare = concatenate((g_output，moco_mem)，axis=0)#[(n + m)，d]
similarities = matmul(f_output，compare.T)(n + m)]
similarities /= temperature
pos_similarities = similarities[mask]#[n，k]
neg similarities = similarities[!mask] exp_pos_sim = exp(pos_similarities)#(n + m -k)]
normalizing_constant = broadcast([n， k]
reduce_sum(exp(neg_similarities)，axis=1),[n，k]
shape(numerator))))
score = exp_pos_sim / (exp_pos_sim + normalizing_constant)
loss = -mean(log(score)) return loss
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CHOICE OF ACTION SPACE MATTER? + +Xue Bin Peng & Michiel van de Panne + +Department of Computer Science University of the British Columbia Vancouver, Canada {xbpeng,van}@cs.ubc.ca + +# ABSTRACT + +The use of deep reinforcement learning allows for high-dimensional state descriptors, but little is known about how the choice of action representation impacts the learning difficulty and the resulting performance. We compare the impact of four different action parameterizations (torques, muscle-activations, target joint angles, and target joint-angle velocities) in terms of learning time, policy robustness, motion quality, and policy query rates. Our results are evaluated on a gaitcycle imitation task for multiple planar articulated figures and multiple gaits. We demonstrate that the local feedback provided by higher-level action parameterizations can significantly impact the learning, robustness, and quality of the resulting policies. + +# 1 INTRODUCTION + +The introduction of deep learning models to reinforcement learning (RL) has enabled policies to operate directly on high-dimensional, low-level state features. As a result, deep reinforcement learning (DeepRL) has demonstrated impressive capabilities, such as developing control policies that can map from input image pixels to output joint torques (Lillicrap et al., 2015). However, the quality and robustness often falls short of what has been achieved with hand-crafted action abstractions, e.g., Coros et al. (2011); Geijtenbeek et al. (2013). While much is known about the learning of state representations, the choice of action parameterization is a design decision whose impact is not yet well understood. + +Joint torques can be thought of as the most basic and generic representation for driving the movement of articulated figures, given that muscles and other actuation models eventually result in joint torques. However this ignores the intrinsic embodied nature of biological systems, particularly the synergy between control and biomechanics. Passive-dynamics, such as elasticity and damping from muscles and tendons, play an integral role in shaping motions: they provide mechanisms for energy storage, and mechanical impedance which generates instantaneous feedback without requiring any explicit computation. Loeb coins the term preflexes (Loeb, 1995) to describe these effects, and their impact on motion control has been described as providing intelligence by mechanics (Blickhan et al., 2007). This can also be thought of as a kind of partitioning of the computations between the control and physical system. + +In this paper we explore the impact of four different actuation models on learning to control dynamic articulated figure locomotion: (1) torques (Tor); (2) activations for musculotendon units (MTU); (3) target joint angles for proportional-derivative controllers (PD); and (4) target joint velocities (Vel). Because Deep RL methods are capable of learning control policies for all these models, it now becomes possible to directly assess how the choice of actuation model affects the learning difficulty. We also assess the learned policies with respect to robustness, motion quality, and policy query rates. We show that action spaces which incorporate local feedback can significantly improve learning speed and performance, while still preserving the generality afforded by torque-level control. Such parameterizations also allow for more complex body structures and subjective improvements in motion quality. + +Our specific contributions are: (1) We introduce a DeepRL framework for motion imitation tasks; (2) We evaluate the impact of four different actuation models on learned control policies according to four criteria; and (3) We propose an optimization approach that combines policy learning and actuator optimization, allowing neural networks to effective control complex muscle models. + +# 2 BACKGROUND + +Our task will be structured as a standard reinforcement problem where an agent interacts with its environment according to a policy in order to maximize a reward signal. The policy $\pi ( s , a ) = p ( a | s )$ represents the conditional probability density function of selecting action $a \in { \mathcal { A } }$ in state $s \in S$ . At each control step $t$ , the agent observes a state $s _ { t }$ and samples an action $a _ { t }$ from $\pi$ . The environment in turn responds with a scalar reward $r _ { t }$ , and a new state $s _ { t } ^ { \prime } \ = \ s _ { t + 1 }$ sampled from its dynamics $p ( s ^ { \prime } | s , a )$ . For a parameterized policy $\pi _ { \boldsymbol { \theta } } ( s , a )$ , the goal of the agent is learn the parameters $\theta$ which maximizes the expected cumulative reward + +$$ +J ( \pi _ { \theta } ) = \mathbb { E } \left[ \sum _ { t = 0 } ^ { T } \gamma ^ { t } r _ { t } \Big | \pi _ { \theta } \right] +$$ + +with $\gamma \in [ 0 , 1 ]$ as the discount factor, and $T$ as the horizon. The gradient of the expected reward $\nabla _ { \boldsymbol { \theta } } J ( \pi _ { \boldsymbol { \theta } } )$ can be determined according to the policy gradient theorem (Sutton et al., 2001), which provides a direction of improvement to adjust the policy parameters $\theta$ . + +$$ +\nabla _ { \theta } J ( \pi _ { \theta } ) = \int _ { S } d _ { \theta } ( s ) \int _ { A } \nabla _ { \theta } \log ( \pi _ { \theta } ( s , a ) ) A ( s , a ) d a d s +$$ + +where $\begin{array} { r } { d _ { \theta } ( s ) = \int _ { S } \sum _ { t = 0 } ^ { T } \gamma ^ { t } p _ { 0 } ( s _ { 0 } ) p ( s _ { 0 } \to s | t , \pi _ { \theta } ) d s _ { 0 } } \end{array}$ is the discounted state distribution, where $p _ { 0 } ( s )$ represents the initial state distribution, and $p ( s _ { 0 } \to s | t , \pi _ { \theta } )$ models the likelihood of reaching state $s$ by starting at $s _ { 0 }$ and following the policy $\pi _ { \theta } ( s , a )$ for $t$ steps (Silver et al., 2014). $A ( s , \bar { a } )$ represents a generalized advantage function. The choice of advantage function gives rise to a family of policy gradient algorithms, but in this work, we will focus on the one-step temporal difference advantage function (Schulman et al., 2015) + +$$ +A ( s _ { t } , a _ { t } ) = r _ { t } + \gamma V ( s _ { t } ^ { \prime } ) - V ( s _ { t } ) +$$ + +where $\begin{array} { r } { V ( s ) = \mathbb { E } \left[ \sum _ { t = 0 } ^ { T } \gamma ^ { t } r _ { t } \Big \rvert s _ { 0 } = s , \pi _ { \theta } \right] } \end{array}$ is the state-value function, and can be defined recursively via the Bellman equation + +$$ +V ( s _ { t } ) = \underset { r _ { t } , s _ { t } ^ { \prime } } { \mathbb { E } } \left[ r _ { t } + \gamma V ( s _ { t } ^ { \prime } ) | s _ { t } , \pi _ { \theta } \right] +$$ + +A parameterized value function $V _ { \phi } ( s )$ , with parameters $\phi$ , can be learned iteratively in a manner similar to Q-Learning by minimizing the Bellman loss, + +$$ +L ( \phi ) = \underset { s _ { t } , r _ { t } , s _ { t } ^ { \prime } } { \mathbb { E } } \left[ \frac { 1 } { 2 } \left( y _ { t } - V _ { \phi } ( s _ { t } ) \right) ^ { 2 } \right] , \qquad y _ { t } = r _ { t } + \gamma V _ { \phi } ( s _ { t } ^ { \prime } ) +$$ + +$\pi _ { \theta }$ and $V _ { \phi }$ can be trained in tandem using an actor-critic framework (Konda & Tsitsiklis, 2000). + +In this work, each policy will be represented as a gaussian distribution with a parameterized mean $\mu _ { \boldsymbol { \theta } } ( s )$ and fixed covariance matrix $\dot { \Sigma } = \mathrm { d i a g } \{ \sigma _ { i } ^ { 2 } \}$ , where $\sigma _ { i }$ is manually specified for each action parameter. Actions can be sampled from the distribution by applying gaussian noise to the mean action + +$$ +a _ { t } = \mu _ { \theta } ( s _ { t } ) + \mathcal { N } ( 0 , \Sigma ) +$$ + +The corresponding policy gradient will assume the form + +$$ +\nabla _ { \theta } J ( \pi _ { \theta } ) = \int _ { S } d _ { \theta } ( s ) \int _ { A } \nabla _ { \theta } \mu _ { \theta } ( s ) \Sigma ^ { - 1 } \left( a - \mu _ { \theta } ( s ) \right) A ( s , a ) d a d s +$$ + +which can be interpreted as shifting the mean of the action distribution towards actions that lead to higher than expected rewards, while moving away from actions that lead to lower than expected rewards. + +# 3 TASK REPRESENTATION + +# 3.1 REFERENCE MOTION + +In our task, the goal of a policy is to imitate a given reference motion $\{ q _ { t } ^ { * } \}$ which consists of a sequence of kinematic poses $q _ { t } ^ { * }$ in reduced coordinates. The reference velocity $\dot { q _ { t } ^ { * } }$ at a given time $t$ is approximated by finite-difference $\begin{array} { r } { \dot { q _ { t } ^ { * } } \approx \frac { q _ { t + \triangle t } ^ { * } - q _ { t } ^ { * } } { \triangle t } } \end{array}$ . Reference motions are generated via either using a recorded simulation result from a preexisting controller $( ^ { 6 6 } \mathrm { S i m } ^ { 7 } )$ , or via manually-authored keyframes. Since hand-crafted reference motions may not be physically realizable, the goal is to closely reproduce a motion while satisfying physical constraints. + +# 3.2 STATES + +To define the state of the agent, a feature transformation $\Phi ( q , \dot { q } )$ is used to extract a set of features from the reduced-coordinate pose $q$ and velocity $\dot { q }$ . The features consist of the height of the root (pelvis) from the ground, the position of each link with respect to the root, and the center of mass velocity of each link. When training a policy to imitate a cyclic reference motion $\{ q _ { t } ^ { * } \}$ , knowledge of the motion phase can help simplify learning. Therefore, we augment the state features with a set of target features $\Phi ( q _ { t } ^ { * } , \dot { q } _ { t } ^ { * } )$ , resulting in a combined state represented by $s _ { t } = ( \Phi ( q _ { t } , \dot { q _ { t } } ) , \Phi ( q _ { t } ^ { * } , { \dot { q _ { t } } } ^ { * } ) )$ . Similar results can also be achieved by providing a single motion phase variable as a state feature, as we show in Figure 15 (supplemental material). + +# 3.3 ACTIONS + +We train separate policies for each of the four actuation models, as described below. Each actuation model also has related actuation parameters, such as feedback gains for PD-controllers and musculotendon properties for MTUs. These parameters can be manually specified, as we do for the PD and Vel models, or they can be optimized for the task at hand, as for the MTU models. Table 1 provides a list of actuator parameters for each actuation model. + +Target Joint Angles (PD): Each action represents a set of target angles $\hat { q }$ , where $\hat { q } ^ { i }$ specifies the target angles for joint $i$ . $\hat { q }$ is applied to PD-controllers which compute torques according to $\tau ^ { i } =$ $k _ { p } ^ { i } ( \hat { q } ^ { i } - q ^ { i } ) + k _ { d } ^ { i } ( \hat { \dot { q } } ^ { i } - \dot { q } ^ { i } )$ , where $\hat { \dot { q } } ^ { i } = 0$ , and $k _ { p } ^ { i }$ and $k _ { d } ^ { i }$ are manually-specified gains. + +Target Joint Velocities (Vel): Each action specifies a set of target velocities $\hat { \dot { q } }$ which are used to compute torques according to $\tau ^ { i } = k _ { d } ^ { i } ( { \hat { \dot { q } } } ^ { i } - { \dot { q } } ^ { i } )$ , where the gains $k _ { d } ^ { i }$ are specified to be the same as those used for target angles. + +Torques (Tor): Each action directly specifies torques for every joint, and constant torques are applied for the duration of a control step. Due to torque limits, actions are bounded by manually specified limits for each joint. Unlike the other actuation models, the torque model does not require additional actuator parameters, and can thus be regarded as requiring the least amount of domain knowledge. Torque limits are excluded from the actuator parameter set as they are common for all parameterizations. + +Muscle Activations (MTU): Each action specifies activations for a set of musculotendon units (MTU). Detailed modeling and implementation information are available in Wang et al. (2012). Each MTU is modeled as a contractile element (CE) attached to a serial elastic element (SE) and parallel elastic element (PE). The force exerted by the MTU can be calculated according to $F _ { M T U } = F _ { S E } = F _ { C E } + F _ { P E }$ . Both $F _ { S E }$ and $F _ { P E }$ are modeled as passive springs, while $F _ { C E }$ is actively controlled according to $F _ { C E } = a _ { M T U } F _ { 0 } f _ { l } ( l _ { C E } ) f _ { v } ( v _ { C E } )$ , with $a _ { M T U }$ being the muscle activation, $F _ { 0 }$ the maximum isometric force, $l _ { C E }$ and $v _ { C E }$ being the length and velocity of the contractile element. The functions $f _ { l } ( l _ { C E } )$ and $f _ { v } ( v _ { C E } )$ represent the force-length and force-velocity relationships, modeling the variations in the maximum force that can be exerted by a muscle as a function of its length and contraction velocity. Analytic forms are available in Geyer et al. (2003). Activations are bounded between [0, 1]. The length of each contractile element $l _ { C E }$ are included as state features. To simplify control and reduce the number of internal state parameters per MTU, the policies directly control muscle activations instead of indirectly through excitations (Wang et al., 2012). + +Table 1: Actuation models and their respective actuator parameters. + +
ActuationModelActuatorParameters
Target Joint Angles (PD) proportional gains kp, derivative gains kd
Target Joint Velocities(Vel)derivative gains kd
Torques (Tor)none
Muscle Activations (MTU)optimal contractile element length, serial elastic element rest length,maximum isometric force, pennation,moment arm,maximum moment arm joint orientation, rest joint orientation.
+ +# 3.4 REWARD + +The reward function consists of a weighted sum of terms that encourage the policy to track a reference motion. + +$$ +\begin{array} { c } { r = w _ { p o s e } r _ { p o s e } + w _ { v e l } r _ { v e l } + w _ { e n d } r _ { e n d } + w _ { r o o t } r _ { r o o t } + w _ { c o m } r _ { c o m } } \\ { ~ } \\ { w _ { p o s e } = 0 . 5 , ~ w _ { v e l } = 0 . 0 5 , ~ w _ { e n d } = 0 . 1 5 , ~ w _ { r o o t } = 0 . 1 , ~ w _ { c o m } = 0 . 2 } \end{array} +$$ + +Details of each term are available in the supplemental material. $r _ { p o s e }$ penalizes deviation of the character pose from the reference pose, and $r _ { v e l }$ penalizes deviation of the joint velocities. $r _ { e n d }$ and $r _ { r o o t }$ accounts for the position error of the end-effectors and root. $r _ { c o m }$ penalizes deviations in the center of mass velocity from that of the reference motion. + +# 3.5 INITIAL STATE DISTRIBUTION + +We design the initial state distribution, $p _ { 0 } ( s )$ , to sample states uniformly along the reference trajectory. At the start of each episode, $q ^ { * }$ and $\dot { q } ^ { * }$ are sampled from the reference trajectory, and used to initialize the pose and velocity of the agent. This helps guide the agent to explore states near the target trajectory. + +# 4 ACTOR-CRITIC LEARNING ALGORITHM + +Instead of directly using the temporal difference advantage function, we adapt a positive temporal difference (PTD) update as proposed by Van Hasselt (2012). + +$$ +A ( s , a ) = I \left[ \delta > 0 \right] = { \left\{ \begin{array} { l l } { 1 , } & { \delta > 0 } \\ { 0 , } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. } +$$ + +$$ +\delta = r + \gamma V ( s ^ { \prime } ) - V ( s ) +$$ + +Unlike more conventional policy gradient methods, PTD is less sensitive to the scale of the advantage function and avoids instabilities that can result from negative TD updates. For a Gaussian policy, a negative TD update moves the mean of the distribution away from an observed action, effectively shifting the mean towards an unknown action that may be no better than the current mean action (Van Hasselt, 2012). In expectation, these updates converges to the true policy gradient, but for stochastic estimates of the policy gradient, these updates can cause the agent to adopt undesirable behaviours which affect subsequent experiences collected by the agent. Furthermore, we incorporate experience replay, which has been demonstrated to improve stability when training neural network policies with Q-learning in discrete action spaces. Experience replay often requires off-policy methods, such as importance weighting, to account for differences between the policy being trained and the behavior policy used to generate experiences (WawrzynSki & Tanwani, 2013).´ However, we have not found importance weighting to be beneficial for PTD. + +Stochastic policies are used during training for exploration, while deterministic policy are deployed for evaluation at runtime. The choice between a stochastic and deterministic policy can be specified by the addition of a binary indicator variable $\lambda \in [ 0 , 1 ]$ + +$$ +a _ { t } = \mu _ { \theta } ( s _ { t } ) + \lambda N ( 0 , \Sigma ) +$$ + +where $\lambda = 1$ corresponds to a stochastic policy with exploration noise, and $\lambda = 0$ corresponds to a deterministic policy that always selects the mean of the distribution. Noise from a stochastic policy will result in a state distribution that differs from that of the deterministic policy at runtime. To imitate this discrepancy, we incorporate $\epsilon$ -greedy exploration in addition to the original Gaussian exploration. During training, $\lambda$ is determined by a Bernoulli random variable $\lambda \sim \operatorname { B e r } ( \epsilon )$ , where $\lambda = 1$ with probability $\epsilon \in [ 0 , 1 ]$ . The exploration rate $\epsilon$ is annealed linearly from 1 to 0.2 over $5 0 0 \mathrm { k }$ iterations, which slowly adjusts the state distribution encountered during training to better resemble the distribution at runtime. Since the policy gradient is defined for stochastic policies, only tuples recorded with exploration noise (i.e. $\lambda = 1$ ) can be used to update the actor, while the critic can be updated using all tuples. + +Training proceeds episodically, where the initial state of each episode is sampled from $p _ { 0 } ( s )$ , and the episode duration is drawn from an exponential distribution with a mean of 2s. To discourage falling, an episode will also terminate if any part of the character’s trunk makes contact with the ground for an extended period of time, leaving the agent with zero reward for all subsequent steps. Algorithm 1 in the supplemental material summarizes the complete learning process. + +MTU Actuator Optimization: Actuation models such as MTUs are defined by further parameters whose values impact performance (Geijtenbeek et al., 2013). Geyer et al. (2003) uses existing anatomical estimates for humans to determine MTU parameters, but such data is not be available for more arbitrary creatures. Alternatively, Geijtenbeek et al. (2013) uses covariance matrix adaptation (CMA), a derivative-free evolutionary search strategy, to simultaneously optimize MTU and policy parameters. This approach is limited to policies with reasonably low dimensional parameter spaces, and is thus ill-suited for neural network models with hundreds of thousands of parameters. To avoid manual-tuning of actuator parameters, we propose a heuristic approach that alternates between policy learning and actuator optimization, as detailed in the supplemental material. + +![](images/37eeafcd2ada57b0a9a694e94f6efe9f905699c1775e021139c85ef284ffc1cf.jpg) +Figure 1: Simulated articulated figures and their state representation. Revolute joints connect all links. From left to right: 7-link biped; 19-link raptor; 21-link dog; State features: root height, relative position (red) of each link with respect to the root and their respective linear velocity (green). + +# 5 RESULTS + +The motions are best seen in the supplemental video https://youtu.be/L3vDo3nLI98. We evaluate the action parameterizations by training policies for a simulated 2D biped, dog, and raptor as shown in Figure 1. Depending on the agent and the actuation model, our systems have 58–214 state dimensions, 6–44 action dimensions, and 0–282 actuator parameters, as summarized in Table 3 (supplemental materials). The MTU models have at least double the number of action parameters because they come in antagonistic pairs. As well, additional MTUs are used for the legs to more accurately reflect bipedal biomechanics. This includes MTUs that span multiple joints. + +Each policy is represented by a three layer neural network, as illustrated in Figure 8 (supplemental material) with 512 and 256 fully-connected units, followed by a linear output layer where the number of output units vary according to the number of action parameters for each character and actuation model. ReLU activation functions are used for both hidden layers. Each network has approximately $2 0 0 \mathrm { k }$ parameters. The value function is represented by a similar network, except having a single linear output unit. The policies are queried at $6 0 \mathrm { H z }$ for a control step of about 0.0167s. Each network is randomly initialized and trained for about 1 million iterations, requiring 32 million tuples, the equivalent of approximately 6 days of simulated time. Each policy requires about 10 hours for the biped, and 20 hours for the raptor and dog on an 8-core Intel Xeon E5-2687W. + +![](images/93b7292fed0365462a02a39731b438eabc707c5107988fe50e2bceec740286b3.jpg) +Figure 2: Learning curves for each policy during 1 million iterations. + +Only the actuator parameters for MTUs are optimized with Algorithm 2, since the parameters for the other actuation models are few and reasonably intuitive to determine. The initial actuator parameters $\psi _ { 0 }$ are manually specified, while the initial policy parameters $\theta _ { 0 }$ are randomly initialized. Each pass optimizes $\psi$ using CMA for 250 generations with 16 samples per generation, and $\theta$ is trained for $2 5 0 \mathrm { k }$ iterations. Parameters are initialized with values from the previous pass. The expected value of each CMA sample of $\psi$ is estimated using the average cumulative reward over 16 rollouts with a duration of 10s each. Separate MTU parameters are optimized for each character and motion. Each set of parameters is optimized for 6 passes following Algorithm 2, requiring approximately 50 hours. Figure 5 illustrates the performance improvement per pass. Figure 6 compares the performance of MTUs before and after optimization. For most examples, the optimized actuator parameters significantly improve learning speed and final performance. For the sake of comparison, after a set of actuator parameters has been optimized, a new policy is retrained with the new actuator parameters and its performance compared to the other actuation models. + +Policy Performance and Learning Speed: Figure 2 shows learning curves for the policies and the performance of the final policies are summarized in Table 4. Performance is evaluated using the normalized cumulative reward (NCR), calculated from the average cumulative reward over 32 episodes with lengths of 10s, and normalized by the maximum and minimum cumulative reward possible for each episode. No discounting is applied when calculating the NCR. The initial state of each episode is sampled from the reference motion according to $p ( s _ { 0 } )$ . To compare learning speeds, we use the normalized area under each learning curve (AUC) as a proxy for the learning speed of a particular actuation model, where 0 represents the worst possible performance and no progress during training, and 1 represents the best possible performance without requiring training. + +PD performs well across all examples, achieving comparable-to-the-best performance for all motions. PD also learns faster than the other parameterizations for 5 of the 7 motions. The final performance of Tor is among the poorest for all the motions. Differences in performance appear more pronounced as characters become more complex. For the simple 7-link biped, most parameterizations achieve similar performance. However, for the more complex dog and raptor, the performance of Tor policies deteriorate with respect to other policies such as PD and Vel. MTU policies often exhibited the slowest learning speed, which may be a consequence of the higher dimensional action spaces, i.e., requiring antagonistic muscle pairs, and complex muscle dynamics. Nonetheless, once optimized, the MTU policies produce more natural motions and responsive behaviors as compared to other parameterizations. We note that the naturalness of motions is not well captured by the reward, since it primarily gauges similarity to the reference motion, which may not be representative of natural responses when perturbed from the nominal trajectory. A sensitivity analysis of the policies’ performance to variations in network architecture and hyperparameters are available in the supplemental material. + +Policy Robustness: To evaluate robustness, we recorded the NCR achieved by each policy when subjected to external perturbations. The perturbations assume the form of random forces applied to the trunk of the characters. Figure 3 illustrates the performance of the policies when subjected to perturbations of different magnitudes. The magnitude of the forces are constant, but direction varies randomly. Each force is applied for 0.1 to 0.4s, with 1 to 4s between each perturbation. Performance is estimated using the average over 128 episodes of length 20s each. For the biped walk, the Tor policy is significantly less robust than those for the other types of actions, while the MTU policy is the least robust for the raptor run. Overall, the PD policies are among the most robust for all the motions. In addition to external forces, we also evaluate robustness over randomly generated terrain consisting of bumps with varying heights and slopes with varying steepness. We evaluate the performance on irregular terrain (Figure 12, supplemental material). There are few discernible patterns for this test. The Vel and MTU policies are significantly worse than the Tor and PD policies for the dog bound on the bumpy terrain. The unnatural jittery behavior of the dog Tor policy proves to be surprisingly robust for this scenario. We suspect that the behavior prevents the trunk from contacting the ground for extended periods for time, and thereby escaping our system’s fall detection. + +![](images/59c61c631803c3c1ef5dced30e6b9d2417658876f02b3fafd5c5920c6ab96ddd.jpg) +Figure 3: Performance when subjected to random perturbation forces of different magnitudes. + +Query Rate: Figure 4 compares the performance of different parameterizations for different policy query rates. Separate policies are trained with queries of 15Hz, $3 0 \mathrm { H z }$ , $6 0 \mathrm { H z }$ , and $1 2 0 \mathrm { H z }$ . Actuation models that incorporate low-level feedback such as PD and Vel, appear to cope more effectively to lower query rates, while the Tor degrades more rapidly at lower query rates. It is not yet obvious to us why MTU policies appear to perform better at lower query rates and worse at higher rates. Lastly, Figure 14 shows the policy outputs as a function of time for the four actuation models, for a particular joint, as well as showing the resulting joint torque. Interestingly, the MTU action is visibly smoother than the other actions and results in joint torques profiles that are smoother than those seen for PD and Vel. + +![](images/d2c70584eca59abbfb1ffb645c9e04955f3a1ad659e5d9061ea8ca9b9960c36c.jpg) +Figure 4: Performance of policies with different query rates for the biped (left) and dog (right). Separate policies are trained for each query rate. + +# 6 RELATED WORK + +DeepRL has driven impressive recent advances in learning motion control, i.e., solving for continuous-action control problems using reinforcement learning. All four of the actions types that we explore have seen previous use in the machine learning literature. WawrzynSki & Tanwani ´ (2013) use an actor-critic approach with experience replay to learn skills for an octopus arm (actuated by a simple muscle model) and a planar half cheetah (actuated by joint-based PD-controllers). + +Recent work on deterministic policy gradients (Lillicrap et al., 2015) and on RL benchmarks, e.g., OpenAI Gym, generally use joint torques as the action space, as do the test suites in recent work (Schulman et al., 2015) on using generalized advantage estimation. Other recent work uses: the PR2 effort control interface as a proxy for torque control (Levine et al., 2015); joint velocities (Gu et al., 2016); velocities under an implicit control policy (Mordatch et al., 2015); or provide abstract actions (Hausknecht & Stone, 2015). Our learning procedures are based on prior work using actorcritic approaches with positive temporal difference updates (Van Hasselt, 2012). + +Work in biomechanics has long recognized the embodied nature of the control problem and the view that musculotendon systems provide “preflexes” (Loeb, 1995) that effectively provide a form intelligence by mechanics (Blickhan et al., 2007), as well as allowing for energy storage. The control strategies for physics-based character simulations in computer animation also use all the forms of actuation that we evaluate in this paper. Representative examples include quadratic programs that solve for joint torques (de Lasa et al., 2010), joint velocities for skilled bicycle stunts (Tan et al., 2014), muscle models for locomotion (Wang et al., 2012; Geijtenbeek et al., 2013), mixed use of feed-forward torques and joint target angles (Coros et al., 2011), and joint target angles computed by learned linear (time-indexed) feedback strategies (Liu et al., 2016). Lastly, control methods in robotics use a mix of actuation types, including direct-drive torques (or their virtualized equivalents), series elastic actuators, PD control, and velocity control. These methods often rely heavily on modelbased solutions and thus we do not describe these in further detail here. + +# 7 CONCLUSIONS + +Our experiments suggest that action parameterizations that include basic local feedback, such as PD target angles, MTU activations, or target velocities, can improve policy performance and learning speed across different motions and character morphologies. Such models more accurately reflect the embodied nature of control in biomechanical systems, and the role of mechanical components in shaping the overall dynamics of motions and their control. The difference between low-level and high-level action parameterizations grow with the complexity of the characters, with high-level parameterizations scaling more gracefully to complex characters. As a caveat, there may well be tasks, such as impedance control, where lower-level action parameterizations such as Tor may prove advantageous. We believe that no single action parameterization will be the best for all problems. However, since objectives for motion control problems are often naturally expressed in terms of kinematic properties, higher-level actions such as target joint angles and velocities may be effective for a wide variety of motion control problems. We hope that our work will help open discussions around the choice of action parameterizations. + +Our results have only been demonstrated on planar articulated figure simulations; the extension to 3D currently remains as future work. Furthermore, our current torque limits are still large as compared to what might be physically realizable. Tuning actuator parameters for complex actuation models such as MTUs remains challenging. Though our actuator optimization technique is able to improve performance as compared to manual tuning, the resulting parameters may still not be optimal for the desired task. Therefore, our comparisons of MTUs to other action parameterizations may not be reflective of the full potential of MTUs with more optimal actuator parameters. Furthermore, our actuator optimization currently tunes parameters for a specific motion, rather than a larger suite of motions, as might be expected in nature. + +Since the reward terms are mainly calculated according to joint positions and velocities, it may seem that it is inherently biased in favour of PD and Vel. However, the real challenges for the control policies lie elsewhere, such as learning to compensate for gravity and ground-reaction forces, and learning foot-placement strategies that are needed to maintain balance for the locomotion gaits. The reference pose terms provide little information on how to achieve these hidden aspects of motion control that will ultimately determine the success of the locomotion policy. While we have yet to provide a concrete answer for the generalization of our results to different reward functions, we believe that the choice of action parameterization is a design decision that deserves greater attention regardless of the choice of reward function. + +Finally, it is reasonable to expect that evolutionary processes would result in the effective co-design of actuation mechanics and control capabilities. Developing optimization and learning algorithms to allow for this kind of co-design is a fascinating possibility for future work. + +# REFERENCES + +Reinhard Blickhan, Andre Seyfarth, Hartmut Geyer, Sten Grimmer, Heiko Wagner, and Michael Gunther. Intelligence by mechanics. ¨ Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 365(1850):199–220, 2007. + +Stelian Coros, Andrej Karpathy, Ben Jones, Lionel Reveret, and Michiel van de Panne. Locomotion skills for simulated quadrupeds. ACM Transactions on Graphics, 30(4):Article TBD, 2011. + +Martin de Lasa, Igor Mordatch, and Aaron Hertzmann. Feature-based locomotion controllers. In ACM Transactions on Graphics (TOG), volume 29, pp. 131. ACM, 2010. + +Thomas Geijtenbeek, Michiel van de Panne, and A. Frank van der Stappen. Flexible muscle-based locomotion for bipedal creatures. ACM Transactions on Graphics, 32(6), 2013. + +Hartmut Geyer, Andre Seyfarth, and Reinhard Blickhan. Positive force feedback in bouncing gaits? Proc. Royal Society of London B: Biological Sciences, 270(1529):2173–2183, 2003. + +Shixiang Gu, Ethan Holly, Timothy Lillicrap, and Sergey Levine. Deep reinforcement learning for robotic manipulation. arXiv preprint arXiv:1610.00633, 2016. + +Matthew J. Hausknecht and Peter Stone. Deep reinforcement learning in parameterized action space. CoRR, abs/1511.04143, 2015. + +Vijay Konda and John Tsitsiklis. Actor-critic algorithms. In SIAM Journal on Control and Optimization, pp. 1008–1014. MIT Press, 2000. + +Sergey Levine, Chelsea Finn, Trevor Darrell, and Pieter Abbeel. End-to-end training of deep visuomotor policies. CoRR, abs/1504.00702, 2015. + +Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. CoRR, abs/1509.02971, 2015. + +Libin Liu, Michiel van de Panne, and KangKang Yin. Guided learning of control graphs for physicsbased characters. ACM Transactions on Graphics, 35(3), 2016. + +GE Loeb. Control implications of musculoskeletal mechanics. In Engineering in Medicine and Biology Society, 1995., IEEE 17th Annual Conference, volume 2, pp. 1393–1394. IEEE, 1995. + +Igor Mordatch, Kendall Lowrey, Galen Andrew, Zoran Popovic, and Emanuel Todorov. Interactive control of diverse complex characters with neural networks. In Advances in Neural Information Processing Systems 28, pp. 3132–3140, 2015. + +John Schulman, Philipp Moritz, Sergey Levine, Michael I. Jordan, and Pieter Abbeel. Highdimensional continuous control using generalized advantage estimation. CoRR, abs/1506.02438, 2015. + +David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In Proc. International Conference on Machine Learning, pp. 387–395, 2014. + +R. Sutton, D. Mcallester, S. Singh, and Y. Mansour. Policy gradient methods for reinforcement learning with function approximation, 2001. + +Jie Tan, Yuting Gu, C. Karen Liu, and Greg Turk. Learning bicycle stunts. ACM Trans. Graph., 33 (4):50:1–50:12, 2014. ISSN 0730-0301. + +Hado Van Hasselt. Reinforcement learning in continuous state and action spaces. In Reinforcement Learning, pp. 207–251. Springer, 2012. + +Jack M. Wang, Samuel R. Hamner, Scott L. Delp, Vladlen Koltun, and More Specifically. Optimizing locomotion controllers using biologically-based actuators and objectives. ACM Trans. Graph, 2012. + +Paweł WawrzynSki and Ajay Kumar Tanwani. Autonomous reinforcement learning with experience ´ replay. Neural Networks, 41:156–167, 2013. + +SUPPLEMENTARY MATERIAL + +# Algorithm 1 Actor-critic Learning Using Positive Temporal Differences + +1: $\theta \gets$ random weights +2: $\phi $ random weights +3: while not done do +4: for $s t e p = 1 , . . . , m$ do +5: $s \gets$ start state +6: $\lambda \operatorname { B e r } ( \epsilon _ { t } )$ +7: $a \gets \mu _ { \theta } ( s ) + \lambda \mathcal { N } ( 0 , \Sigma )$ +8: Apply $a$ and simulate forward 1 step +9: $s ^ { \prime } \gets$ end state +10: $r \gets$ reward +11: $\tau \gets ( s , a , r , s ^ { \prime } , \lambda )$ +12: store $\tau$ in replay memory +13: if episode terminated then +14: Sample $s _ { 0 }$ from $p _ { 0 } ( s )$ +15: Reinitialize state s to s0 +16: end if +17: end for +18: Update critic: +19: Sample minibatch of $n$ tuples $\{ \tau _ { i } = ( s _ { i } , a _ { i } , r _ { i } , \lambda _ { i } , s _ { i } ^ { \prime } ) \}$ from replay memory +20: for each $\tau _ { i }$ do +21: $\delta _ { i } \gets r _ { i } + \gamma V _ { \phi } ( s _ { i } ^ { \prime } ) - V _ { \phi } ( s _ { i } )$ +22: $\begin{array} { r } { \phi \phi + \alpha _ { V } \frac { 1 } { n } \delta _ { i } \nabla _ { \phi } V _ { \phi } ( s _ { i } ) } \end{array}$ +23: end for +24: Update actor: +25: Sample minibatch of $n$ tuples $\{ \tau _ { j } = ( s _ { j } , a _ { j } , r _ { j } , \lambda _ { j } , s _ { j } ^ { \prime } ) \}$ from replay memory where $\lambda _ { j } = 1$ +26: for each $\tau _ { j }$ do +27: $\delta _ { j } r _ { j } ^ { - } + \gamma V _ { \phi } ( s _ { j } ^ { \prime } ) - V _ { \phi } ( s _ { j } )$ +28: if $\delta _ { j } > 0$ then +29: Oaj ← aj − µθ(sj ) +30: Oa˜j ← BoundActionGradient(Oaj , µθ(sj )) +31: θ ← θ + απ 1n Oθµθ(sj )Σ−1Oa˜j +32: end if +33: end for +34: end while + +# Algorithm 2 Alternating Actuator Optimization + +1: θ ← θ0 +2: ψ ← ψ0 +3: while not done do +4: $\theta \gets$ argmaxθ0 $J ( \pi _ { \theta ^ { \prime } } , \psi )$ with Algorithm 1 +5: $\psi \mathrm { a r g m a x } _ { \psi ^ { \prime } } J ( \pi _ { \theta } , \psi ^ { \prime } )$ with CMA +6: end while + +# MTU Actuator Optimization + +The actuator parameters $\psi$ can be interpreted as a parameterization of the dynamics of the system $p ( s ^ { \prime } | s , a , \psi )$ . The expected cumulative reward can then be re-parameterized according to + +$$ +J ( \pi _ { \theta } , \psi ) = \int _ { S } d _ { \theta } ( s | \psi ) \int _ { A } \pi _ { \theta } ( s , a ) A ( s , a ) d a d s +$$ + +where $\begin{array} { r } { d _ { \theta } ( s | \psi ) = \int _ { S } \sum _ { t = 0 } ^ { T } \gamma ^ { t } p _ { 0 } ( s _ { 0 } ) p ( s _ { 0 } \to s | t , \pi _ { \theta } , \psi ) d s _ { 0 } } \end{array}$ . $\theta$ and $\psi$ are then learned in tandem following Algorithm 2. This alternating method optimizes both the control and dynamics in order to maximize the expected value of the agent, as analogous to the role of evolution in biomechanics. During each pass, the policy parameters $\theta$ are trained to improve the agent’s expected value for a fixed set of actuator parameters $\psi$ . Next, $\psi$ is optimized using CMA to improve performance while keeping $\theta$ fixed. The expected value of each CMA sample of $\psi$ is estimated using the average cumulative reward over multiple rollouts. + +Figure 5 illustrates the improvement in performance during the optimization process, as applied to motions for three different agents. Figure 6 compares the learning curves for the initial and final MTU parameters, for the same three motions. + +![](images/2aa165cccc6359401c538e389058e6300bc5e8138336d124477ef18a0e095674.jpg) +Figure 5: Performance of intermediate MTU policies and actuator parameters per pass of actuator optimization following Algorithm 2. + +![](images/6c476091ce400a69ffdd6088f73cb4ca31db9148402a5816b61e98671118be8d.jpg) +Figure 6: Learning curves comparing initial and optimized MTU parameters. + +# Bounded Action Space + +Properties such as torque and neural activation limits result in bounds on the range of values that can be assumed by actions for a particular parameterization. Improper enforcement of these bounds can lead to unstable learning as the gradient information outside the bounds may not be reliable (Hausknecht $\&$ Stone, 2015). To ensure that all actions respect their bounds, we adopt a method similar to the inverting gradients approach proposed by Hausknecht & Stone (2015). Let $\nabla a \ : =$ $( a - \mu ( s ) ) A ( s , a )$ be the empirical action gradient from the policy gradient estimate of a Gaussian policy. Given the lower and upper bounds $\bar { [ } l ^ { i } , u ^ { i } ]$ of the ith action parameter, the bounded gradient of the $i$ th action parameter $\nabla \tilde { a } ^ { i }$ is determined according to + +$$ +\begin{array} { r } { \nabla \tilde { a } ^ { i } = \left\{ \begin{array} { l l } { l ^ { i } - \mu ^ { i } ( s ) , } & { \mu ^ { i } ( s ) < l ^ { i } \mathrm { ~ a n d ~ } \nabla a ^ { i } < 0 } \\ { u ^ { i } - \mu ^ { i } ( s ) , } & { \mu ^ { i } ( s ) > u ^ { i } \mathrm { ~ a n d ~ } \nabla a ^ { i } > 0 } \\ { \nabla a ^ { i } , } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \end{array} +$$ + +Unlike the inverting gradients approach, which scales all gradients depending on proximity to the bounds, this method preserves the empirical gradients when bounds are respected, and alters the gradients only when bounds are violated. + +# Reward + +The terms of the reward function are defined as follows: + +$$ +\begin{array} { c } { { r _ { p o s e } = \displaystyle \exp \left( - | | q ^ { * } - q | | _ { W } ^ { 2 } \right) , \qquad r _ { v e l } = \exp \left( - | | \dot { q } ^ { * } - \dot { q } | | _ { W } ^ { 2 } \right) } } \\ { { r _ { e n d } = \exp \left( - 4 0 \displaystyle \sum _ { e } | | x _ { e } ^ { * } - x _ { e } | | ^ { 2 } \right) } } \\ { { r _ { r o o t } = \exp \left( - 1 0 ( h _ { r o o t } ^ { * } - h _ { r o o t } ) ^ { 2 } \right) , \qquad r _ { c o m } = \exp \left( - 1 0 | | \dot { x } _ { c o m } ^ { * } - \dot { x } _ { c o m } | | ^ { 2 } \right) } } \end{array} +$$ + +$q$ and $q ^ { * }$ denotes the character pose and reference pose represented in reduced-coordinates, while $\dot { q }$ and $\dot { q } ^ { * }$ are the respective joints velocities. $W$ is a manually-specified per joint diagonal weighting matrix. $h _ { r o o t }$ is the height of the root from the ground, and $\dot { x } _ { c o m }$ is the center of mass velocity. + +![](images/6051f1be1329d01271438efb16ecddd05702aeeb72f660b322b3b9b628b86633.jpg) +Figure 7: Left: fixed initial state biases agent to regions of the state space near the initial state, particular during early iterations of training. Right: initial states sampled from reference trajectory allows agent to explore state space more uniformly around reference trajectory. + +![](images/a65fe6924458b4e1ce4e13840ab449634cd9b7ea950838cff36b93251c1fdcf1.jpg) +Figure 8: Neural Network Architecture. Each policy is represented by a three layered network, with 512 and 256 fully-connected hidden units, followed by a linear output layer. + +Table 2: Training hyperparameters. + +
ParameterValueDescription
Y0.9cumulative reward discount factor
απ0.001actor learning rate
αv0.01critic learning rate
momentum0.9stochastic gradient descent momentum
weight decay0L2 regularizer for critic parameters
θ weight decay0.0005L2 regularizer for actor parameters
minibatch size32tuples per stochastic gradient descent step
replay memory size500000number of the most recent tuples stored for future updates
+ +Table 3: The number of state, action, and actuation model parameters for different characters and actuation models. + +
Character+ Actuation ModelState ParametersAction ParametersActuator Parameters
Biped + TorBiped + VelBiped + PDBiped +MTU5858587460612114
6616
Raptor + TorRaptor + VelRaptor + PDRaptor + MTU1541541541941801836258
18
1840
Dog + TorDog + VelDog + PDDog + MTU1701701702142002040282
20
20
44
+ +Table 4: Performance of policies trained for the various characters and actuation models. Performance is measured using the normalized cumulative reward (NCR) and learning speed is represented by the normalized area under each learning curve (AUC). The best performing parameterizations for each character and motion are in bold. + +
Character+ActuationMotionPerformance (NCR)Learning Speed (AUC)
Biped + Tor Biped + Vel Biped + PDWalk Walk Walk0.7662 ± 0.3117 0.9520 ± 0.0034 0.9524 ± 0.0034 0.9584 ± 0.00650.4788 0.6308 0.6997 0.7165
Biped + MTU Biped +Tor Biped + Vel Biped +PDWalk March March March March0.9353 ±0.0072 0.9784 ± 0.0018 0.9767 士 0.0068 0.9484 ±0.00210.7478 0.9035 0.9136 0.5587
Biped + MTU Biped + Tor Biped + Vel Biped + PDRun Run Run0.9032± 0.0102 0.9070 ± 0.0106 0.9057 ± 0.00560.6938 0.7301 0.7880
Biped +MTU Raptor+ Tor Raptor + VelRun Run (Sim) Run (Sim)0.8988 ± 0.0094 0.7265 士 0.0037 0.9612 土 0.00550.5360 0.5061 0.8118
Raptor + PDRun (Sim)0.9863 ± 0.00170.9282
Raptor + MTURun (Sim)0.9708 ± 0.00230.6330
Raptor +TorRun
0.6141 ± 0.00910.3814
Raptor + VelRun0.8732 ±0.00370.7008
Raptor + PDRun0.9548 ± 0.00100.8372
Raptor + MTURun0.9533 ± 0.0015
Dog + Tor0.7258
Bound (Sim)0.7888 ± 0.00460.4895
Dog + VelBound (Sim)0.9788 ±0.00440.7862
Dog +PDBound (Sim)0.9797 ± 0.00120.9280
Dog+ MTUBound (Sim)0.9033 ±0.00290.6825
Dog + TorRear-Up0.8151 士 0.01130.5550
Dog + VelRear-Up0.7364 ± 0.27070.7454
Dog + PDRear-Up0.9565 ± 0.00580.8701
Dog +MTURear-Up0.8744 ± 0.25660.7932
+ +# Sensitivity Analysis + +We further analyze the sensitivity of the results to different initializations and design decisions. Figure 9 compares the learning curves from multiple policies trained using different random initializations of the networks. Four policies are trained for each actuation model. The results for a particular actuation model are similar across different runs, and the trends between the various actuation models also appear to be consistent. To evaluate the sensitivity to the amount of exploration noise applied during training, we trained policies where the standard deviation of the action distribution is twice and half of the default values. Figure 10 illustrates the learning curves for each policy. Overall, the performance of the policies do not appear to change significantly for the particular range of values. Finally, Figure 11 compares the results using different network architectures. The network variations include doubling the number of units in both hidden layers, halving the number of hidden units, and inserting an additional layer with 512 units between the two existing hidden layers. The choice of network structure does not appear to have a noticeable impact on the results, and the differences between the actuation models appear to be consistent across the different networks. + +![](images/a4122f84a35d17b435985fd9066bfb9e44ca5abd22dbbfe7c0468a11a3984ddb.jpg) +Figure 9: Learning curves from different random network initializations. Four policies are trained for each actuation model. + +![](images/ecc0556e6aa6b37e854123df001f02073985ee151ec65d8e57c9e40b0010d5db.jpg) +Figure 10: Learning curves comparing the effects of scaling the standard deviation of the action distribution by 1x, $2 \mathbf { x }$ , and $1 / 2 \mathbf { x }$ . + +![](images/81a0d5f7bf058539a618f806f9cf036cc6447092bac4b8944cce3c08ad5027fc.jpg) +Figure 11: Learning curves for different network architectures. The network structures include, doubling the number of units in each hidden layer, halving the number of units, and inserting an additional hidden layer with 512 units between the two existing hidden layers. + +![](images/96e285809966aa4462223b8ef89ba17a7af3dedd24fa36b8b2bb490481102985.jpg) +Figure 12: Performance of different action parameterizations when traveling across randomly generated irregular terrain. (left) Dog running across bumpy terrain, where the height of each bump varies uniformly between 0 and a specified maximum height. (middle) and (right) biped and dog traveling across randomly generated slopes with bounded maximum steepness. + +![](images/e5bb1f37053981e4f42fa138e3225788bb00a225a75572ffbdbc9b523e4f5c4a.jpg) +Figure 13: Simulated Motions Using the PD Action Representation. The top row uses an MTU action space while the remainder are driven by a PD action space. + +![](images/878b406dd0a5f4dd939f54654f51993f4acb9a3796220905c1584a0eb1ae2125.jpg) +Figure 14: Policy actions over time and the resulting torques for the four action types. Data is from one biped walk cycle (1s). Left: Actions $( 6 0 \mathrm { H z } )$ , for the right hip for PD, Vel, and Tor, and the right gluteal muscle for MTU. Right: Torques applied to the right hip joint, sampled at $6 0 0 \mathrm { H z }$ . + +![](images/2ae25d079af3301717e68708ae16d52963be7f22046b2e82c2f3f40ec67dd423.jpg) +Figure 15: Learning curves for different state representations including state $^ +$ target state, state $^ +$ phase, and only state. \ No newline at end of file diff --git a/parse/train/r1kGbydxg/r1kGbydxg_content_list.json b/parse/train/r1kGbydxg/r1kGbydxg_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..dd3dfa5e1e837eaac7049117f49435a4deff277d --- /dev/null +++ b/parse/train/r1kGbydxg/r1kGbydxg_content_list.json @@ -0,0 +1,1583 @@ +[ + { + "type": "text", + "text": "LEARNING LOCOMOTION SKILLS USING DEEPRL: DOES THE CHOICE OF ACTION SPACE MATTER? ", + "text_level": 1, + "bbox": [ + 176, + 98, + 779, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Xue Bin Peng & Michiel van de Panne ", + "bbox": [ + 186, + 170, + 450, + 184 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Department of Computer Science University of the British Columbia Vancouver, Canada {xbpeng,van}@cs.ubc.ca ", + "bbox": [ + 184, + 185, + 411, + 239 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 276, + 544, + 291 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The use of deep reinforcement learning allows for high-dimensional state descriptors, but little is known about how the choice of action representation impacts the learning difficulty and the resulting performance. We compare the impact of four different action parameterizations (torques, muscle-activations, target joint angles, and target joint-angle velocities) in terms of learning time, policy robustness, motion quality, and policy query rates. Our results are evaluated on a gaitcycle imitation task for multiple planar articulated figures and multiple gaits. We demonstrate that the local feedback provided by higher-level action parameterizations can significantly impact the learning, robustness, and quality of the resulting policies. ", + "bbox": [ + 233, + 311, + 764, + 450 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 486, + 336, + 502 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "The introduction of deep learning models to reinforcement learning (RL) has enabled policies to operate directly on high-dimensional, low-level state features. As a result, deep reinforcement learning (DeepRL) has demonstrated impressive capabilities, such as developing control policies that can map from input image pixels to output joint torques (Lillicrap et al., 2015). However, the quality and robustness often falls short of what has been achieved with hand-crafted action abstractions, e.g., Coros et al. (2011); Geijtenbeek et al. (2013). While much is known about the learning of state representations, the choice of action parameterization is a design decision whose impact is not yet well understood. ", + "bbox": [ + 174, + 520, + 825, + 631 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Joint torques can be thought of as the most basic and generic representation for driving the movement of articulated figures, given that muscles and other actuation models eventually result in joint torques. However this ignores the intrinsic embodied nature of biological systems, particularly the synergy between control and biomechanics. Passive-dynamics, such as elasticity and damping from muscles and tendons, play an integral role in shaping motions: they provide mechanisms for energy storage, and mechanical impedance which generates instantaneous feedback without requiring any explicit computation. Loeb coins the term preflexes (Loeb, 1995) to describe these effects, and their impact on motion control has been described as providing intelligence by mechanics (Blickhan et al., 2007). This can also be thought of as a kind of partitioning of the computations between the control and physical system. ", + "bbox": [ + 174, + 638, + 825, + 777 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper we explore the impact of four different actuation models on learning to control dynamic articulated figure locomotion: (1) torques (Tor); (2) activations for musculotendon units (MTU); (3) target joint angles for proportional-derivative controllers (PD); and (4) target joint velocities (Vel). Because Deep RL methods are capable of learning control policies for all these models, it now becomes possible to directly assess how the choice of actuation model affects the learning difficulty. We also assess the learned policies with respect to robustness, motion quality, and policy query rates. We show that action spaces which incorporate local feedback can significantly improve learning speed and performance, while still preserving the generality afforded by torque-level control. Such parameterizations also allow for more complex body structures and subjective improvements in motion quality. ", + "bbox": [ + 174, + 785, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Our specific contributions are: (1) We introduce a DeepRL framework for motion imitation tasks; (2) We evaluate the impact of four different actuation models on learned control policies according to four criteria; and (3) We propose an optimization approach that combines policy learning and actuator optimization, allowing neural networks to effective control complex muscle models. ", + "bbox": [ + 173, + 103, + 825, + 160 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 BACKGROUND ", + "text_level": 1, + "bbox": [ + 174, + 179, + 326, + 195 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our task will be structured as a standard reinforcement problem where an agent interacts with its environment according to a policy in order to maximize a reward signal. The policy $\\pi ( s , a ) = p ( a | s )$ represents the conditional probability density function of selecting action $a \\in { \\mathcal { A } }$ in state $s \\in S$ . At each control step $t$ , the agent observes a state $s _ { t }$ and samples an action $a _ { t }$ from $\\pi$ . The environment in turn responds with a scalar reward $r _ { t }$ , and a new state $s _ { t } ^ { \\prime } \\ = \\ s _ { t + 1 }$ sampled from its dynamics $p ( s ^ { \\prime } | s , a )$ . For a parameterized policy $\\pi _ { \\boldsymbol { \\theta } } ( s , a )$ , the goal of the agent is learn the parameters $\\theta$ which maximizes the expected cumulative reward ", + "bbox": [ + 173, + 209, + 825, + 308 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/6b6ed355ac17b67747564631730a02f6d68b8b9fe17b94730cabd67e8d92d77d.jpg", + "text": "$$\nJ ( \\pi _ { \\theta } ) = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { T } \\gamma ^ { t } r _ { t } \\Big | \\pi _ { \\theta } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 411, + 310, + 584, + 356 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "with $\\gamma \\in [ 0 , 1 ]$ as the discount factor, and $T$ as the horizon. The gradient of the expected reward $\\nabla _ { \\boldsymbol { \\theta } } J ( \\pi _ { \\boldsymbol { \\theta } } )$ can be determined according to the policy gradient theorem (Sutton et al., 2001), which provides a direction of improvement to adjust the policy parameters $\\theta$ . ", + "bbox": [ + 173, + 364, + 826, + 409 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/8271fccebfdd479538cfb57c0c7679feeb5b22de2b20da47fc5e77f3232036fb.jpg", + "text": "$$\n\\nabla _ { \\theta } J ( \\pi _ { \\theta } ) = \\int _ { S } d _ { \\theta } ( s ) \\int _ { A } \\nabla _ { \\theta } \\log ( \\pi _ { \\theta } ( s , a ) ) A ( s , a ) d a d s\n$$", + "text_format": "latex", + "bbox": [ + 318, + 424, + 679, + 457 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { d _ { \\theta } ( s ) = \\int _ { S } \\sum _ { t = 0 } ^ { T } \\gamma ^ { t } p _ { 0 } ( s _ { 0 } ) p ( s _ { 0 } \\to s | t , \\pi _ { \\theta } ) d s _ { 0 } } \\end{array}$ is the discounted state distribution, where $p _ { 0 } ( s )$ represents the initial state distribution, and $p ( s _ { 0 } \\to s | t , \\pi _ { \\theta } )$ models the likelihood of reaching state $s$ by starting at $s _ { 0 }$ and following the policy $\\pi _ { \\theta } ( s , a )$ for $t$ steps (Silver et al., 2014). $A ( s , \\bar { a } )$ represents a generalized advantage function. The choice of advantage function gives rise to a family of policy gradient algorithms, but in this work, we will focus on the one-step temporal difference advantage function (Schulman et al., 2015) ", + "bbox": [ + 173, + 465, + 826, + 554 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/7970ef16a13e3c42cf836087d5a74db4b6a9702ab990dbcfe12218ac3b17c718.jpg", + "text": "$$\nA ( s _ { t } , a _ { t } ) = r _ { t } + \\gamma V ( s _ { t } ^ { \\prime } ) - V ( s _ { t } )\n$$", + "text_format": "latex", + "bbox": [ + 385, + 556, + 612, + 574 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { V ( s ) = \\mathbb { E } \\left[ \\sum _ { t = 0 } ^ { T } \\gamma ^ { t } r _ { t } \\Big \\rvert s _ { 0 } = s , \\pi _ { \\theta } \\right] } \\end{array}$ is the state-value function, and can be defined recursively via the Bellman equation ", + "bbox": [ + 174, + 578, + 820, + 613 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/8fafe83ea8a1863d96375ba967100f0c6eb2e15c5bb6fbdfb218b860fcc7f442.jpg", + "text": "$$\nV ( s _ { t } ) = \\underset { r _ { t } , s _ { t } ^ { \\prime } } { \\mathbb { E } } \\left[ r _ { t } + \\gamma V ( s _ { t } ^ { \\prime } ) | s _ { t } , \\pi _ { \\theta } \\right]\n$$", + "text_format": "latex", + "bbox": [ + 385, + 612, + 612, + 640 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "A parameterized value function $V _ { \\phi } ( s )$ , with parameters $\\phi$ , can be learned iteratively in a manner similar to Q-Learning by minimizing the Bellman loss, ", + "bbox": [ + 173, + 648, + 825, + 678 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/fbe51251860dc9a410f24cbe6593d4897db16ff3e83537481b0a6b56abcf187f.jpg", + "text": "$$\nL ( \\phi ) = \\underset { s _ { t } , r _ { t } , s _ { t } ^ { \\prime } } { \\mathbb { E } } \\left[ \\frac { 1 } { 2 } \\left( y _ { t } - V _ { \\phi } ( s _ { t } ) \\right) ^ { 2 } \\right] , \\qquad y _ { t } = r _ { t } + \\gamma V _ { \\phi } ( s _ { t } ^ { \\prime } )\n$$", + "text_format": "latex", + "bbox": [ + 299, + 681, + 699, + 717 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "$\\pi _ { \\theta }$ and $V _ { \\phi }$ can be trained in tandem using an actor-critic framework (Konda & Tsitsiklis, 2000). ", + "bbox": [ + 166, + 727, + 799, + 742 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In this work, each policy will be represented as a gaussian distribution with a parameterized mean $\\mu _ { \\boldsymbol { \\theta } } ( s )$ and fixed covariance matrix $\\dot { \\Sigma } = \\mathrm { d i a g } \\{ \\sigma _ { i } ^ { 2 } \\}$ , where $\\sigma _ { i }$ is manually specified for each action parameter. Actions can be sampled from the distribution by applying gaussian noise to the mean action ", + "bbox": [ + 173, + 748, + 825, + 804 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/b55c7f136e66d4ee965abce306dc1f40951ababd94ffbc3fdf33ee1f0ef47fe1.jpg", + "text": "$$\na _ { t } = \\mu _ { \\theta } ( s _ { t } ) + \\mathcal { N } ( 0 , \\Sigma )\n$$", + "text_format": "latex", + "bbox": [ + 419, + 803, + 578, + 820 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The corresponding policy gradient will assume the form ", + "bbox": [ + 176, + 820, + 544, + 835 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/d5581a3db1a0ec6a5221e8a0aac193bb94ef37a2d0b1c8e573b0af46921fb8d2.jpg", + "text": "$$\n\\nabla _ { \\theta } J ( \\pi _ { \\theta } ) = \\int _ { S } d _ { \\theta } ( s ) \\int _ { A } \\nabla _ { \\theta } \\mu _ { \\theta } ( s ) \\Sigma ^ { - 1 } \\left( a - \\mu _ { \\theta } ( s ) \\right) A ( s , a ) d a d s\n$$", + "text_format": "latex", + "bbox": [ + 287, + 838, + 710, + 872 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "which can be interpreted as shifting the mean of the action distribution towards actions that lead to higher than expected rewards, while moving away from actions that lead to lower than expected rewards. ", + "bbox": [ + 174, + 881, + 823, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "3 TASK REPRESENTATION ", + "text_level": 1, + "bbox": [ + 176, + 102, + 406, + 118 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 REFERENCE MOTION ", + "text_level": 1, + "bbox": [ + 176, + 133, + 362, + 148 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In our task, the goal of a policy is to imitate a given reference motion $\\{ q _ { t } ^ { * } \\}$ which consists of a sequence of kinematic poses $q _ { t } ^ { * }$ in reduced coordinates. The reference velocity $\\dot { q _ { t } ^ { * } }$ at a given time $t$ is approximated by finite-difference $\\begin{array} { r } { \\dot { q _ { t } ^ { * } } \\approx \\frac { q _ { t + \\triangle t } ^ { * } - q _ { t } ^ { * } } { \\triangle t } } \\end{array}$ . Reference motions are generated via either using a recorded simulation result from a preexisting controller $( ^ { 6 6 } \\mathrm { S i m } ^ { 7 } )$ , or via manually-authored keyframes. Since hand-crafted reference motions may not be physically realizable, the goal is to closely reproduce a motion while satisfying physical constraints. ", + "bbox": [ + 173, + 160, + 825, + 253 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.2 STATES ", + "text_level": 1, + "bbox": [ + 174, + 272, + 266, + 286 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "To define the state of the agent, a feature transformation $\\Phi ( q , \\dot { q } )$ is used to extract a set of features from the reduced-coordinate pose $q$ and velocity $\\dot { q }$ . The features consist of the height of the root (pelvis) from the ground, the position of each link with respect to the root, and the center of mass velocity of each link. When training a policy to imitate a cyclic reference motion $\\{ q _ { t } ^ { * } \\}$ , knowledge of the motion phase can help simplify learning. Therefore, we augment the state features with a set of target features $\\Phi ( q _ { t } ^ { * } , \\dot { q } _ { t } ^ { * } )$ , resulting in a combined state represented by $s _ { t } = ( \\Phi ( q _ { t } , \\dot { q _ { t } } ) , \\Phi ( q _ { t } ^ { * } , { \\dot { q _ { t } } } ^ { * } ) )$ . Similar results can also be achieved by providing a single motion phase variable as a state feature, as we show in Figure 15 (supplemental material). ", + "bbox": [ + 174, + 299, + 825, + 411 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.3 ACTIONS ", + "text_level": 1, + "bbox": [ + 174, + 429, + 279, + 444 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We train separate policies for each of the four actuation models, as described below. Each actuation model also has related actuation parameters, such as feedback gains for PD-controllers and musculotendon properties for MTUs. These parameters can be manually specified, as we do for the PD and Vel models, or they can be optimized for the task at hand, as for the MTU models. Table 1 provides a list of actuator parameters for each actuation model. ", + "bbox": [ + 174, + 455, + 825, + 526 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Target Joint Angles (PD): Each action represents a set of target angles $\\hat { q }$ , where $\\hat { q } ^ { i }$ specifies the target angles for joint $i$ . $\\hat { q }$ is applied to PD-controllers which compute torques according to $\\tau ^ { i } =$ $k _ { p } ^ { i } ( \\hat { q } ^ { i } - q ^ { i } ) + k _ { d } ^ { i } ( \\hat { \\dot { q } } ^ { i } - \\dot { q } ^ { i } )$ , where $\\hat { \\dot { q } } ^ { i } = 0$ , and $k _ { p } ^ { i }$ and $k _ { d } ^ { i }$ are manually-specified gains. ", + "bbox": [ + 174, + 534, + 825, + 578 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Target Joint Velocities (Vel): Each action specifies a set of target velocities $\\hat { \\dot { q } }$ which are used to compute torques according to $\\tau ^ { i } = k _ { d } ^ { i } ( { \\hat { \\dot { q } } } ^ { i } - { \\dot { q } } ^ { i } )$ , where the gains $k _ { d } ^ { i }$ are specified to be the same as those used for target angles. ", + "bbox": [ + 174, + 587, + 825, + 632 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Torques (Tor): Each action directly specifies torques for every joint, and constant torques are applied for the duration of a control step. Due to torque limits, actions are bounded by manually specified limits for each joint. Unlike the other actuation models, the torque model does not require additional actuator parameters, and can thus be regarded as requiring the least amount of domain knowledge. Torque limits are excluded from the actuator parameter set as they are common for all parameterizations. ", + "bbox": [ + 173, + 638, + 825, + 722 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Muscle Activations (MTU): Each action specifies activations for a set of musculotendon units (MTU). Detailed modeling and implementation information are available in Wang et al. (2012). Each MTU is modeled as a contractile element (CE) attached to a serial elastic element (SE) and parallel elastic element (PE). The force exerted by the MTU can be calculated according to $F _ { M T U } = F _ { S E } = F _ { C E } + F _ { P E }$ . Both $F _ { S E }$ and $F _ { P E }$ are modeled as passive springs, while $F _ { C E }$ is actively controlled according to $F _ { C E } = a _ { M T U } F _ { 0 } f _ { l } ( l _ { C E } ) f _ { v } ( v _ { C E } )$ , with $a _ { M T U }$ being the muscle activation, $F _ { 0 }$ the maximum isometric force, $l _ { C E }$ and $v _ { C E }$ being the length and velocity of the contractile element. The functions $f _ { l } ( l _ { C E } )$ and $f _ { v } ( v _ { C E } )$ represent the force-length and force-velocity relationships, modeling the variations in the maximum force that can be exerted by a muscle as a function of its length and contraction velocity. Analytic forms are available in Geyer et al. (2003). Activations are bounded between [0, 1]. The length of each contractile element $l _ { C E }$ are included as state features. To simplify control and reduce the number of internal state parameters per MTU, the policies directly control muscle activations instead of indirectly through excitations (Wang et al., 2012). ", + "bbox": [ + 173, + 729, + 825, + 924 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/9faf58aa97a2cbbbbb1a11007bafc15617c0928f4a1d46ac6c54b1bf24b386d1.jpg", + "table_caption": [ + "Table 1: Actuation models and their respective actuator parameters. " + ], + "table_footnote": [], + "table_body": "
ActuationModelActuatorParameters
Target Joint Angles (PD) proportional gains kp, derivative gains kd
Target Joint Velocities(Vel)derivative gains kd
Torques (Tor)none
Muscle Activations (MTU)optimal contractile element length, serial elastic element rest length,maximum isometric force, pennation,moment arm,maximum moment arm joint orientation, rest joint orientation.
", + "bbox": [ + 173, + 101, + 841, + 204 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.4 REWARD ", + "text_level": 1, + "bbox": [ + 174, + 255, + 277, + 268 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The reward function consists of a weighted sum of terms that encourage the policy to track a reference motion. ", + "bbox": [ + 173, + 281, + 823, + 309 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/7ed72042a39f1e0c19dfafcaeca5f6589359e008f5983457afbccae85628b183.jpg", + "text": "$$\n\\begin{array} { c } { r = w _ { p o s e } r _ { p o s e } + w _ { v e l } r _ { v e l } + w _ { e n d } r _ { e n d } + w _ { r o o t } r _ { r o o t } + w _ { c o m } r _ { c o m } } \\\\ { ~ } \\\\ { w _ { p o s e } = 0 . 5 , ~ w _ { v e l } = 0 . 0 5 , ~ w _ { e n d } = 0 . 1 5 , ~ w _ { r o o t } = 0 . 1 , ~ w _ { c o m } = 0 . 2 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 272, + 318, + 725, + 361 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Details of each term are available in the supplemental material. $r _ { p o s e }$ penalizes deviation of the character pose from the reference pose, and $r _ { v e l }$ penalizes deviation of the joint velocities. $r _ { e n d }$ and $r _ { r o o t }$ accounts for the position error of the end-effectors and root. $r _ { c o m }$ penalizes deviations in the center of mass velocity from that of the reference motion. ", + "bbox": [ + 174, + 371, + 826, + 428 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.5 INITIAL STATE DISTRIBUTION ", + "text_level": 1, + "bbox": [ + 176, + 445, + 423, + 460 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We design the initial state distribution, $p _ { 0 } ( s )$ , to sample states uniformly along the reference trajectory. At the start of each episode, $q ^ { * }$ and $\\dot { q } ^ { * }$ are sampled from the reference trajectory, and used to initialize the pose and velocity of the agent. This helps guide the agent to explore states near the target trajectory. ", + "bbox": [ + 173, + 470, + 825, + 529 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 ACTOR-CRITIC LEARNING ALGORITHM ", + "text_level": 1, + "bbox": [ + 174, + 547, + 539, + 565 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Instead of directly using the temporal difference advantage function, we adapt a positive temporal difference (PTD) update as proposed by Van Hasselt (2012). ", + "bbox": [ + 171, + 580, + 823, + 609 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/817c049178dadd041fc4f8bbd27bc88ee4048bae55f476073c229dc8ae687a55.jpg", + "text": "$$\nA ( s , a ) = I \\left[ \\delta > 0 \\right] = { \\left\\{ \\begin{array} { l l } { 1 , } & { \\delta > 0 } \\\\ { 0 , } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. }\n$$", + "text_format": "latex", + "bbox": [ + 369, + 617, + 629, + 654 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/f3fc7ac2153613e4aedf2591b96e4d41d03e084c7906aa175bb593437dc3fcb9.jpg", + "text": "$$\n\\delta = r + \\gamma V ( s ^ { \\prime } ) - V ( s )\n$$", + "text_format": "latex", + "bbox": [ + 416, + 662, + 580, + 680 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Unlike more conventional policy gradient methods, PTD is less sensitive to the scale of the advantage function and avoids instabilities that can result from negative TD updates. For a Gaussian policy, a negative TD update moves the mean of the distribution away from an observed action, effectively shifting the mean towards an unknown action that may be no better than the current mean action (Van Hasselt, 2012). In expectation, these updates converges to the true policy gradient, but for stochastic estimates of the policy gradient, these updates can cause the agent to adopt undesirable behaviours which affect subsequent experiences collected by the agent. Furthermore, we incorporate experience replay, which has been demonstrated to improve stability when training neural network policies with Q-learning in discrete action spaces. Experience replay often requires off-policy methods, such as importance weighting, to account for differences between the policy being trained and the behavior policy used to generate experiences (WawrzynSki & Tanwani, 2013).´ However, we have not found importance weighting to be beneficial for PTD. ", + "bbox": [ + 173, + 684, + 825, + 852 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Stochastic policies are used during training for exploration, while deterministic policy are deployed for evaluation at runtime. The choice between a stochastic and deterministic policy can be specified by the addition of a binary indicator variable $\\lambda \\in [ 0 , 1 ]$ ", + "bbox": [ + 174, + 858, + 823, + 901 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d58b48f9314873eac7db0cf90f14d2de138da83d991d87f0c5df009d9926b846.jpg", + "text": "$$\na _ { t } = \\mu _ { \\theta } ( s _ { t } ) + \\lambda N ( 0 , \\Sigma )\n$$", + "text_format": "latex", + "bbox": [ + 415, + 907, + 583, + 925 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $\\lambda = 1$ corresponds to a stochastic policy with exploration noise, and $\\lambda = 0$ corresponds to a deterministic policy that always selects the mean of the distribution. Noise from a stochastic policy will result in a state distribution that differs from that of the deterministic policy at runtime. To imitate this discrepancy, we incorporate $\\epsilon$ -greedy exploration in addition to the original Gaussian exploration. During training, $\\lambda$ is determined by a Bernoulli random variable $\\lambda \\sim \\operatorname { B e r } ( \\epsilon )$ , where $\\lambda = 1$ with probability $\\epsilon \\in [ 0 , 1 ]$ . The exploration rate $\\epsilon$ is annealed linearly from 1 to 0.2 over $5 0 0 \\mathrm { k }$ iterations, which slowly adjusts the state distribution encountered during training to better resemble the distribution at runtime. Since the policy gradient is defined for stochastic policies, only tuples recorded with exploration noise (i.e. $\\lambda = 1$ ) can be used to update the actor, while the critic can be updated using all tuples. ", + "bbox": [ + 173, + 103, + 825, + 242 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Training proceeds episodically, where the initial state of each episode is sampled from $p _ { 0 } ( s )$ , and the episode duration is drawn from an exponential distribution with a mean of 2s. To discourage falling, an episode will also terminate if any part of the character’s trunk makes contact with the ground for an extended period of time, leaving the agent with zero reward for all subsequent steps. Algorithm 1 in the supplemental material summarizes the complete learning process. ", + "bbox": [ + 174, + 250, + 825, + 319 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "MTU Actuator Optimization: Actuation models such as MTUs are defined by further parameters whose values impact performance (Geijtenbeek et al., 2013). Geyer et al. (2003) uses existing anatomical estimates for humans to determine MTU parameters, but such data is not be available for more arbitrary creatures. Alternatively, Geijtenbeek et al. (2013) uses covariance matrix adaptation (CMA), a derivative-free evolutionary search strategy, to simultaneously optimize MTU and policy parameters. This approach is limited to policies with reasonably low dimensional parameter spaces, and is thus ill-suited for neural network models with hundreds of thousands of parameters. To avoid manual-tuning of actuator parameters, we propose a heuristic approach that alternates between policy learning and actuator optimization, as detailed in the supplemental material. ", + "bbox": [ + 174, + 325, + 825, + 452 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/37eeafcd2ada57b0a9a694e94f6efe9f905699c1775e021139c85ef284ffc1cf.jpg", + "image_caption": [ + "Figure 1: Simulated articulated figures and their state representation. Revolute joints connect all links. From left to right: 7-link biped; 19-link raptor; 21-link dog; State features: root height, relative position (red) of each link with respect to the root and their respective linear velocity (green). " + ], + "image_footnote": [], + "bbox": [ + 251, + 470, + 799, + 568 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "5 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 662, + 281, + 679 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The motions are best seen in the supplemental video https://youtu.be/L3vDo3nLI98. We evaluate the action parameterizations by training policies for a simulated 2D biped, dog, and raptor as shown in Figure 1. Depending on the agent and the actuation model, our systems have 58–214 state dimensions, 6–44 action dimensions, and 0–282 actuator parameters, as summarized in Table 3 (supplemental materials). The MTU models have at least double the number of action parameters because they come in antagonistic pairs. As well, additional MTUs are used for the legs to more accurately reflect bipedal biomechanics. This includes MTUs that span multiple joints. ", + "bbox": [ + 174, + 694, + 825, + 791 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Each policy is represented by a three layer neural network, as illustrated in Figure 8 (supplemental material) with 512 and 256 fully-connected units, followed by a linear output layer where the number of output units vary according to the number of action parameters for each character and actuation model. ReLU activation functions are used for both hidden layers. Each network has approximately $2 0 0 \\mathrm { k }$ parameters. The value function is represented by a similar network, except having a single linear output unit. The policies are queried at $6 0 \\mathrm { H z }$ for a control step of about 0.0167s. Each network is randomly initialized and trained for about 1 million iterations, requiring 32 million tuples, the equivalent of approximately 6 days of simulated time. Each policy requires about 10 hours for the biped, and 20 hours for the raptor and dog on an 8-core Intel Xeon E5-2687W. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/93b7292fed0365462a02a39731b438eabc707c5107988fe50e2bceec740286b3.jpg", + "image_caption": [ + "Figure 2: Learning curves for each policy during 1 million iterations. " + ], + "image_footnote": [], + "bbox": [ + 183, + 102, + 820, + 314 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Only the actuator parameters for MTUs are optimized with Algorithm 2, since the parameters for the other actuation models are few and reasonably intuitive to determine. The initial actuator parameters $\\psi _ { 0 }$ are manually specified, while the initial policy parameters $\\theta _ { 0 }$ are randomly initialized. Each pass optimizes $\\psi$ using CMA for 250 generations with 16 samples per generation, and $\\theta$ is trained for $2 5 0 \\mathrm { k }$ iterations. Parameters are initialized with values from the previous pass. The expected value of each CMA sample of $\\psi$ is estimated using the average cumulative reward over 16 rollouts with a duration of 10s each. Separate MTU parameters are optimized for each character and motion. Each set of parameters is optimized for 6 passes following Algorithm 2, requiring approximately 50 hours. Figure 5 illustrates the performance improvement per pass. Figure 6 compares the performance of MTUs before and after optimization. For most examples, the optimized actuator parameters significantly improve learning speed and final performance. For the sake of comparison, after a set of actuator parameters has been optimized, a new policy is retrained with the new actuator parameters and its performance compared to the other actuation models. ", + "bbox": [ + 174, + 375, + 825, + 554 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Policy Performance and Learning Speed: Figure 2 shows learning curves for the policies and the performance of the final policies are summarized in Table 4. Performance is evaluated using the normalized cumulative reward (NCR), calculated from the average cumulative reward over 32 episodes with lengths of 10s, and normalized by the maximum and minimum cumulative reward possible for each episode. No discounting is applied when calculating the NCR. The initial state of each episode is sampled from the reference motion according to $p ( s _ { 0 } )$ . To compare learning speeds, we use the normalized area under each learning curve (AUC) as a proxy for the learning speed of a particular actuation model, where 0 represents the worst possible performance and no progress during training, and 1 represents the best possible performance without requiring training. ", + "bbox": [ + 174, + 561, + 825, + 688 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "PD performs well across all examples, achieving comparable-to-the-best performance for all motions. PD also learns faster than the other parameterizations for 5 of the 7 motions. The final performance of Tor is among the poorest for all the motions. Differences in performance appear more pronounced as characters become more complex. For the simple 7-link biped, most parameterizations achieve similar performance. However, for the more complex dog and raptor, the performance of Tor policies deteriorate with respect to other policies such as PD and Vel. MTU policies often exhibited the slowest learning speed, which may be a consequence of the higher dimensional action spaces, i.e., requiring antagonistic muscle pairs, and complex muscle dynamics. Nonetheless, once optimized, the MTU policies produce more natural motions and responsive behaviors as compared to other parameterizations. We note that the naturalness of motions is not well captured by the reward, since it primarily gauges similarity to the reference motion, which may not be representative of natural responses when perturbed from the nominal trajectory. A sensitivity analysis of the policies’ performance to variations in network architecture and hyperparameters are available in the supplemental material. ", + "bbox": [ + 174, + 694, + 825, + 887 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Policy Robustness: To evaluate robustness, we recorded the NCR achieved by each policy when subjected to external perturbations. The perturbations assume the form of random forces applied to the trunk of the characters. Figure 3 illustrates the performance of the policies when subjected to perturbations of different magnitudes. The magnitude of the forces are constant, but direction varies randomly. Each force is applied for 0.1 to 0.4s, with 1 to 4s between each perturbation. Performance is estimated using the average over 128 episodes of length 20s each. For the biped walk, the Tor policy is significantly less robust than those for the other types of actions, while the MTU policy is the least robust for the raptor run. Overall, the PD policies are among the most robust for all the motions. In addition to external forces, we also evaluate robustness over randomly generated terrain consisting of bumps with varying heights and slopes with varying steepness. We evaluate the performance on irregular terrain (Figure 12, supplemental material). There are few discernible patterns for this test. The Vel and MTU policies are significantly worse than the Tor and PD policies for the dog bound on the bumpy terrain. The unnatural jittery behavior of the dog Tor policy proves to be surprisingly robust for this scenario. We suspect that the behavior prevents the trunk from contacting the ground for extended periods for time, and thereby escaping our system’s fall detection. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 297 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/59c61c631803c3c1ef5dced30e6b9d2417658876f02b3fafd5c5920c6ab96ddd.jpg", + "image_caption": [ + "Figure 3: Performance when subjected to random perturbation forces of different magnitudes. " + ], + "image_footnote": [], + "bbox": [ + 178, + 310, + 821, + 436 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Query Rate: Figure 4 compares the performance of different parameterizations for different policy query rates. Separate policies are trained with queries of 15Hz, $3 0 \\mathrm { H z }$ , $6 0 \\mathrm { H z }$ , and $1 2 0 \\mathrm { H z }$ . Actuation models that incorporate low-level feedback such as PD and Vel, appear to cope more effectively to lower query rates, while the Tor degrades more rapidly at lower query rates. It is not yet obvious to us why MTU policies appear to perform better at lower query rates and worse at higher rates. Lastly, Figure 14 shows the policy outputs as a function of time for the four actuation models, for a particular joint, as well as showing the resulting joint torque. Interestingly, the MTU action is visibly smoother than the other actions and results in joint torques profiles that are smoother than those seen for PD and Vel. ", + "bbox": [ + 173, + 481, + 825, + 606 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/d2c70584eca59abbfb1ffb645c9e04955f3a1ad659e5d9061ea8ca9b9960c36c.jpg", + "image_caption": [ + "Figure 4: Performance of policies with different query rates for the biped (left) and dog (right). Separate policies are trained for each query rate. " + ], + "image_footnote": [], + "bbox": [ + 277, + 618, + 718, + 752 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "6 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 823, + 344, + 839 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "DeepRL has driven impressive recent advances in learning motion control, i.e., solving for continuous-action control problems using reinforcement learning. All four of the actions types that we explore have seen previous use in the machine learning literature. WawrzynSki & Tanwani ´ (2013) use an actor-critic approach with experience replay to learn skills for an octopus arm (actuated by a simple muscle model) and a planar half cheetah (actuated by joint-based PD-controllers). ", + "bbox": [ + 174, + 854, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Recent work on deterministic policy gradients (Lillicrap et al., 2015) and on RL benchmarks, e.g., OpenAI Gym, generally use joint torques as the action space, as do the test suites in recent work (Schulman et al., 2015) on using generalized advantage estimation. Other recent work uses: the PR2 effort control interface as a proxy for torque control (Levine et al., 2015); joint velocities (Gu et al., 2016); velocities under an implicit control policy (Mordatch et al., 2015); or provide abstract actions (Hausknecht & Stone, 2015). Our learning procedures are based on prior work using actorcritic approaches with positive temporal difference updates (Van Hasselt, 2012). ", + "bbox": [ + 174, + 103, + 823, + 200 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Work in biomechanics has long recognized the embodied nature of the control problem and the view that musculotendon systems provide “preflexes” (Loeb, 1995) that effectively provide a form intelligence by mechanics (Blickhan et al., 2007), as well as allowing for energy storage. The control strategies for physics-based character simulations in computer animation also use all the forms of actuation that we evaluate in this paper. Representative examples include quadratic programs that solve for joint torques (de Lasa et al., 2010), joint velocities for skilled bicycle stunts (Tan et al., 2014), muscle models for locomotion (Wang et al., 2012; Geijtenbeek et al., 2013), mixed use of feed-forward torques and joint target angles (Coros et al., 2011), and joint target angles computed by learned linear (time-indexed) feedback strategies (Liu et al., 2016). Lastly, control methods in robotics use a mix of actuation types, including direct-drive torques (or their virtualized equivalents), series elastic actuators, PD control, and velocity control. These methods often rely heavily on modelbased solutions and thus we do not describe these in further detail here. ", + "bbox": [ + 174, + 208, + 825, + 375 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "7 CONCLUSIONS ", + "text_level": 1, + "bbox": [ + 176, + 395, + 328, + 411 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Our experiments suggest that action parameterizations that include basic local feedback, such as PD target angles, MTU activations, or target velocities, can improve policy performance and learning speed across different motions and character morphologies. Such models more accurately reflect the embodied nature of control in biomechanical systems, and the role of mechanical components in shaping the overall dynamics of motions and their control. The difference between low-level and high-level action parameterizations grow with the complexity of the characters, with high-level parameterizations scaling more gracefully to complex characters. As a caveat, there may well be tasks, such as impedance control, where lower-level action parameterizations such as Tor may prove advantageous. We believe that no single action parameterization will be the best for all problems. However, since objectives for motion control problems are often naturally expressed in terms of kinematic properties, higher-level actions such as target joint angles and velocities may be effective for a wide variety of motion control problems. We hope that our work will help open discussions around the choice of action parameterizations. ", + "bbox": [ + 174, + 426, + 825, + 607 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Our results have only been demonstrated on planar articulated figure simulations; the extension to 3D currently remains as future work. Furthermore, our current torque limits are still large as compared to what might be physically realizable. Tuning actuator parameters for complex actuation models such as MTUs remains challenging. Though our actuator optimization technique is able to improve performance as compared to manual tuning, the resulting parameters may still not be optimal for the desired task. Therefore, our comparisons of MTUs to other action parameterizations may not be reflective of the full potential of MTUs with more optimal actuator parameters. Furthermore, our actuator optimization currently tunes parameters for a specific motion, rather than a larger suite of motions, as might be expected in nature. ", + "bbox": [ + 174, + 614, + 825, + 738 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Since the reward terms are mainly calculated according to joint positions and velocities, it may seem that it is inherently biased in favour of PD and Vel. However, the real challenges for the control policies lie elsewhere, such as learning to compensate for gravity and ground-reaction forces, and learning foot-placement strategies that are needed to maintain balance for the locomotion gaits. The reference pose terms provide little information on how to achieve these hidden aspects of motion control that will ultimately determine the success of the locomotion policy. While we have yet to provide a concrete answer for the generalization of our results to different reward functions, we believe that the choice of action parameterization is a design decision that deserves greater attention regardless of the choice of reward function. ", + "bbox": [ + 174, + 746, + 825, + 871 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Finally, it is reasonable to expect that evolutionary processes would result in the effective co-design of actuation mechanics and control capabilities. 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", + "bbox": [ + 169, + 892, + 823, + 921 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "SUPPLEMENTARY MATERIAL ", + "bbox": [ + 176, + 102, + 416, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Algorithm 1 Actor-critic Learning Using Positive Temporal Differences ", + "text_level": 1, + "bbox": [ + 174, + 140, + 648, + 155 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "1: $\\theta \\gets$ random weights \n2: $\\phi $ random weights \n3: while not done do \n4: for $s t e p = 1 , . . . , m$ do \n5: $s \\gets$ start state \n6: $\\lambda \\operatorname { B e r } ( \\epsilon _ { t } )$ \n7: $a \\gets \\mu _ { \\theta } ( s ) + \\lambda \\mathcal { N } ( 0 , \\Sigma )$ \n8: Apply $a$ and simulate forward 1 step \n9: $s ^ { \\prime } \\gets$ end state \n10: $r \\gets$ reward \n11: $\\tau \\gets ( s , a , r , s ^ { \\prime } , \\lambda )$ \n12: store $\\tau$ in replay memory \n13: if episode terminated then \n14: Sample $s _ { 0 }$ from $p _ { 0 } ( s )$ \n15: Reinitialize state s to s0 \n16: end if \n17: end for \n18: Update critic: \n19: Sample minibatch of $n$ tuples $\\{ \\tau _ { i } = ( s _ { i } , a _ { i } , r _ { i } , \\lambda _ { i } , s _ { i } ^ { \\prime } ) \\}$ from replay memory \n20: for each $\\tau _ { i }$ do \n21: $\\delta _ { i } \\gets r _ { i } + \\gamma V _ { \\phi } ( s _ { i } ^ { \\prime } ) - V _ { \\phi } ( s _ { i } )$ \n22: $\\begin{array} { r } { \\phi \\phi + \\alpha _ { V } \\frac { 1 } { n } \\delta _ { i } \\nabla _ { \\phi } V _ { \\phi } ( s _ { i } ) } \\end{array}$ \n23: end for \n24: Update actor: \n25: Sample minibatch of $n$ tuples $\\{ \\tau _ { j } = ( s _ { j } , a _ { j } , r _ { j } , \\lambda _ { j } , s _ { j } ^ { \\prime } ) \\}$ from replay memory where $\\lambda _ { j } = 1$ \n26: for each $\\tau _ { j }$ do \n27: $\\delta _ { j } r _ { j } ^ { - } + \\gamma V _ { \\phi } ( s _ { j } ^ { \\prime } ) - V _ { \\phi } ( s _ { j } )$ \n28: if $\\delta _ { j } > 0$ then \n29: Oaj ← aj − µθ(sj ) \n30: Oa˜j ← BoundActionGradient(Oaj , µθ(sj )) \n31: θ ← θ + απ 1n Oθµθ(sj )Σ−1Oa˜j \n32: end if \n33: end for \n34: end while ", + "bbox": [ + 171, + 156, + 821, + 695 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Algorithm 2 Alternating Actuator Optimization ", + "text_level": 1, + "bbox": [ + 176, + 712, + 486, + 727 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "1: θ ← θ0 \n2: ψ ← ψ0 \n3: while not done do \n4: $\\theta \\gets$ argmaxθ0 $J ( \\pi _ { \\theta ^ { \\prime } } , \\psi )$ with Algorithm 1 \n5: $\\psi \\mathrm { a r g m a x } _ { \\psi ^ { \\prime } } J ( \\pi _ { \\theta } , \\psi ^ { \\prime } )$ with CMA \n6: end while ", + "bbox": [ + 179, + 731, + 500, + 813 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "MTU Actuator Optimization ", + "text_level": 1, + "bbox": [ + 174, + 98, + 465, + 119 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "The actuator parameters $\\psi$ can be interpreted as a parameterization of the dynamics of the system $p ( s ^ { \\prime } | s , a , \\psi )$ . The expected cumulative reward can then be re-parameterized according to ", + "bbox": [ + 173, + 125, + 825, + 154 + ], + "page_idx": 10 + }, + { + "type": "equation", + "img_path": "images/ca2632ce1af05513b7b3f6ee21003713d41b9bb490c0dc8bf2a18e3cdfa5aa8c.jpg", + "text": "$$\nJ ( \\pi _ { \\theta } , \\psi ) = \\int _ { S } d _ { \\theta } ( s | \\psi ) \\int _ { A } \\pi _ { \\theta } ( s , a ) A ( s , a ) d a d s\n$$", + "text_format": "latex", + "bbox": [ + 338, + 170, + 660, + 203 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "where $\\begin{array} { r } { d _ { \\theta } ( s | \\psi ) = \\int _ { S } \\sum _ { t = 0 } ^ { T } \\gamma ^ { t } p _ { 0 } ( s _ { 0 } ) p ( s _ { 0 } \\to s | t , \\pi _ { \\theta } , \\psi ) d s _ { 0 } } \\end{array}$ . $\\theta$ and $\\psi$ are then learned in tandem following Algorithm 2. This alternating method optimizes both the control and dynamics in order to maximize the expected value of the agent, as analogous to the role of evolution in biomechanics. During each pass, the policy parameters $\\theta$ are trained to improve the agent’s expected value for a fixed set of actuator parameters $\\psi$ . Next, $\\psi$ is optimized using CMA to improve performance while keeping $\\theta$ fixed. The expected value of each CMA sample of $\\psi$ is estimated using the average cumulative reward over multiple rollouts. ", + "bbox": [ + 173, + 213, + 825, + 314 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Figure 5 illustrates the improvement in performance during the optimization process, as applied to motions for three different agents. Figure 6 compares the learning curves for the initial and final MTU parameters, for the same three motions. ", + "bbox": [ + 173, + 320, + 825, + 363 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/2aa165cccc6359401c538e389058e6300bc5e8138336d124477ef18a0e095674.jpg", + "image_caption": [ + "Figure 5: Performance of intermediate MTU policies and actuator parameters per pass of actuator optimization following Algorithm 2. " + ], + "image_footnote": [], + "bbox": [ + 176, + 378, + 823, + 508 + ], + "page_idx": 10 + }, + { + "type": "image", + "img_path": "images/6c476091ce400a69ffdd6088f73cb4ca31db9148402a5816b61e98671118be8d.jpg", + "image_caption": [ + "Figure 6: Learning curves comparing initial and optimized MTU parameters. " + ], + "image_footnote": [], + "bbox": [ + 176, + 575, + 823, + 705 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Bounded Action Space ", + "text_level": 1, + "bbox": [ + 174, + 98, + 403, + 119 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Properties such as torque and neural activation limits result in bounds on the range of values that can be assumed by actions for a particular parameterization. Improper enforcement of these bounds can lead to unstable learning as the gradient information outside the bounds may not be reliable (Hausknecht $\\&$ Stone, 2015). To ensure that all actions respect their bounds, we adopt a method similar to the inverting gradients approach proposed by Hausknecht & Stone (2015). Let $\\nabla a \\ : =$ $( a - \\mu ( s ) ) A ( s , a )$ be the empirical action gradient from the policy gradient estimate of a Gaussian policy. Given the lower and upper bounds $\\bar { [ } l ^ { i } , u ^ { i } ]$ of the ith action parameter, the bounded gradient of the $i$ th action parameter $\\nabla \\tilde { a } ^ { i }$ is determined according to ", + "bbox": [ + 173, + 125, + 826, + 237 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/a418eb156dcac8d93028280af0051a08b1e6735916bea769999e60dd33bb6164.jpg", + "text": "$$\n\\begin{array} { r } { \\nabla \\tilde { a } ^ { i } = \\left\\{ \\begin{array} { l l } { l ^ { i } - \\mu ^ { i } ( s ) , } & { \\mu ^ { i } ( s ) < l ^ { i } \\mathrm { ~ a n d ~ } \\nabla a ^ { i } < 0 } \\\\ { u ^ { i } - \\mu ^ { i } ( s ) , } & { \\mu ^ { i } ( s ) > u ^ { i } \\mathrm { ~ a n d ~ } \\nabla a ^ { i } > 0 } \\\\ { \\nabla a ^ { i } , } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 339, + 253, + 658, + 306 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Unlike the inverting gradients approach, which scales all gradients depending on proximity to the bounds, this method preserves the empirical gradients when bounds are respected, and alters the gradients only when bounds are violated. ", + "bbox": [ + 176, + 316, + 825, + 359 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Reward ", + "text_level": 1, + "bbox": [ + 173, + 390, + 256, + 409 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "The terms of the reward function are defined as follows: ", + "bbox": [ + 174, + 415, + 540, + 429 + ], + "page_idx": 11 + }, + { + "type": "equation", + "img_path": "images/540ab27ebae4e94a1ab39beddd6675377698ca16c3c1f2c381e3161ea9933a97.jpg", + "text": "$$\n\\begin{array} { c } { { r _ { p o s e } = \\displaystyle \\exp \\left( - | | q ^ { * } - q | | _ { W } ^ { 2 } \\right) , \\qquad r _ { v e l } = \\exp \\left( - | | \\dot { q } ^ { * } - \\dot { q } | | _ { W } ^ { 2 } \\right) } } \\\\ { { r _ { e n d } = \\exp \\left( - 4 0 \\displaystyle \\sum _ { e } | | x _ { e } ^ { * } - x _ { e } | | ^ { 2 } \\right) } } \\\\ { { r _ { r o o t } = \\exp \\left( - 1 0 ( h _ { r o o t } ^ { * } - h _ { r o o t } ) ^ { 2 } \\right) , \\qquad r _ { c o m } = \\exp \\left( - 1 0 | | \\dot { x } _ { c o m } ^ { * } - \\dot { x } _ { c o m } | | ^ { 2 } \\right) } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 236, + 433, + 759, + 526 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "$q$ and $q ^ { * }$ denotes the character pose and reference pose represented in reduced-coordinates, while $\\dot { q }$ and $\\dot { q } ^ { * }$ are the respective joints velocities. $W$ is a manually-specified per joint diagonal weighting matrix. $h _ { r o o t }$ is the height of the root from the ground, and $\\dot { x } _ { c o m }$ is the center of mass velocity. ", + "bbox": [ + 176, + 532, + 821, + 577 + ], + "page_idx": 11 + }, + { + "type": "image", + "img_path": "images/6051f1be1329d01271438efb16ecddd05702aeeb72f660b322b3b9b628b86633.jpg", + "image_caption": [ + "Figure 7: Left: fixed initial state biases agent to regions of the state space near the initial state, particular during early iterations of training. Right: initial states sampled from reference trajectory allows agent to explore state space more uniformly around reference trajectory. " + ], + "image_footnote": [], + "bbox": [ + 233, + 106, + 767, + 239 + ], + "page_idx": 12 + }, + { + "type": "image", + "img_path": "images/a65fe6924458b4e1ce4e13840ab449634cd9b7ea950838cff36b93251c1fdcf1.jpg", + "image_caption": [ + "Figure 8: Neural Network Architecture. Each policy is represented by a three layered network, with 512 and 256 fully-connected hidden units, followed by a linear output layer. " + ], + "image_footnote": [], + "bbox": [ + 339, + 318, + 653, + 467 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/19d756059d4c4e1c99fccfe1d4c274183eb93916269310ed858f9db51b6f443f.jpg", + "table_caption": [ + "Table 2: Training hyperparameters. " + ], + "table_footnote": [], + "table_body": "
ParameterValueDescription
Y0.9cumulative reward discount factor
απ0.001actor learning rate
αv0.01critic learning rate
momentum0.9stochastic gradient descent momentum
weight decay0L2 regularizer for critic parameters
θ weight decay0.0005L2 regularizer for actor parameters
minibatch size32tuples per stochastic gradient descent step
replay memory size500000number of the most recent tuples stored for future updates
", + "bbox": [ + 192, + 534, + 805, + 662 + ], + "page_idx": 12 + }, + { + "type": "table", + "img_path": "images/9d77ec04893570f2bc00f7d10e17c470521cebf763f612c01cf39f61589696dc.jpg", + "table_caption": [ + "Table 3: The number of state, action, and actuation model parameters for different characters and actuation models. " + ], + "table_footnote": [], + "table_body": "
Character+ Actuation ModelState ParametersAction ParametersActuator Parameters
Biped + TorBiped + VelBiped + PDBiped +MTU5858587460612114
6616
Raptor + TorRaptor + VelRaptor + PDRaptor + MTU1541541541941801836258
18
1840
Dog + TorDog + VelDog + PDDog + MTU1701701702142002040282
20
20
44
", + "bbox": [ + 181, + 101, + 816, + 285 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/5b7a5a24dbd5087b6cf550426d8cfe4678c2c779f9f34704f8d2577679fe9274.jpg", + "table_caption": [ + "Table 4: Performance of policies trained for the various characters and actuation models. Performance is measured using the normalized cumulative reward (NCR) and learning speed is represented by the normalized area under each learning curve (AUC). The best performing parameterizations for each character and motion are in bold. " + ], + "table_footnote": [], + "table_body": "
Character+ActuationMotionPerformance (NCR)Learning Speed (AUC)
Biped + Tor Biped + Vel Biped + PDWalk Walk Walk0.7662 ± 0.3117 0.9520 ± 0.0034 0.9524 ± 0.0034 0.9584 ± 0.00650.4788 0.6308 0.6997 0.7165
Biped + MTU Biped +Tor Biped + Vel Biped +PDWalk March March March March0.9353 ±0.0072 0.9784 ± 0.0018 0.9767 士 0.0068 0.9484 ±0.00210.7478 0.9035 0.9136 0.5587
Biped + MTU Biped + Tor Biped + Vel Biped + PDRun Run Run0.9032± 0.0102 0.9070 ± 0.0106 0.9057 ± 0.00560.6938 0.7301 0.7880
Biped +MTU Raptor+ Tor Raptor + VelRun Run (Sim) Run (Sim)0.8988 ± 0.0094 0.7265 士 0.0037 0.9612 土 0.00550.5360 0.5061 0.8118
Raptor + PDRun (Sim)0.9863 ± 0.00170.9282
Raptor + MTURun (Sim)0.9708 ± 0.00230.6330
Raptor +TorRun
0.6141 ± 0.00910.3814
Raptor + VelRun0.8732 ±0.00370.7008
Raptor + PDRun0.9548 ± 0.00100.8372
Raptor + MTURun0.9533 ± 0.0015
Dog + Tor0.7258
Bound (Sim)0.7888 ± 0.00460.4895
Dog + VelBound (Sim)0.9788 ±0.00440.7862
Dog +PDBound (Sim)0.9797 ± 0.00120.9280
Dog+ MTUBound (Sim)0.9033 ±0.00290.6825
Dog + TorRear-Up0.8151 士 0.01130.5550
Dog + VelRear-Up0.7364 ± 0.27070.7454
Dog + PDRear-Up0.9565 ± 0.00580.8701
Dog +MTURear-Up0.8744 ± 0.25660.7932
", + "bbox": [ + 200, + 343, + 795, + 751 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Sensitivity Analysis ", + "text_level": 1, + "bbox": [ + 174, + 98, + 370, + 119 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We further analyze the sensitivity of the results to different initializations and design decisions. Figure 9 compares the learning curves from multiple policies trained using different random initializations of the networks. Four policies are trained for each actuation model. The results for a particular actuation model are similar across different runs, and the trends between the various actuation models also appear to be consistent. To evaluate the sensitivity to the amount of exploration noise applied during training, we trained policies where the standard deviation of the action distribution is twice and half of the default values. Figure 10 illustrates the learning curves for each policy. Overall, the performance of the policies do not appear to change significantly for the particular range of values. Finally, Figure 11 compares the results using different network architectures. The network variations include doubling the number of units in both hidden layers, halving the number of hidden units, and inserting an additional layer with 512 units between the two existing hidden layers. The choice of network structure does not appear to have a noticeable impact on the results, and the differences between the actuation models appear to be consistent across the different networks. ", + "bbox": [ + 173, + 125, + 825, + 305 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/a4122f84a35d17b435985fd9066bfb9e44ca5abd22dbbfe7c0468a11a3984ddb.jpg", + "image_caption": [ + "Figure 9: Learning curves from different random network initializations. Four policies are trained for each actuation model. " + ], + "image_footnote": [], + "bbox": [ + 300, + 316, + 689, + 542 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/ecc0556e6aa6b37e854123df001f02073985ee151ec65d8e57c9e40b0010d5db.jpg", + "image_caption": [ + "Figure 10: Learning curves comparing the effects of scaling the standard deviation of the action distribution by 1x, $2 \\mathbf { x }$ , and $1 / 2 \\mathbf { x }$ . " + ], + "image_footnote": [], + "bbox": [ + 196, + 611, + 799, + 799 + ], + "page_idx": 14 + }, + { + "type": "image", + "img_path": "images/81a0d5f7bf058539a618f806f9cf036cc6447092bac4b8944cce3c08ad5027fc.jpg", + "image_caption": [ + "Figure 11: Learning curves for different network architectures. The network structures include, doubling the number of units in each hidden layer, halving the number of units, and inserting an additional hidden layer with 512 units between the two existing hidden layers. " + ], + "image_footnote": [], + "bbox": [ + 215, + 104, + 781, + 280 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/96e285809966aa4462223b8ef89ba17a7af3dedd24fa36b8b2bb490481102985.jpg", + "image_caption": [ + "Figure 12: Performance of different action parameterizations when traveling across randomly generated irregular terrain. (left) Dog running across bumpy terrain, where the height of each bump varies uniformly between 0 and a specified maximum height. (middle) and (right) biped and dog traveling across randomly generated slopes with bounded maximum steepness. " + ], + "image_footnote": [], + "bbox": [ + 176, + 349, + 821, + 477 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/e5bb1f37053981e4f42fa138e3225788bb00a225a75572ffbdbc9b523e4f5c4a.jpg", + "image_caption": [ + "Figure 13: Simulated Motions Using the PD Action Representation. The top row uses an MTU action space while the remainder are driven by a PD action space. " + ], + "image_footnote": [], + "bbox": [ + 289, + 558, + 712, + 875 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/878b406dd0a5f4dd939f54654f51993f4acb9a3796220905c1584a0eb1ae2125.jpg", + "image_caption": [ + "Figure 14: Policy actions over time and the resulting torques for the four action types. Data is from one biped walk cycle (1s). Left: Actions $( 6 0 \\mathrm { H z } )$ , for the right hip for PD, Vel, and Tor, and the right gluteal muscle for MTU. Right: Torques applied to the right hip joint, sampled at $6 0 0 \\mathrm { H z }$ . " + ], + "image_footnote": [], + "bbox": [ + 222, + 82, + 789, + 595 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/2ae25d079af3301717e68708ae16d52963be7f22046b2e82c2f3f40ec67dd423.jpg", + "image_caption": [ + "Figure 15: Learning curves for different state representations including state $^ +$ target state, state $^ +$ phase, and only state. 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We compare the impact of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 280, + 470, + 293 + ], + "spans": [ + { + "bbox": [ + 141, + 280, + 470, + 293 + ], + "score": 1.0, + "content": "four different action parameterizations (torques, muscle-activations, target joint", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 291, + 470, + 304 + ], + "spans": [ + { + "bbox": [ + 141, + 291, + 470, + 304 + ], + "score": 1.0, + "content": "angles, and target joint-angle velocities) in terms of learning time, policy robust-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 303, + 469, + 314 + ], + "spans": [ + { + "bbox": [ + 141, + 303, + 469, + 314 + ], + "score": 1.0, + "content": "ness, motion quality, and policy query rates. Our results are evaluated on a gait-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 313, + 469, + 325 + ], + "spans": [ + { + "bbox": [ + 142, + 313, + 469, + 325 + ], + "score": 1.0, + "content": "cycle imitation task for multiple planar articulated figures and multiple gaits. We", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 324, + 469, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 324, + 469, + 336 + ], + "score": 1.0, + "content": "demonstrate that the local feedback provided by higher-level action parameteriza-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 334, + 469, + 348 + ], + "spans": [ + { + "bbox": [ + 141, + 334, + 469, + 348 + ], + "score": 1.0, + "content": "tions can significantly impact the learning, robustness, and quality of the resulting", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 140, + 346, + 179, + 358 + ], + "spans": [ + { + "bbox": [ + 140, + 346, + 179, + 358 + ], + "score": 1.0, + "content": "policies.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12.5, + "bbox_fs": [ + 140, + 247, + 470, + 358 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 385, + 206, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 208, + 401 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 208, + 401 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 412, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 411, + 505, + 426 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 426 + ], + "score": 1.0, + "content": "The introduction of deep learning models to reinforcement learning (RL) has enabled policies to op-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 506, + 437 + ], + "score": 1.0, + "content": "erate directly on high-dimensional, low-level state features. As a result, deep reinforcement learning", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 106, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "(DeepRL) has demonstrated impressive capabilities, such as developing control policies that can", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 505, + 459 + ], + "score": 1.0, + "content": "map from input image pixels to output joint torques (Lillicrap et al., 2015). However, the quality", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 455, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 470 + ], + "score": 1.0, + "content": "and robustness often falls short of what has been achieved with hand-crafted action abstractions,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 479 + ], + "score": 1.0, + "content": "e.g., Coros et al. (2011); Geijtenbeek et al. (2013). While much is known about the learning of state", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "representations, the choice of action parameterization is a design decision whose impact is not yet", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 489, + 175, + 501 + ], + "spans": [ + { + "bbox": [ + 106, + 489, + 175, + 501 + ], + "score": 1.0, + "content": "well understood.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 411, + 506, + 501 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 506, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "Joint torques can be thought of as the most basic and generic representation for driving the move-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "ment of articulated figures, given that muscles and other actuation models eventually result in joint", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 529, + 505, + 540 + ], + "score": 1.0, + "content": "torques. 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The features consist of the height of the root", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "score": 1.0, + "content": "(pelvis) from the ground, the position of each link with respect to the root, and the center of mass", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 425, + 283 + ], + "score": 1.0, + "content": "velocity of each link. When training a policy to imitate a cyclic reference motion", + "type": "text" + }, + { + "bbox": [ + 425, + 270, + 445, + 282 + ], + "score": 0.92, + "content": "\\{ q _ { t } ^ { * } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 271, + 506, + 283 + ], + "score": 1.0, + "content": ", knowledge of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "the motion phase can help simplify learning. Therefore, we augment the state features with a set of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 291, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 165, + 306 + ], + "score": 1.0, + "content": "target features", + "type": "text" + }, + { + "bbox": [ + 166, + 292, + 208, + 304 + ], + "score": 0.93, + "content": "\\Phi ( q _ { t } ^ { * } , \\dot { q } _ { t } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 291, + 390, + 306 + ], + "score": 1.0, + "content": ", resulting in a combined state represented by", + "type": "text" + }, + { + "bbox": [ + 390, + 292, + 501, + 304 + ], + "score": 0.92, + "content": "s _ { t } = ( \\Phi ( q _ { t } , \\dot { q _ { t } } ) , \\Phi ( q _ { t } ^ { * } , { \\dot { q _ { t } } } ^ { * } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 291, + 505, + 306 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "Similar results can also be achieved by providing a single motion phase variable as a state feature,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 315, + 306, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 306, + 327 + ], + "score": 1.0, + "content": "as we show in Figure 15 (supplemental material).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 107, + 340, + 171, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 172, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 172, + 354 + ], + "score": 1.0, + "content": "3.3 ACTIONS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 107, + 362, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 107, + 362, + 505, + 373 + ], + "score": 1.0, + "content": "We train separate policies for each of the four actuation models, as described below. Each actuation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 371, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 385 + ], + "score": 1.0, + "content": "model also has related actuation parameters, such as feedback gains for PD-controllers and muscu-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "lotendon properties for MTUs. These parameters can be manually specified, as we do for the PD", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "and Vel models, or they can be optimized for the task at hand, as for the MTU models. Table 1", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 406, + 360, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 360, + 418 + ], + "score": 1.0, + "content": "provides a list of actuator parameters for each actuation model.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 402, + 435 + ], + "score": 1.0, + "content": "Target Joint Angles (PD): Each action represents a set of target angles", + "type": "text" + }, + { + "bbox": [ + 403, + 423, + 409, + 434 + ], + "score": 0.83, + "content": "\\hat { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 423, + 441, + 435 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 441, + 422, + 451, + 434 + ], + "score": 0.87, + "content": "\\hat { q } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "specifies the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 197, + 446 + ], + "score": 1.0, + "content": "target angles for joint", + "type": "text" + }, + { + "bbox": [ + 197, + 434, + 201, + 443 + ], + "score": 0.53, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 433, + 208, + 446 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 208, + 434, + 214, + 445 + ], + "score": 0.78, + "content": "\\hat { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 433, + 482, + 446 + ], + "score": 1.0, + "content": "is applied to PD-controllers which compute torques according to", + "type": "text" + }, + { + "bbox": [ + 482, + 433, + 505, + 445 + ], + "score": 0.87, + "content": "\\tau ^ { i } =", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 444, + 452, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 212, + 460 + ], + "score": 0.92, + "content": "k _ { p } ^ { i } ( \\hat { q } ^ { i } - q ^ { i } ) + k _ { d } ^ { i } ( \\hat { \\dot { q } } ^ { i } - \\dot { q } ^ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 445, + 242, + 460 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 243, + 444, + 270, + 458 + ], + "score": 0.93, + "content": "\\hat { \\dot { q } } ^ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 445, + 290, + 460 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 291, + 445, + 302, + 459 + ], + "score": 0.89, + "content": "k _ { p } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 445, + 320, + 460 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 320, + 445, + 331, + 458 + ], + "score": 0.9, + "content": "k _ { d } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 445, + 452, + 460 + ], + "score": 1.0, + "content": "are manually-specified gains.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 422, + 479 + ], + "score": 1.0, + "content": "Target Joint Velocities (Vel): Each action specifies a set of target velocities", + "type": "text" + }, + { + "bbox": [ + 422, + 464, + 429, + 478 + ], + "score": 0.85, + "content": "\\hat { \\dot { q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 465, + 505, + 479 + ], + "score": 1.0, + "content": "which are used to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 477, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 227, + 492 + ], + "score": 1.0, + "content": "compute torques according to", + "type": "text" + }, + { + "bbox": [ + 228, + 477, + 298, + 491 + ], + "score": 0.94, + "content": "\\tau ^ { i } = k _ { d } ^ { i } ( { \\hat { \\dot { q } } } ^ { i } - { \\dot { q } } ^ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 477, + 368, + 492 + ], + "score": 1.0, + "content": ", where the gains", + "type": "text" + }, + { + "bbox": [ + 368, + 478, + 379, + 491 + ], + "score": 0.9, + "content": "k _ { d } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 477, + 505, + 492 + ], + "score": 1.0, + "content": "are specified to be the same as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 220, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 220, + 502 + ], + "score": 1.0, + "content": "those used for target angles.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "Torques (Tor): Each action directly specifies torques for every joint, and constant torques are ap-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 504, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 504, + 529 + ], + "score": 1.0, + "content": "plied for the duration of a control step. Due to torque limits, actions are bounded by manually", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "specified limits for each joint. Unlike the other actuation models, the torque model does not require", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "additional actuator parameters, and can thus be regarded as requiring the least amount of domain", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "knowledge. Torque limits are excluded from the actuator parameter set as they are common for all", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 563, + 182, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 182, + 573 + ], + "score": 1.0, + "content": "parameterizations.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "Muscle Activations (MTU): Each action specifies activations for a set of musculotendon units", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "(MTU). Detailed modeling and implementation information are available in Wang et al. (2012).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "Each MTU is modeled as a contractile element (CE) attached to a serial elastic element (SE)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 624 + ], + "score": 1.0, + "content": "and parallel elastic element (PE). The force exerted by the MTU can be calculated according to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 621, + 504, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 236, + 633 + ], + "score": 0.92, + "content": "F _ { M T U } = F _ { S E } = F _ { C E } + F _ { P E }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 621, + 265, + 636 + ], + "score": 1.0, + "content": ". Both", + "type": "text" + }, + { + "bbox": [ + 266, + 622, + 285, + 633 + ], + "score": 0.89, + "content": "F _ { S E }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 621, + 304, + 636 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 305, + 622, + 325, + 633 + ], + "score": 0.9, + "content": "F _ { P E }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 621, + 483, + 636 + ], + "score": 1.0, + "content": "are modeled as passive springs, while", + "type": "text" + }, + { + "bbox": [ + 483, + 622, + 504, + 633 + ], + "score": 0.89, + "content": "F _ { C E }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 244, + 646 + ], + "score": 1.0, + "content": "is actively controlled according to", + "type": "text" + }, + { + "bbox": [ + 245, + 633, + 382, + 645 + ], + "score": 0.9, + "content": "F _ { C E } = a _ { M T U } F _ { 0 } f _ { l } ( l _ { C E } ) f _ { v } ( v _ { C E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 632, + 406, + 646 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 406, + 634, + 433, + 644 + ], + "score": 0.82, + "content": "a _ { M T U }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "being the muscle", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 150, + 657 + ], + "score": 1.0, + "content": "activation,", + "type": "text" + }, + { + "bbox": [ + 150, + 644, + 162, + 655 + ], + "score": 0.88, + "content": "F _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 644, + 286, + 657 + ], + "score": 1.0, + "content": "the maximum isometric force,", + "type": "text" + }, + { + "bbox": [ + 286, + 645, + 303, + 655 + ], + "score": 0.88, + "content": "l _ { C E }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 644, + 321, + 657 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 321, + 646, + 340, + 655 + ], + "score": 0.87, + "content": "v _ { C E }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "being the length and velocity of the con-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 234, + 668 + ], + "score": 1.0, + "content": "tractile element. The functions", + "type": "text" + }, + { + "bbox": [ + 234, + 655, + 266, + 667 + ], + "score": 0.93, + "content": "f _ { l } ( l _ { C E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 654, + 285, + 668 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 286, + 655, + 321, + 667 + ], + "score": 0.92, + "content": "f _ { v } ( v _ { C E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "represent the force-length and force-velocity", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "relationships, modeling the variations in the maximum force that can be exerted by a muscle as a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "function of its length and contraction velocity. Analytic forms are available in Geyer et al. (2003).", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 433, + 699 + ], + "score": 1.0, + "content": "Activations are bounded between [0, 1]. The length of each contractile element", + "type": "text" + }, + { + "bbox": [ + 434, + 688, + 451, + 699 + ], + "score": 0.9, + "content": "l _ { C E }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 687, + 506, + 699 + ], + "score": 1.0, + "content": "are included", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "as state features. 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The reference velocity", + "type": "text" + }, + { + "bbox": [ + 430, + 140, + 440, + 153 + ], + "score": 0.88, + "content": "\\dot { q _ { t } ^ { * } }", + "type": "inline_equation" + }, + { + "bbox": [ + 441, + 141, + 505, + 154 + ], + "score": 1.0, + "content": "at a given time", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 152, + 506, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 111, + 165 + ], + "score": 0.71, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 152, + 260, + 171 + ], + "score": 1.0, + "content": "is approximated by finite-difference", + "type": "text" + }, + { + "bbox": [ + 261, + 152, + 323, + 169 + ], + "score": 0.94, + "content": "\\begin{array} { r } { \\dot { q _ { t } ^ { * } } \\approx \\frac { q _ { t + \\triangle t } ^ { * } - q _ { t } ^ { * } } { \\triangle t } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 324, + 153, + 506, + 168 + ], + "score": 1.0, + "content": ". 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Since hand-crafted reference motions may not be physically realizable, the goal is to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 190, + 367, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 190, + 367, + 202 + ], + "score": 1.0, + "content": "closely reproduce a motion while satisfying physical constraints.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 128, + 506, + 202 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 216, + 163, + 227 + ], + "lines": [ + { + "bbox": [ + 105, + 215, + 165, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 215, + 165, + 229 + ], + "score": 1.0, + "content": "3.2 STATES", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 107, + 237, + 505, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 505, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 335, + 250 + ], + "score": 1.0, + "content": "To define the state of the agent, a feature transformation", + "type": "text" + }, + { + "bbox": [ + 336, + 237, + 366, + 250 + ], + "score": 0.93, + "content": "\\Phi ( q , \\dot { q } )", + "type": "inline_equation" + }, + { + "bbox": [ + 366, + 237, + 505, + 250 + ], + "score": 1.0, + "content": "is used to extract a set of features", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 247, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 246, + 261 + ], + "score": 1.0, + "content": "from the reduced-coordinate pose", + "type": "text" + }, + { + "bbox": [ + 246, + 250, + 253, + 260 + ], + "score": 0.78, + "content": "q", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 247, + 307, + 261 + ], + "score": 1.0, + "content": "and velocity", + "type": "text" + }, + { + "bbox": [ + 307, + 249, + 313, + 260 + ], + "score": 0.76, + "content": "\\dot { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 247, + 506, + 261 + ], + "score": 1.0, + "content": ". The features consist of the height of the root", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 260, + 506, + 272 + ], + "score": 1.0, + "content": "(pelvis) from the ground, the position of each link with respect to the root, and the center of mass", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 425, + 283 + ], + "score": 1.0, + "content": "velocity of each link. When training a policy to imitate a cyclic reference motion", + "type": "text" + }, + { + "bbox": [ + 425, + 270, + 445, + 282 + ], + "score": 0.92, + "content": "\\{ q _ { t } ^ { * } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 271, + 506, + 283 + ], + "score": 1.0, + "content": ", knowledge of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 506, + 294 + ], + "score": 1.0, + "content": "the motion phase can help simplify learning. Therefore, we augment the state features with a set of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 291, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 165, + 306 + ], + "score": 1.0, + "content": "target features", + "type": "text" + }, + { + "bbox": [ + 166, + 292, + 208, + 304 + ], + "score": 0.93, + "content": "\\Phi ( q _ { t } ^ { * } , \\dot { q } _ { t } ^ { * } )", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 291, + 390, + 306 + ], + "score": 1.0, + "content": ", resulting in a combined state represented by", + "type": "text" + }, + { + "bbox": [ + 390, + 292, + 501, + 304 + ], + "score": 0.92, + "content": "s _ { t } = ( \\Phi ( q _ { t } , \\dot { q _ { t } } ) , \\Phi ( q _ { t } ^ { * } , { \\dot { q _ { t } } } ^ { * } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 291, + 505, + 306 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 316 + ], + "score": 1.0, + "content": "Similar results can also be achieved by providing a single motion phase variable as a state feature,", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 315, + 306, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 306, + 327 + ], + "score": 1.0, + "content": "as we show in Figure 15 (supplemental material).", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 237, + 506, + 327 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 340, + 171, + 352 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 172, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 172, + 354 + ], + "score": 1.0, + "content": "3.3 ACTIONS", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 361, + 505, + 417 + ], + "lines": [ + { + "bbox": [ + 107, + 362, + 505, + 373 + ], + "spans": [ + { + "bbox": [ + 107, + 362, + 505, + 373 + ], + "score": 1.0, + "content": "We train separate policies for each of the four actuation models, as described below. Each actuation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 371, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 505, + 385 + ], + "score": 1.0, + "content": "model also has related actuation parameters, such as feedback gains for PD-controllers and muscu-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "lotendon properties for MTUs. These parameters can be manually specified, as we do for the PD", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "and Vel models, or they can be optimized for the task at hand, as for the MTU models. Table 1", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 406, + 360, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 360, + 418 + ], + "score": 1.0, + "content": "provides a list of actuator parameters for each actuation model.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 362, + 505, + 418 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 423, + 505, + 458 + ], + "lines": [ + { + "bbox": [ + 106, + 422, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 402, + 435 + ], + "score": 1.0, + "content": "Target Joint Angles (PD): Each action represents a set of target angles", + "type": "text" + }, + { + "bbox": [ + 403, + 423, + 409, + 434 + ], + "score": 0.83, + "content": "\\hat { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 423, + 441, + 435 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 441, + 422, + 451, + 434 + ], + "score": 0.87, + "content": "\\hat { q } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "specifies the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 197, + 446 + ], + "score": 1.0, + "content": "target angles for joint", + "type": "text" + }, + { + "bbox": [ + 197, + 434, + 201, + 443 + ], + "score": 0.53, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 433, + 208, + 446 + ], + "score": 1.0, + "content": ".", + "type": "text" + }, + { + "bbox": [ + 208, + 434, + 214, + 445 + ], + "score": 0.78, + "content": "\\hat { q }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 433, + 482, + 446 + ], + "score": 1.0, + "content": "is applied to PD-controllers which compute torques according to", + "type": "text" + }, + { + "bbox": [ + 482, + 433, + 505, + 445 + ], + "score": 0.87, + "content": "\\tau ^ { i } =", + "type": "inline_equation" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 444, + 452, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 212, + 460 + ], + "score": 0.92, + "content": "k _ { p } ^ { i } ( \\hat { q } ^ { i } - q ^ { i } ) + k _ { d } ^ { i } ( \\hat { \\dot { q } } ^ { i } - \\dot { q } ^ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 445, + 242, + 460 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 243, + 444, + 270, + 458 + ], + "score": 0.93, + "content": "\\hat { \\dot { q } } ^ { i } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 445, + 290, + 460 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 291, + 445, + 302, + 459 + ], + "score": 0.89, + "content": "k _ { p } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 445, + 320, + 460 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 320, + 445, + 331, + 458 + ], + "score": 0.9, + "content": "k _ { d } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 445, + 452, + 460 + ], + "score": 1.0, + "content": "are manually-specified gains.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 422, + 505, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 465, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 464, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 465, + 422, + 479 + ], + "score": 1.0, + "content": "Target Joint Velocities (Vel): Each action specifies a set of target velocities", + "type": "text" + }, + { + "bbox": [ + 422, + 464, + 429, + 478 + ], + "score": 0.85, + "content": "\\hat { \\dot { q } }", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 465, + 505, + 479 + ], + "score": 1.0, + "content": "which are used to", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 477, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 477, + 227, + 492 + ], + "score": 1.0, + "content": "compute torques according to", + "type": "text" + }, + { + "bbox": [ + 228, + 477, + 298, + 491 + ], + "score": 0.94, + "content": "\\tau ^ { i } = k _ { d } ^ { i } ( { \\hat { \\dot { q } } } ^ { i } - { \\dot { q } } ^ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 477, + 368, + 492 + ], + "score": 1.0, + "content": ", where the gains", + "type": "text" + }, + { + "bbox": [ + 368, + 478, + 379, + 491 + ], + "score": 0.9, + "content": "k _ { d } ^ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 477, + 505, + 492 + ], + "score": 1.0, + "content": "are specified to be the same as", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 488, + 220, + 502 + ], + "spans": [ + { + "bbox": [ + 106, + 488, + 220, + 502 + ], + "score": 1.0, + "content": "those used for target angles.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 464, + 505, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 572 + ], + "lines": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "Torques (Tor): Each action directly specifies torques for every joint, and constant torques are ap-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 516, + 504, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 504, + 529 + ], + "score": 1.0, + "content": "plied for the duration of a control step. Due to torque limits, actions are bounded by manually", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 528, + 506, + 541 + ], + "spans": [ + { + "bbox": [ + 104, + 528, + 506, + 541 + ], + "score": 1.0, + "content": "specified limits for each joint. Unlike the other actuation models, the torque model does not require", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 505, + 551 + ], + "score": 1.0, + "content": "additional actuator parameters, and can thus be regarded as requiring the least amount of domain", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "knowledge. Torque limits are excluded from the actuator parameter set as they are common for all", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 563, + 182, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 563, + 182, + 573 + ], + "score": 1.0, + "content": "parameterizations.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 505, + 506, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 578, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 505, + 590 + ], + "score": 1.0, + "content": "Muscle Activations (MTU): Each action specifies activations for a set of musculotendon units", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 602 + ], + "score": 1.0, + "content": "(MTU). Detailed modeling and implementation information are available in Wang et al. (2012).", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "Each MTU is modeled as a contractile element (CE) attached to a serial elastic element (SE)", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 609, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 506, + 624 + ], + "score": 1.0, + "content": "and parallel elastic element (PE). The force exerted by the MTU can be calculated according to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 621, + 504, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 236, + 633 + ], + "score": 0.92, + "content": "F _ { M T U } = F _ { S E } = F _ { C E } + F _ { P E }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 621, + 265, + 636 + ], + "score": 1.0, + "content": ". Both", + "type": "text" + }, + { + "bbox": [ + 266, + 622, + 285, + 633 + ], + "score": 0.89, + "content": "F _ { S E }", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 621, + 304, + 636 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 305, + 622, + 325, + 633 + ], + "score": 0.9, + "content": "F _ { P E }", + "type": "inline_equation" + }, + { + "bbox": [ + 325, + 621, + 483, + 636 + ], + "score": 1.0, + "content": "are modeled as passive springs, while", + "type": "text" + }, + { + "bbox": [ + 483, + 622, + 504, + 633 + ], + "score": 0.89, + "content": "F _ { C E }", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 244, + 646 + ], + "score": 1.0, + "content": "is actively controlled according to", + "type": "text" + }, + { + "bbox": [ + 245, + 633, + 382, + 645 + ], + "score": 0.9, + "content": "F _ { C E } = a _ { M T U } F _ { 0 } f _ { l } ( l _ { C E } ) f _ { v } ( v _ { C E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 632, + 406, + 646 + ], + "score": 1.0, + "content": ", with", + "type": "text" + }, + { + "bbox": [ + 406, + 634, + 433, + 644 + ], + "score": 0.82, + "content": "a _ { M T U }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 632, + 506, + 646 + ], + "score": 1.0, + "content": "being the muscle", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 644, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 150, + 657 + ], + "score": 1.0, + "content": "activation,", + "type": "text" + }, + { + "bbox": [ + 150, + 644, + 162, + 655 + ], + "score": 0.88, + "content": "F _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 644, + 286, + 657 + ], + "score": 1.0, + "content": "the maximum isometric force,", + "type": "text" + }, + { + "bbox": [ + 286, + 645, + 303, + 655 + ], + "score": 0.88, + "content": "l _ { C E }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 644, + 321, + 657 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 321, + 646, + 340, + 655 + ], + "score": 0.87, + "content": "v _ { C E }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 644, + 505, + 657 + ], + "score": 1.0, + "content": "being the length and velocity of the con-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 234, + 668 + ], + "score": 1.0, + "content": "tractile element. The functions", + "type": "text" + }, + { + "bbox": [ + 234, + 655, + 266, + 667 + ], + "score": 0.93, + "content": "f _ { l } ( l _ { C E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 654, + 285, + 668 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 286, + 655, + 321, + 667 + ], + "score": 0.92, + "content": "f _ { v } ( v _ { C E } )", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "represent the force-length and force-velocity", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "score": 1.0, + "content": "relationships, modeling the variations in the maximum force that can be exerted by a muscle as a", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "function of its length and contraction velocity. Analytic forms are available in Geyer et al. (2003).", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 687, + 506, + 699 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 433, + 699 + ], + "score": 1.0, + "content": "Activations are bounded between [0, 1]. The length of each contractile element", + "type": "text" + }, + { + "bbox": [ + 434, + 688, + 451, + 699 + ], + "score": 0.9, + "content": "l _ { C E }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 687, + 506, + 699 + ], + "score": 1.0, + "content": "are included", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "as state features. 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ActuationModelActuatorParameters
Target Joint Angles (PD) proportional gains kp, derivative gains kd
Target Joint Velocities(Vel)derivative gains kd
Torques (Tor)none
Muscle Activations (MTU)optimal contractile element length, serial elastic element rest length,maximum isometric force, pennation,moment arm,maximum moment arm joint orientation, rest joint orientation.
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"\\begin{array} { c } { r = w _ { p o s e } r _ { p o s e } + w _ { v e l } r _ { v e l } + w _ { e n d } r _ { e n d } + w _ { r o o t } r _ { r o o t } + w _ { c o m } r _ { c o m } } \\\\ { ~ } \\\\ { w _ { p o s e } = 0 . 5 , ~ w _ { v e l } = 0 . 0 5 , ~ w _ { e n d } = 0 . 1 5 , ~ w _ { r o o t } = 0 . 1 , ~ w _ { c o m } = 0 . 2 } \\end{array}", + "type": "interline_equation", + "image_path": "7ed72042a39f1e0c19dfafcaeca5f6589359e008f5983457afbccae85628b183.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 167, + 252, + 444, + 263.3333333333333 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 167, + 263.3333333333333, + 444, + 274.66666666666663 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 167, + 274.66666666666663, + 444, + 285.99999999999994 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 294, + 506, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 506, + 308 + ], + "spans": [ + { + 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ActuationModelActuatorParameters
Target Joint Angles (PD) proportional gains kp, derivative gains kd
Target Joint Velocities(Vel)derivative gains kd
Torques (Tor)none
Muscle Activations (MTU)optimal contractile element length, serial elastic element rest length,maximum isometric force, pennation,moment arm,maximum moment arm joint orientation, rest joint orientation.
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286 + ], + "score": 0.88, + "content": "\\begin{array} { c } { r = w _ { p o s e } r _ { p o s e } + w _ { v e l } r _ { v e l } + w _ { e n d } r _ { e n d } + w _ { r o o t } r _ { r o o t } + w _ { c o m } r _ { c o m } } \\\\ { ~ } \\\\ { w _ { p o s e } = 0 . 5 , ~ w _ { v e l } = 0 . 0 5 , ~ w _ { e n d } = 0 . 1 5 , ~ w _ { r o o t } = 0 . 1 , ~ w _ { c o m } = 0 . 2 } \\end{array}", + "type": "interline_equation", + "image_path": "7ed72042a39f1e0c19dfafcaeca5f6589359e008f5983457afbccae85628b183.jpg" + } + ] + } + ], + "index": 8, + "virtual_lines": [ + { + "bbox": [ + 167, + 252, + 444, + 263.3333333333333 + ], + "spans": [], + "index": 7 + }, + { + "bbox": [ + 167, + 263.3333333333333, + 444, + 274.66666666666663 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 167, + 274.66666666666663, + 444, + 285.99999999999994 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 294, + 506, + 339 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 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At the start of each episode,", + "type": "text" + }, + { + "bbox": [ + 244, + 385, + 255, + 396 + ], + "score": 0.87, + "content": "q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 385, + 273, + 397 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 273, + 386, + 284, + 397 + ], + "score": 0.88, + "content": "\\dot { q } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 385, + 505, + 397 + ], + "score": 1.0, + "content": "are sampled from the reference trajectory, and used to", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 408 + ], + "score": 1.0, + "content": "initialize the pose and velocity of the agent. This helps guide the agent to explore states near the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 407, + 174, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 174, + 421 + ], + "score": 1.0, + "content": "target trajectory.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5, + "bbox_fs": [ + 105, + 372, + 505, + 421 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 434, + 330, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 434, + 331, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 331, + 449 + ], + "score": 1.0, + "content": "4 ACTOR-CRITIC LEARNING ALGORITHM", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 105, + 460, + 504, + 483 + ], + "lines": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 473 + ], + "score": 1.0, + "content": "Instead of directly using the temporal difference advantage function, we adapt a positive temporal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 471, + 349, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 349, + 484 + ], + "score": 1.0, + "content": "difference (PTD) update as proposed by Van Hasselt (2012).", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 106, + 460, + 505, + 484 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 226, + 489, + 385, + 518 + ], + "lines": [ + { + "bbox": [ + 226, + 489, + 385, + 518 + ], + "spans": [ + { + "bbox": [ + 226, + 489, + 385, + 518 + ], + "score": 0.92, + "content": "A ( s , a ) = I \\left[ \\delta > 0 \\right] = { \\left\\{ \\begin{array} { l l } { 1 , } & { \\delta > 0 } \\\\ { 0 , } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. }", + "type": "interline_equation", + "image_path": "817c049178dadd041fc4f8bbd27bc88ee4048bae55f476073c229dc8ae687a55.jpg" + } + ] + } + ], + "index": 22.5, + "virtual_lines": [ + { + "bbox": [ + 226, + 489, + 385, + 503.5 + ], + "spans": [], + "index": 22 + }, + { + "bbox": [ + 226, + 503.5, + 385, + 518.0 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "interline_equation", + "bbox": [ + 255, + 525, + 355, + 539 + ], + "lines": [ + { + "bbox": [ + 255, + 525, + 355, + 539 + ], + "spans": [ + { + "bbox": [ + 255, + 525, + 355, + 539 + ], + "score": 0.89, + "content": "\\delta = r + \\gamma V ( s ^ { \\prime } ) - V ( s )", + "type": "interline_equation", + "image_path": "f3fc7ac2153613e4aedf2591b96e4d41d03e084c7906aa175bb593437dc3fcb9.jpg" + } + ] + } + ], + "index": 24, + "virtual_lines": [ + { + "bbox": [ + 255, + 525, + 355, + 539 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 542, + 505, + 675 + ], + "lines": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 555 + ], + "score": 1.0, + "content": "Unlike more conventional policy gradient methods, PTD is less sensitive to the scale of the ad-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "spans": [ + { + "bbox": [ + 106, + 553, + 505, + 565 + ], + "score": 1.0, + "content": "vantage function and avoids instabilities that can result from negative TD updates. For a Gaussian", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 106, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "policy, a negative TD update moves the mean of the distribution away from an observed action,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 576, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 505, + 587 + ], + "score": 1.0, + "content": "effectively shifting the mean towards an unknown action that may be no better than the current", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 586, + 504, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 504, + 599 + ], + "score": 1.0, + "content": "mean action (Van Hasselt, 2012). 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The choice between a stochastic and deterministic policy can be specified", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 702, + 326, + 715 + ], + "spans": [ + { + "bbox": [ + 105, + 702, + 286, + 715 + ], + "score": 1.0, + "content": "by the addition of a binary indicator variable", + "type": "text" + }, + { + "bbox": [ + 287, + 702, + 326, + 714 + ], + "score": 0.9, + "content": "\\lambda \\in [ 0 , 1 ]", + "type": "inline_equation" + } + ], + "index": 39 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 679, + 506, + 715 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 254, + 719, + 357, + 733 + ], + "lines": [ + { + "bbox": [ + 254, + 719, + 357, + 733 + ], + "spans": [ + { + "bbox": [ + 254, + 719, + 357, + 733 + ], + "score": 0.9, + "content": "a _ { t } = \\mu _ { \\theta } ( s _ { t } ) + \\lambda N ( 0 , \\Sigma )", + "type": "interline_equation", + "image_path": "d58b48f9314873eac7db0cf90f14d2de138da83d991d87f0c5df009d9926b846.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 254, + 719, + 357, + 733 + ], + "spans": [], + "index": 40 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 134, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 83, + 164, + 93 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 82, + 423, + 95 + ], + "score": 1.0, + "content": "corresponds to a stochastic policy with exploration noise, and", + "type": "text" + }, + { + "bbox": [ + 423, + 83, + 452, + 93 + ], + "score": 0.9, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "corresponds", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 504, + 106 + ], + "score": 1.0, + "content": "to a deterministic policy that always selects the mean of the distribution. Noise from a stochastic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 504, + 117 + ], + "score": 1.0, + "content": "policy will result in a state distribution that differs from that of the deterministic policy at runtime.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 279, + 128 + ], + "score": 1.0, + "content": "To imitate this discrepancy, we incorporate", + "type": "text" + }, + { + "bbox": [ + 279, + 118, + 284, + 126 + ], + "score": 0.61, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "-greedy exploration in addition to the original Gaussian", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 227, + 138 + ], + "score": 1.0, + "content": "exploration. 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Since the policy gradient is defined for stochastic policies, only tuples", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 254, + 183 + ], + "score": 1.0, + "content": "recorded with exploration noise (i.e.", + "type": "text" + }, + { + "bbox": [ + 255, + 171, + 281, + 181 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 170, + 505, + 183 + ], + "score": 1.0, + "content": ") can be used to update the actor, while the critic can be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 206, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 206, + 194 + ], + "score": 1.0, + "content": "updated using all tuples.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 460, + 211 + ], + "score": 1.0, + "content": "Training proceeds episodically, where the initial state of each episode is sampled from", + "type": "text" + }, + { + "bbox": [ + 460, + 198, + 483, + 210 + ], + "score": 0.91, + "content": "p _ { 0 } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 198, + 505, + 211 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "the episode duration is drawn from an exponential distribution with a mean of 2s. To discourage", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "falling, an episode will also terminate if any part of the character’s trunk makes contact with the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 504, + 244 + ], + "score": 1.0, + "content": "ground for an extended period of time, leaving the agent with zero reward for all subsequent steps.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 448, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 448, + 255 + ], + "score": 1.0, + "content": "Algorithm 1 in the supplemental material summarizes the complete learning process.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "MTU Actuator Optimization: Actuation models such as MTUs are defined by further parameters", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 268, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 506, + 284 + ], + "score": 1.0, + "content": "whose values impact performance (Geijtenbeek et al., 2013). Geyer et al. (2003) uses existing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "anatomical estimates for humans to determine MTU parameters, but such data is not be available for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "more arbitrary creatures. Alternatively, Geijtenbeek et al. (2013) uses covariance matrix adaptation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "(CMA), a derivative-free evolutionary search strategy, to simultaneously optimize MTU and policy", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "parameters. This approach is limited to policies with reasonably low dimensional parameter spaces,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "and is thus ill-suited for neural network models with hundreds of thousands of parameters. To", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 350 + ], + "score": 1.0, + "content": "avoid manual-tuning of actuator parameters, we propose a heuristic approach that alternates between", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 439, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 439, + 360 + ], + "score": 1.0, + "content": "policy learning and actuator optimization, as detailed in the supplemental material.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19 + }, + { + "type": "image", + "bbox": [ + 154, + 373, + 489, + 450 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 154, + 373, + 489, + 450 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 154, + 373, + 489, + 450 + ], + "spans": [ + { + "bbox": [ + 154, + 373, + 489, + 450 + ], + "score": 0.964, + "type": "image", + "image_path": "37eeafcd2ada57b0a9a694e94f6efe9f905699c1775e021139c85ef284ffc1cf.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 154, + 373, + 489, + 398.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 154, + 398.6666666666667, + 489, + 424.33333333333337 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 154, + 424.33333333333337, + 489, + 450.00000000000006 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 462, + 504, + 496 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "Figure 1: Simulated articulated figures and their state representation. Revolute joints connect all", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "links. From left to right: 7-link biped; 19-link raptor; 21-link dog; State features: root height,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "relative position (red) of each link with respect to the root and their respective linear velocity (green).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 525, + 172, + 538 + ], + "lines": [ + { + "bbox": [ + 104, + 524, + 174, + 541 + ], + "spans": [ + { + "bbox": [ + 104, + 524, + 174, + 541 + ], + "score": 1.0, + "content": "5 RESULTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "The motions are best seen in the supplemental video https://youtu.be/L3vDo3nLI98. We evalu-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "ate the action parameterizations by training policies for a simulated 2D biped, dog, and raptor as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "shown in Figure 1. Depending on the agent and the actuation model, our systems have 58–214 state", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "dimensions, 6–44 action dimensions, and 0–282 actuator parameters, as summarized in Table 3", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "(supplemental materials). The MTU models have at least double the number of action parameters", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 606, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 505, + 617 + ], + "score": 1.0, + "content": "because they come in antagonistic pairs. As well, additional MTUs are used for the legs to more", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 617, + 455, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 455, + 629 + ], + "score": 1.0, + "content": "accurately reflect bipedal biomechanics. This includes MTUs that span multiple joints.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Each policy is represented by a three layer neural network, as illustrated in Figure 8 (supplemental", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "material) with 512 and 256 fully-connected units, followed by a linear output layer where the number", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "of output units vary according to the number of action parameters for each character and actuation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "model. 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The policies are queried at", + "type": "text" + }, + { + "bbox": [ + 302, + 688, + 326, + 698 + ], + "score": 0.43, + "content": "6 0 \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "for a control step of about 0.0167s. Each", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "network is randomly initialized and trained for about 1 million iterations, requiring 32 million tuples,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "the equivalent of approximately 6 days of simulated time. Each policy requires about 10 hours for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 437, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 437, + 733 + ], + "score": 1.0, + "content": "the biped, and 20 hours for the raptor and dog on an 8-core Intel Xeon E5-2687W.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 134, + 95 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 134, + 83, + 164, + 93 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 82, + 423, + 95 + ], + "score": 1.0, + "content": "corresponds to a stochastic policy with exploration noise, and", + "type": "text" + }, + { + "bbox": [ + 423, + 83, + 452, + 93 + ], + "score": 0.9, + "content": "\\lambda = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "corresponds", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 504, + 106 + ], + "score": 1.0, + "content": "to a deterministic policy that always selects the mean of the distribution. Noise from a stochastic", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 504, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 504, + 117 + ], + "score": 1.0, + "content": "policy will result in a state distribution that differs from that of the deterministic policy at runtime.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 279, + 128 + ], + "score": 1.0, + "content": "To imitate this discrepancy, we incorporate", + "type": "text" + }, + { + "bbox": [ + 279, + 118, + 284, + 126 + ], + "score": 0.61, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 114, + 505, + 128 + ], + "score": 1.0, + "content": "-greedy exploration in addition to the original Gaussian", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 227, + 138 + ], + "score": 1.0, + "content": "exploration. 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The exploration rate", + "type": "text" + }, + { + "bbox": [ + 323, + 139, + 329, + 147 + ], + "score": 0.71, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 138, + 482, + 149 + ], + "score": 1.0, + "content": "is annealed linearly from 1 to 0.2 over", + "type": "text" + }, + { + "bbox": [ + 483, + 137, + 504, + 148 + ], + "score": 0.44, + "content": "5 0 0 \\mathrm { k }", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 506, + 161 + ], + "score": 1.0, + "content": "iterations, which slowly adjusts the state distribution encountered during training to better resemble", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 505, + 172 + ], + "score": 1.0, + "content": "the distribution at runtime. Since the policy gradient is defined for stochastic policies, only tuples", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 505, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 254, + 183 + ], + "score": 1.0, + "content": "recorded with exploration noise (i.e.", + "type": "text" + }, + { + "bbox": [ + 255, + 171, + 281, + 181 + ], + "score": 0.9, + "content": "\\lambda = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 170, + 505, + 183 + ], + "score": 1.0, + "content": ") can be used to update the actor, while the critic can be", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 206, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 206, + 194 + ], + "score": 1.0, + "content": "updated using all tuples.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 82, + 506, + 194 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 198, + 505, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 106, + 198, + 460, + 211 + ], + "score": 1.0, + "content": "Training proceeds episodically, where the initial state of each episode is sampled from", + "type": "text" + }, + { + "bbox": [ + 460, + 198, + 483, + 210 + ], + "score": 0.91, + "content": "p _ { 0 } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 198, + 505, + 211 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 505, + 222 + ], + "score": 1.0, + "content": "the episode duration is drawn from an exponential distribution with a mean of 2s. To discourage", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "falling, an episode will also terminate if any part of the character’s trunk makes contact with the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 231, + 504, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 504, + 244 + ], + "score": 1.0, + "content": "ground for an extended period of time, leaving the agent with zero reward for all subsequent steps.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 448, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 448, + 255 + ], + "score": 1.0, + "content": "Algorithm 1 in the supplemental material summarizes the complete learning process.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 198, + 505, + 255 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 258, + 505, + 358 + ], + "lines": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 258, + 506, + 272 + ], + "score": 1.0, + "content": "MTU Actuator Optimization: Actuation models such as MTUs are defined by further parameters", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 268, + 506, + 284 + ], + "spans": [ + { + "bbox": [ + 106, + 268, + 506, + 284 + ], + "score": 1.0, + "content": "whose values impact performance (Geijtenbeek et al., 2013). Geyer et al. (2003) uses existing", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "spans": [ + { + "bbox": [ + 106, + 281, + 505, + 293 + ], + "score": 1.0, + "content": "anatomical estimates for humans to determine MTU parameters, but such data is not be available for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 505, + 304 + ], + "score": 1.0, + "content": "more arbitrary creatures. Alternatively, Geijtenbeek et al. (2013) uses covariance matrix adaptation", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 505, + 316 + ], + "score": 1.0, + "content": "(CMA), a derivative-free evolutionary search strategy, to simultaneously optimize MTU and policy", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 327 + ], + "score": 1.0, + "content": "parameters. This approach is limited to policies with reasonably low dimensional parameter spaces,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 106, + 325, + 505, + 337 + ], + "score": 1.0, + "content": "and is thus ill-suited for neural network models with hundreds of thousands of parameters. To", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 506, + 350 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 350 + ], + "score": 1.0, + "content": "avoid manual-tuning of actuator parameters, we propose a heuristic approach that alternates between", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 439, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 439, + 360 + ], + "score": 1.0, + "content": "policy learning and actuator optimization, as detailed in the supplemental material.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 258, + 506, + 360 + ] + }, + { + "type": "image", + "bbox": [ + 154, + 373, + 489, + 450 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 154, + 373, + 489, + 450 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 154, + 373, + 489, + 450 + ], + "spans": [ + { + "bbox": [ + 154, + 373, + 489, + 450 + ], + "score": 0.964, + "type": "image", + "image_path": "37eeafcd2ada57b0a9a694e94f6efe9f905699c1775e021139c85ef284ffc1cf.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 154, + 373, + 489, + 398.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 154, + 398.6666666666667, + 489, + 424.33333333333337 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 154, + 424.33333333333337, + 489, + 450.00000000000006 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 462, + 504, + 496 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "Figure 1: Simulated articulated figures and their state representation. Revolute joints connect all", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 472, + 505, + 486 + ], + "score": 1.0, + "content": "links. From left to right: 7-link biped; 19-link raptor; 21-link dog; State features: root height,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "relative position (red) of each link with respect to the root and their respective linear velocity (green).", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 28 + } + ], + "index": 26.5 + }, + { + "type": "title", + "bbox": [ + 107, + 525, + 172, + 538 + ], + "lines": [ + { + "bbox": [ + 104, + 524, + 174, + 541 + ], + "spans": [ + { + "bbox": [ + 104, + 524, + 174, + 541 + ], + "score": 1.0, + "content": "5 RESULTS", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 563 + ], + "score": 1.0, + "content": "The motions are best seen in the supplemental video https://youtu.be/L3vDo3nLI98. We evalu-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 505, + 574 + ], + "score": 1.0, + "content": "ate the action parameterizations by training policies for a simulated 2D biped, dog, and raptor as", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "shown in Figure 1. Depending on the agent and the actuation model, our systems have 58–214 state", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "dimensions, 6–44 action dimensions, and 0–282 actuator parameters, as summarized in Table 3", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 506, + 607 + ], + "score": 1.0, + "content": "(supplemental materials). The MTU models have at least double the number of action parameters", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 606, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 505, + 617 + ], + "score": 1.0, + "content": "because they come in antagonistic pairs. As well, additional MTUs are used for the legs to more", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 617, + 455, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 455, + 629 + ], + "score": 1.0, + "content": "accurately reflect bipedal biomechanics. This includes MTUs that span multiple joints.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34, + "bbox_fs": [ + 105, + 549, + 506, + 629 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 632, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 505, + 645 + ], + "score": 1.0, + "content": "Each policy is represented by a three layer neural network, as illustrated in Figure 8 (supplemental", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 505, + 656 + ], + "score": 1.0, + "content": "material) with 512 and 256 fully-connected units, followed by a linear output layer where the number", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "of output units vary according to the number of action parameters for each character and actuation", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 505, + 679 + ], + "score": 1.0, + "content": "model. 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Each policy requires about 10 hours for", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 720, + 437, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 437, + 733 + ], + "score": 1.0, + "content": "the biped, and 20 hours for the raptor and dog on an 8-core Intel Xeon E5-2687W.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 633, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 81, + 502, + 249 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 81, + 502, + 249 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 81, + 502, + 249 + ], + "spans": [ + { + "bbox": [ + 112, + 81, + 502, + 249 + ], + "score": 0.969, + "type": "image", + "image_path": "93b7292fed0365462a02a39731b438eabc707c5107988fe50e2bceec740286b3.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 81, + 502, + 137.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 137.0, + 502, + 193.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 193.0, + 502, + 249.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 165, + 261, + 442, + 273 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 261, + 443, + 275 + ], + "spans": [ + { + "bbox": [ + 165, + 261, + 443, + 275 + ], + "score": 1.0, + "content": "Figure 2: Learning curves for each policy during 1 million iterations.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 296, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 310 + ], + "score": 1.0, + "content": "Only the actuator parameters for MTUs are optimized with Algorithm 2, since the parameters for the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "other actuation models are few and reasonably intuitive to determine. 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Each pass", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 148, + 343 + ], + "score": 1.0, + "content": "optimizes", + "type": "text" + }, + { + "bbox": [ + 148, + 331, + 157, + 342 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 330, + 442, + 343 + ], + "score": 1.0, + "content": "using CMA for 250 generations with 16 samples per generation, and", + "type": "text" + }, + { + "bbox": [ + 442, + 331, + 448, + 340 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "is trained for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 128, + 352 + ], + "score": 0.64, + "content": "2 5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "iterations. Parameters are initialized with values from the previous pass. The expected value", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 205, + 364 + ], + "score": 1.0, + "content": "of each CMA sample of", + "type": "text" + }, + { + "bbox": [ + 205, + 352, + 213, + 363 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "is estimated using the average cumulative reward over 16 rollouts with a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "duration of 10s each. Separate MTU parameters are optimized for each character and motion. Each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "set of parameters is optimized for 6 passes following Algorithm 2, requiring approximately 50 hours.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "Figure 5 illustrates the performance improvement per pass. Figure 6 compares the performance", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "of MTUs before and after optimization. For most examples, the optimized actuator parameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "significantly improve learning speed and final performance. For the sake of comparison, after a set of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "actuator parameters has been optimized, a new policy is retrained with the new actuator parameters", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 429, + 351, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 351, + 441 + ], + "score": 1.0, + "content": "and its performance compared to the other actuation models.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "Policy Performance and Learning Speed: Figure 2 shows learning curves for the policies and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 455, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 470 + ], + "score": 1.0, + "content": "the performance of the final policies are summarized in Table 4. Performance is evaluated using", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "the normalized cumulative reward (NCR), calculated from the average cumulative reward over 32", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "episodes with lengths of 10s, and normalized by the maximum and minimum cumulative reward", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 489, + 507, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 507, + 502 + ], + "score": 1.0, + "content": "possible for each episode. No discounting is applied when calculating the NCR. The initial state of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 362, + 514 + ], + "score": 1.0, + "content": "each episode is sampled from the reference motion according to", + "type": "text" + }, + { + "bbox": [ + 362, + 500, + 385, + 513 + ], + "score": 0.92, + "content": "p ( s _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 501, + 505, + 514 + ], + "score": 1.0, + "content": ". To compare learning speeds,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "we use the normalized area under each learning curve (AUC) as a proxy for the learning speed of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "a particular actuation model, where 0 represents the worst possible performance and no progress", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 532, + 467, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 467, + 547 + ], + "score": 1.0, + "content": "during training, and 1 represents the best possible performance without requiring training.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21 + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "PD performs well across all examples, achieving comparable-to-the-best performance for all mo-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "tions. PD also learns faster than the other parameterizations for 5 of the 7 motions. The final per-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "formance of Tor is among the poorest for all the motions. Differences in performance appear more", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "pronounced as characters become more complex. For the simple 7-link biped, most parameteriza-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "tions achieve similar performance. However, for the more complex dog and raptor, the performance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "of Tor policies deteriorate with respect to other policies such as PD and Vel. MTU policies often", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "exhibited the slowest learning speed, which may be a consequence of the higher dimensional action", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "spaces, i.e., requiring antagonistic muscle pairs, and complex muscle dynamics. Nonetheless, once", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "optimized, the MTU policies produce more natural motions and responsive behaviors as compared", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "to other parameterizations. We note that the naturalness of motions is not well captured by the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "reward, since it primarily gauges similarity to the reference motion, which may not be representa-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "tive of natural responses when perturbed from the nominal trajectory. A sensitivity analysis of the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "policies’ performance to variations in network architecture and hyperparameters are available in the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 694, + 199, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 199, + 704 + ], + "score": 1.0, + "content": "supplemental material.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 32.5 + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Policy Robustness: To evaluate robustness, we recorded the NCR achieved by each policy when", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "subjected to external perturbations. The perturbations assume the form of random forces applied", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 307, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 308, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 112, + 81, + 502, + 249 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 112, + 81, + 502, + 249 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 112, + 81, + 502, + 249 + ], + "spans": [ + { + "bbox": [ + 112, + 81, + 502, + 249 + ], + "score": 0.969, + "type": "image", + "image_path": "93b7292fed0365462a02a39731b438eabc707c5107988fe50e2bceec740286b3.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 112, + 81, + 502, + 137.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 112, + 137.0, + 502, + 193.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 112, + 193.0, + 502, + 249.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 165, + 261, + 442, + 273 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 165, + 261, + 443, + 275 + ], + "spans": [ + { + "bbox": [ + 165, + 261, + 443, + 275 + ], + "score": 1.0, + "content": "Figure 2: Learning curves for each policy during 1 million iterations.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + } + ], + "index": 2.0 + }, + { + "type": "text", + "bbox": [ + 107, + 297, + 505, + 439 + ], + "lines": [ + { + "bbox": [ + 106, + 296, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 296, + 505, + 310 + ], + "score": 1.0, + "content": "Only the actuator parameters for MTUs are optimized with Algorithm 2, since the parameters for the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 506, + 321 + ], + "score": 1.0, + "content": "other actuation models are few and reasonably intuitive to determine. The initial actuator parameters", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 318, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 119, + 331 + ], + "score": 0.88, + "content": "\\psi _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 119, + 318, + 351, + 332 + ], + "score": 1.0, + "content": "are manually specified, while the initial policy parameters", + "type": "text" + }, + { + "bbox": [ + 351, + 319, + 362, + 330 + ], + "score": 0.88, + "content": "\\theta _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 318, + 505, + 332 + ], + "score": 1.0, + "content": "are randomly initialized. Each pass", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 330, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 148, + 343 + ], + "score": 1.0, + "content": "optimizes", + "type": "text" + }, + { + "bbox": [ + 148, + 331, + 157, + 342 + ], + "score": 0.85, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 330, + 442, + 343 + ], + "score": 1.0, + "content": "using CMA for 250 generations with 16 samples per generation, and", + "type": "text" + }, + { + "bbox": [ + 442, + 331, + 448, + 340 + ], + "score": 0.78, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 330, + 505, + 343 + ], + "score": 1.0, + "content": "is trained for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 340, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 106, + 341, + 128, + 352 + ], + "score": 0.64, + "content": "2 5 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 340, + 506, + 354 + ], + "score": 1.0, + "content": "iterations. Parameters are initialized with values from the previous pass. The expected value", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 351, + 506, + 364 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 205, + 364 + ], + "score": 1.0, + "content": "of each CMA sample of", + "type": "text" + }, + { + "bbox": [ + 205, + 352, + 213, + 363 + ], + "score": 0.86, + "content": "\\psi", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 351, + 506, + 364 + ], + "score": 1.0, + "content": "is estimated using the average cumulative reward over 16 rollouts with a", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "spans": [ + { + "bbox": [ + 106, + 363, + 505, + 375 + ], + "score": 1.0, + "content": "duration of 10s each. Separate MTU parameters are optimized for each character and motion. Each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "spans": [ + { + "bbox": [ + 106, + 374, + 505, + 387 + ], + "score": 1.0, + "content": "set of parameters is optimized for 6 passes following Algorithm 2, requiring approximately 50 hours.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 505, + 398 + ], + "score": 1.0, + "content": "Figure 5 illustrates the performance improvement per pass. Figure 6 compares the performance", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "of MTUs before and after optimization. For most examples, the optimized actuator parameters", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 506, + 419 + ], + "score": 1.0, + "content": "significantly improve learning speed and final performance. For the sake of comparison, after a set of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 506, + 430 + ], + "score": 1.0, + "content": "actuator parameters has been optimized, a new policy is retrained with the new actuator parameters", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 429, + 351, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 351, + 441 + ], + "score": 1.0, + "content": "and its performance compared to the other actuation models.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 296, + 506, + 441 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 445, + 505, + 545 + ], + "lines": [ + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "Policy Performance and Learning Speed: Figure 2 shows learning curves for the policies and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 455, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 506, + 470 + ], + "score": 1.0, + "content": "the performance of the final policies are summarized in Table 4. Performance is evaluated using", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "spans": [ + { + "bbox": [ + 106, + 468, + 505, + 479 + ], + "score": 1.0, + "content": "the normalized cumulative reward (NCR), calculated from the average cumulative reward over 32", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 491 + ], + "score": 1.0, + "content": "episodes with lengths of 10s, and normalized by the maximum and minimum cumulative reward", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 489, + 507, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 507, + 502 + ], + "score": 1.0, + "content": "possible for each episode. No discounting is applied when calculating the NCR. The initial state of", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 362, + 514 + ], + "score": 1.0, + "content": "each episode is sampled from the reference motion according to", + "type": "text" + }, + { + "bbox": [ + 362, + 500, + 385, + 513 + ], + "score": 0.92, + "content": "p ( s _ { 0 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 501, + 505, + 514 + ], + "score": 1.0, + "content": ". To compare learning speeds,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 505, + 524 + ], + "score": 1.0, + "content": "we use the normalized area under each learning curve (AUC) as a proxy for the learning speed of", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "a particular actuation model, where 0 represents the worst possible performance and no progress", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 532, + 467, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 467, + 547 + ], + "score": 1.0, + "content": "during training, and 1 represents the best possible performance without requiring training.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 444, + 507, + 547 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 550, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "PD performs well across all examples, achieving comparable-to-the-best performance for all mo-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 505, + 574 + ], + "score": 1.0, + "content": "tions. PD also learns faster than the other parameterizations for 5 of the 7 motions. The final per-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 505, + 585 + ], + "score": 1.0, + "content": "formance of Tor is among the poorest for all the motions. Differences in performance appear more", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "pronounced as characters become more complex. For the simple 7-link biped, most parameteriza-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "tions achieve similar performance. However, for the more complex dog and raptor, the performance", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "of Tor policies deteriorate with respect to other policies such as PD and Vel. MTU policies often", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 506, + 628 + ], + "score": 1.0, + "content": "exhibited the slowest learning speed, which may be a consequence of the higher dimensional action", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 506, + 639 + ], + "score": 1.0, + "content": "spaces, i.e., requiring antagonistic muscle pairs, and complex muscle dynamics. Nonetheless, once", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "optimized, the MTU policies produce more natural motions and responsive behaviors as compared", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "to other parameterizations. We note that the naturalness of motions is not well captured by the", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 659, + 505, + 673 + ], + "score": 1.0, + "content": "reward, since it primarily gauges similarity to the reference motion, which may not be representa-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "tive of natural responses when perturbed from the nominal trajectory. A sensitivity analysis of the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 506, + 694 + ], + "score": 1.0, + "content": "policies’ performance to variations in network architecture and hyperparameters are available in the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 694, + 199, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 694, + 199, + 704 + ], + "score": 1.0, + "content": "supplemental material.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 550, + 506, + 704 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "Policy Robustness: To evaluate robustness, we recorded the NCR achieved by each policy when", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "subjected to external perturbations. The perturbations assume the form of random forces applied", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "to the trunk of the characters. Figure 3 illustrates the performance of the policies when subjected", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "to perturbations of different magnitudes. The magnitude of the forces are constant, but direction", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "varies randomly. Each force is applied for 0.1 to 0.4s, with 1 to 4s between each perturbation.", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Performance is estimated using the average over 128 episodes of length 20s each. For the biped", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "walk, the Tor policy is significantly less robust than those for the other types of actions, while the", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "MTU policy is the least robust for the raptor run. Overall, the PD policies are among the most", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 504, + 160 + ], + "score": 1.0, + "content": "robust for all the motions. In addition to external forces, we also evaluate robustness over randomly", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "generated terrain consisting of bumps with varying heights and slopes with varying steepness. We", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 182 + ], + "score": 1.0, + "content": "evaluate the performance on irregular terrain (Figure 12, supplemental material). There are few", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "discernible patterns for this test. The Vel and MTU policies are significantly worse than the Tor and", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 204 + ], + "score": 1.0, + "content": "PD policies for the dog bound on the bumpy terrain. The unnatural jittery behavior of the dog Tor", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "policy proves to be surprisingly robust for this scenario. We suspect that the behavior prevents the", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "trunk from contacting the ground for extended periods for time, and thereby escaping our system’s", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 164, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 164, + 237 + ], + "score": 1.0, + "content": "fall detection.", + "type": "text", + "cross_page": true + } + ], + "index": 13 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 709, + 505, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 236 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "to the trunk of the characters. Figure 3 illustrates the performance of the policies when subjected", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "to perturbations of different magnitudes. The magnitude of the forces are constant, but direction", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 505, + 116 + ], + "score": 1.0, + "content": "varies randomly. Each force is applied for 0.1 to 0.4s, with 1 to 4s between each perturbation.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "Performance is estimated using the average over 128 episodes of length 20s each. For the biped", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "walk, the Tor policy is significantly less robust than those for the other types of actions, while the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 150 + ], + "score": 1.0, + "content": "MTU policy is the least robust for the raptor run. Overall, the PD policies are among the most", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 504, + 160 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 504, + 160 + ], + "score": 1.0, + "content": "robust for all the motions. In addition to external forces, we also evaluate robustness over randomly", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 506, + 172 + ], + "score": 1.0, + "content": "generated terrain consisting of bumps with varying heights and slopes with varying steepness. We", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 170, + 506, + 182 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 182 + ], + "score": 1.0, + "content": "evaluate the performance on irregular terrain (Figure 12, supplemental material). There are few", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 180, + 506, + 194 + ], + "score": 1.0, + "content": "discernible patterns for this test. The Vel and MTU policies are significantly worse than the Tor and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 506, + 204 + ], + "score": 1.0, + "content": "PD policies for the dog bound on the bumpy terrain. The unnatural jittery behavior of the dog Tor", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "spans": [ + { + "bbox": [ + 105, + 203, + 505, + 216 + ], + "score": 1.0, + "content": "policy proves to be surprisingly robust for this scenario. We suspect that the behavior prevents the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "spans": [ + { + "bbox": [ + 105, + 214, + 506, + 227 + ], + "score": 1.0, + "content": "trunk from contacting the ground for extended periods for time, and thereby escaping our system’s", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 224, + 164, + 237 + ], + "spans": [ + { + "bbox": [ + 105, + 224, + 164, + 237 + ], + "score": 1.0, + "content": "fall detection.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 6.5 + }, + { + "type": "image", + "bbox": [ + 109, + 246, + 503, + 346 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 246, + 503, + 346 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 246, + 503, + 346 + ], + "spans": [ + { + "bbox": [ + 109, + 246, + 503, + 346 + ], + "score": 0.967, + "type": "image", + "image_path": "59c61c631803c3c1ef5dced30e6b9d2417658876f02b3fafd5c5920c6ab96ddd.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 109, + 246, + 503, + 279.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 109, + 279.3333333333333, + 503, + 312.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 109, + 312.66666666666663, + 503, + 345.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 113, + 358, + 492, + 370 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 357, + 492, + 372 + ], + "spans": [ + { + "bbox": [ + 115, + 357, + 492, + 372 + ], + "score": 1.0, + "content": "Figure 3: Performance when subjected to random perturbation forces of different magnitudes.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + } + ], + "index": 16.0 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 395 + ], + "score": 1.0, + "content": "Query Rate: Figure 4 compares the performance of different parameterizations for different policy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 104, + 393, + 360, + 406 + ], + "score": 1.0, + "content": "query rates. Separate policies are trained with queries of 15Hz,", + "type": "text" + }, + { + "bbox": [ + 361, + 393, + 383, + 403 + ], + "score": 0.36, + "content": "3 0 \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 393, + 387, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 387, + 393, + 410, + 403 + ], + "score": 0.35, + "content": "6 0 \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 393, + 431, + 406 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 431, + 393, + 459, + 403 + ], + "score": 0.62, + "content": "1 2 0 \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 393, + 506, + 406 + ], + "score": 1.0, + "content": ". Actuation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "models that incorporate low-level feedback such as PD and Vel, appear to cope more effectively to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "lower query rates, while the Tor degrades more rapidly at lower query rates. It is not yet obvious", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "to us why MTU policies appear to perform better at lower query rates and worse at higher rates.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "Lastly, Figure 14 shows the policy outputs as a function of time for the four actuation models, for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "a particular joint, as well as showing the resulting joint torque. Interestingly, the MTU action is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "visibly smoother than the other actions and results in joint torques profiles that are smoother than", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 214, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 214, + 480 + ], + "score": 1.0, + "content": "those seen for PD and Vel.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22 + }, + { + "type": "image", + "bbox": [ + 170, + 490, + 440, + 596 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 170, + 490, + 440, + 596 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 170, + 490, + 440, + 596 + ], + "spans": [ + { + "bbox": [ + 170, + 490, + 440, + 596 + ], + "score": 0.966, + "type": "image", + "image_path": "d2c70584eca59abbfb1ffb645c9e04955f3a1ad659e5d9061ea8ca9b9960c36c.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 170, + 490, + 440, + 525.3333333333334 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 170, + 525.3333333333334, + 440, + 560.6666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 170, + 560.6666666666667, + 440, + 596.0000000000001 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 609, + 503, + 631 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 609, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 504, + 621 + ], + "score": 1.0, + "content": "Figure 4: Performance of policies with different query rates for the biped (left) and dog (right).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 619, + 302, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 302, + 632 + ], + "score": 1.0, + "content": "Separate policies are trained for each query rate.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + } + ], + "index": 29.25 + }, + { + "type": "title", + "bbox": [ + 108, + 652, + 211, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 213, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 213, + 667 + ], + "score": 1.0, + "content": "6 RELATED WORK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "DeepRL has driven impressive recent advances in learning motion control, i.e., solving for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "continuous-action control problems using reinforcement learning. All four of the actions types that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "we explore have seen previous use in the machine learning literature. WawrzynSki & Tanwani ´", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "(2013) use an actor-critic approach with experience replay to learn skills for an octopus arm (actu-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "score": 1.0, + "content": "ated by a simple muscle model) and a planar half cheetah (actuated by joint-based PD-controllers).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2017", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 236 + ], + "lines": [], + "index": 6.5, + "bbox_fs": [ + 105, + 82, + 506, + 237 + ], + "lines_deleted": true + }, + { + "type": "image", + "bbox": [ + 109, + 246, + 503, + 346 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 109, + 246, + 503, + 346 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 246, + 503, + 346 + ], + "spans": [ + { + "bbox": [ + 109, + 246, + 503, + 346 + ], + "score": 0.967, + "type": "image", + "image_path": "59c61c631803c3c1ef5dced30e6b9d2417658876f02b3fafd5c5920c6ab96ddd.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 109, + 246, + 503, + 279.3333333333333 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 109, + 279.3333333333333, + 503, + 312.66666666666663 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 109, + 312.66666666666663, + 503, + 345.99999999999994 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 113, + 358, + 492, + 370 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 357, + 492, + 372 + ], + "spans": [ + { + "bbox": [ + 115, + 357, + 492, + 372 + ], + "score": 1.0, + "content": "Figure 3: Performance when subjected to random perturbation forces of different magnitudes.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + } + ], + "index": 16.0 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 505, + 480 + ], + "lines": [ + { + "bbox": [ + 106, + 382, + 505, + 395 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 395 + ], + "score": 1.0, + "content": "Query Rate: Figure 4 compares the performance of different parameterizations for different policy", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 393, + 506, + 406 + ], + "spans": [ + { + "bbox": [ + 104, + 393, + 360, + 406 + ], + "score": 1.0, + "content": "query rates. Separate policies are trained with queries of 15Hz,", + "type": "text" + }, + { + "bbox": [ + 361, + 393, + 383, + 403 + ], + "score": 0.36, + "content": "3 0 \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 393, + 387, + 406 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 387, + 393, + 410, + 403 + ], + "score": 0.35, + "content": "6 0 \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 393, + 431, + 406 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 431, + 393, + 459, + 403 + ], + "score": 0.62, + "content": "1 2 0 \\mathrm { H z }", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 393, + 506, + 406 + ], + "score": 1.0, + "content": ". Actuation", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 417 + ], + "score": 1.0, + "content": "models that incorporate low-level feedback such as PD and Vel, appear to cope more effectively to", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "lower query rates, while the Tor degrades more rapidly at lower query rates. It is not yet obvious", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "to us why MTU policies appear to perform better at lower query rates and worse at higher rates.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 450 + ], + "score": 1.0, + "content": "Lastly, Figure 14 shows the policy outputs as a function of time for the four actuation models, for", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 447, + 506, + 460 + ], + "spans": [ + { + "bbox": [ + 104, + 447, + 506, + 460 + ], + "score": 1.0, + "content": "a particular joint, as well as showing the resulting joint torque. Interestingly, the MTU action is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 471 + ], + "score": 1.0, + "content": "visibly smoother than the other actions and results in joint torques profiles that are smoother than", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 469, + 214, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 214, + 480 + ], + "score": 1.0, + "content": "those seen for PD and Vel.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 22, + "bbox_fs": [ + 104, + 382, + 506, + 480 + ] + }, + { + "type": "image", + "bbox": [ + 170, + 490, + 440, + 596 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 170, + 490, + 440, + 596 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 170, + 490, + 440, + 596 + ], + "spans": [ + { + "bbox": [ + 170, + 490, + 440, + 596 + ], + "score": 0.966, + "type": "image", + "image_path": "d2c70584eca59abbfb1ffb645c9e04955f3a1ad659e5d9061ea8ca9b9960c36c.jpg" + } + ] + } + ], + "index": 28, + "virtual_lines": [ + { + "bbox": [ + 170, + 490, + 440, + 525.3333333333334 + ], + "spans": [], + "index": 27 + }, + { + "bbox": [ + 170, + 525.3333333333334, + 440, + 560.6666666666667 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 170, + 560.6666666666667, + 440, + 596.0000000000001 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 609, + 503, + 631 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 106, + 609, + 504, + 621 + ], + "spans": [ + { + "bbox": [ + 106, + 609, + 504, + 621 + ], + "score": 1.0, + "content": "Figure 4: Performance of policies with different query rates for the biped (left) and dog (right).", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 619, + 302, + 632 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 302, + 632 + ], + "score": 1.0, + "content": "Separate policies are trained for each query rate.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 30.5 + } + ], + "index": 29.25 + }, + { + "type": "title", + "bbox": [ + 108, + 652, + 211, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 651, + 213, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 213, + 667 + ], + "score": 1.0, + "content": "6 RELATED WORK", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 677, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "DeepRL has driven impressive recent advances in learning motion control, i.e., solving for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "continuous-action control problems using reinforcement learning. All four of the actions types that", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "score": 1.0, + "content": "we explore have seen previous use in the machine learning literature. WawrzynSki & Tanwani ´", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "(2013) use an actor-critic approach with experience replay to learn skills for an octopus arm (actu-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 504, + 732 + ], + "score": 1.0, + "content": "ated by a simple muscle model) and a planar half cheetah (actuated by joint-based PD-controllers).", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 677, + 506, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 159 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 506, + 96 + ], + "score": 1.0, + "content": "Recent work on deterministic policy gradients (Lillicrap et al., 2015) and on RL benchmarks, e.g.,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "OpenAI Gym, generally use joint torques as the action space, as do the test suites in recent work", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 506, + 118 + ], + "score": 1.0, + "content": "(Schulman et al., 2015) on using generalized advantage estimation. Other recent work uses: the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "PR2 effort control interface as a proxy for torque control (Levine et al., 2015); joint velocities (Gu", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "et al., 2016); velocities under an implicit control policy (Mordatch et al., 2015); or provide abstract", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "actions (Hausknecht & Stone, 2015). Our learning procedures are based on prior work using actor-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 428, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 428, + 161 + ], + "score": 1.0, + "content": "critic approaches with positive temporal difference updates (Van Hasselt, 2012).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "Work in biomechanics has long recognized the embodied nature of the control problem and the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "view that musculotendon systems provide “preflexes” (Loeb, 1995) that effectively provide a form", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "intelligence by mechanics (Blickhan et al., 2007), as well as allowing for energy storage. The control", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 210 + ], + "score": 1.0, + "content": "strategies for physics-based character simulations in computer animation also use all the forms of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "actuation that we evaluate in this paper. Representative examples include quadratic programs that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "solve for joint torques (de Lasa et al., 2010), joint velocities for skilled bicycle stunts (Tan et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "2014), muscle models for locomotion (Wang et al., 2012; Geijtenbeek et al., 2013), mixed use of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "feed-forward torques and joint target angles (Coros et al., 2011), and joint target angles computed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "by learned linear (time-indexed) feedback strategies (Liu et al., 2016). Lastly, control methods in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "robotics use a mix of actuation types, including direct-drive torques (or their virtualized equivalents),", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "series elastic actuators, PD control, and velocity control. These methods often rely heavily on model-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 393, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 393, + 298 + ], + "score": 1.0, + "content": "based solutions and thus we do not describe these in further detail here.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 108, + 313, + 201, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 204, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 204, + 329 + ], + "score": 1.0, + "content": "7 CONCLUSIONS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "Our experiments suggest that action parameterizations that include basic local feedback, such as PD", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "target angles, MTU activations, or target velocities, can improve policy performance and learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "speed across different motions and character morphologies. Such models more accurately reflect", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "the embodied nature of control in biomechanical systems, and the role of mechanical components", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "in shaping the overall dynamics of motions and their control. The difference between low-level", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "and high-level action parameterizations grow with the complexity of the characters, with high-level", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "parameterizations scaling more gracefully to complex characters. As a caveat, there may well be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "tasks, such as impedance control, where lower-level action parameterizations such as Tor may prove", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "advantageous. We believe that no single action parameterization will be the best for all problems.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "However, since objectives for motion control problems are often naturally expressed in terms of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "kinematic properties, higher-level actions such as target joint angles and velocities may be effective", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "score": 1.0, + "content": "for a wide variety of motion control problems. We hope that our work will help open discussions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 470, + 293, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 293, + 482 + ], + "score": 1.0, + "content": "around the choice of action parameterizations.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 107, + 487, + 504, + 497 + ], + "spans": [ + { + "bbox": [ + 107, + 487, + 504, + 497 + ], + "score": 1.0, + "content": "Our results have only been demonstrated on planar articulated figure simulations; the extension to 3D", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 509 + ], + "score": 1.0, + "content": "currently remains as future work. Furthermore, our current torque limits are still large as compared", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "to what might be physically realizable. Tuning actuator parameters for complex actuation models", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "such as MTUs remains challenging. Though our actuator optimization technique is able to improve", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "performance as compared to manual tuning, the resulting parameters may still not be optimal for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "score": 1.0, + "content": "the desired task. Therefore, our comparisons of MTUs to other action parameterizations may not be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "reflective of the full potential of MTUs with more optimal actuator parameters. Furthermore, our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 564, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 575 + ], + "score": 1.0, + "content": "actuator optimization currently tunes parameters for a specific motion, rather than a larger suite of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 575, + 270, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 270, + 586 + ], + "score": 1.0, + "content": "motions, as might be expected in nature.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 592, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 505, + 603 + ], + "score": 1.0, + "content": "Since the reward terms are mainly calculated according to joint positions and velocities, it may seem", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "that it is inherently biased in favour of PD and Vel. However, the real challenges for the control", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "score": 1.0, + "content": "policies lie elsewhere, such as learning to compensate for gravity and ground-reaction forces, and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "learning foot-placement strategies that are needed to maintain balance for the locomotion gaits. The", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "reference pose terms provide little information on how to achieve these hidden aspects of motion", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "control that will ultimately determine the success of the locomotion policy. While we have yet to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "provide a concrete answer for the generalization of our results to different reward functions, we", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "believe that the choice of action parameterization is a design decision that deserves greater attention", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 678, + 281, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 281, + 691 + ], + "score": 1.0, + "content": "regardless of the choice of reward function.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 106, + 696, + 504, + 730 + ], + "lines": [ + { + "bbox": [ + 105, + 695, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 505, + 709 + ], + "score": 1.0, + "content": "Finally, it is reasonable to expect that evolutionary processes would result in the effective co-design", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 707, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 505, + 720 + ], + "score": 1.0, + "content": "of actuation mechanics and control capabilities. 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Other recent work uses: the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 104, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "PR2 effort control interface as a proxy for torque control (Levine et al., 2015); joint velocities (Gu", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "et al., 2016); velocities under an implicit control policy (Mordatch et al., 2015); or provide abstract", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 149 + ], + "score": 1.0, + "content": "actions (Hausknecht & Stone, 2015). Our learning procedures are based on prior work using actor-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 428, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 428, + 161 + ], + "score": 1.0, + "content": "critic approaches with positive temporal difference updates (Van Hasselt, 2012).", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 3, + "bbox_fs": [ + 104, + 81, + 506, + 161 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 165, + 505, + 297 + ], + "lines": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 506, + 178 + ], + "score": 1.0, + "content": "Work in biomechanics has long recognized the embodied nature of the control problem and the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "view that musculotendon systems provide “preflexes” (Loeb, 1995) that effectively provide a form", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 187, + 506, + 200 + ], + "score": 1.0, + "content": "intelligence by mechanics (Blickhan et al., 2007), as well as allowing for energy storage. The control", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 199, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 199, + 506, + 210 + ], + "score": 1.0, + "content": "strategies for physics-based character simulations in computer animation also use all the forms of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 506, + 222 + ], + "score": 1.0, + "content": "actuation that we evaluate in this paper. Representative examples include quadratic programs that", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "solve for joint torques (de Lasa et al., 2010), joint velocities for skilled bicycle stunts (Tan et al.,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 506, + 244 + ], + "score": 1.0, + "content": "2014), muscle models for locomotion (Wang et al., 2012; Geijtenbeek et al., 2013), mixed use of", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 104, + 241, + 506, + 255 + ], + "score": 1.0, + "content": "feed-forward torques and joint target angles (Coros et al., 2011), and joint target angles computed", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "by learned linear (time-indexed) feedback strategies (Liu et al., 2016). Lastly, control methods in", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 263, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 505, + 277 + ], + "score": 1.0, + "content": "robotics use a mix of actuation types, including direct-drive torques (or their virtualized equivalents),", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "series elastic actuators, PD control, and velocity control. These methods often rely heavily on model-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 393, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 393, + 298 + ], + "score": 1.0, + "content": "based solutions and thus we do not describe these in further detail here.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5, + "bbox_fs": [ + 104, + 164, + 506, + 298 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 313, + 201, + 326 + ], + "lines": [ + { + "bbox": [ + 105, + 311, + 204, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 311, + 204, + 329 + ], + "score": 1.0, + "content": "7 CONCLUSIONS", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 107, + 338, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 351 + ], + "score": 1.0, + "content": "Our experiments suggest that action parameterizations that include basic local feedback, such as PD", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 105, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "target angles, MTU activations, or target velocities, can improve policy performance and learning", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "speed across different motions and character morphologies. Such models more accurately reflect", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "the embodied nature of control in biomechanical systems, and the role of mechanical components", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "in shaping the overall dynamics of motions and their control. The difference between low-level", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 505, + 405 + ], + "score": 1.0, + "content": "and high-level action parameterizations grow with the complexity of the characters, with high-level", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 404, + 505, + 416 + ], + "score": 1.0, + "content": "parameterizations scaling more gracefully to complex characters. As a caveat, there may well be", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 428 + ], + "score": 1.0, + "content": "tasks, such as impedance control, where lower-level action parameterizations such as Tor may prove", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "advantageous. We believe that no single action parameterization will be the best for all problems.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 437, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 437, + 505, + 448 + ], + "score": 1.0, + "content": "However, since objectives for motion control problems are often naturally expressed in terms of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "kinematic properties, higher-level actions such as target joint angles and velocities may be effective", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 505, + 470 + ], + "score": 1.0, + "content": "for a wide variety of motion control problems. We hope that our work will help open discussions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 470, + 293, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 293, + 482 + ], + "score": 1.0, + "content": "around the choice of action parameterizations.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 338, + 506, + 482 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 487, + 505, + 585 + ], + "lines": [ + { + "bbox": [ + 107, + 487, + 504, + 497 + ], + "spans": [ + { + "bbox": [ + 107, + 487, + 504, + 497 + ], + "score": 1.0, + "content": "Our results have only been demonstrated on planar articulated figure simulations; the extension to 3D", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 498, + 505, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 498, + 505, + 509 + ], + "score": 1.0, + "content": "currently remains as future work. Furthermore, our current torque limits are still large as compared", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "to what might be physically realizable. Tuning actuator parameters for complex actuation models", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "such as MTUs remains challenging. Though our actuator optimization technique is able to improve", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 530, + 505, + 543 + ], + "score": 1.0, + "content": "performance as compared to manual tuning, the resulting parameters may still not be optimal for", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 542, + 505, + 553 + ], + "score": 1.0, + "content": "the desired task. Therefore, our comparisons of MTUs to other action parameterizations may not be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 552, + 505, + 564 + ], + "score": 1.0, + "content": "reflective of the full potential of MTUs with more optimal actuator parameters. Furthermore, our", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 564, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 505, + 575 + ], + "score": 1.0, + "content": "actuator optimization currently tunes parameters for a specific motion, rather than a larger suite of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 575, + 270, + 586 + ], + "spans": [ + { + "bbox": [ + 106, + 575, + 270, + 586 + ], + "score": 1.0, + "content": "motions, as might be expected in nature.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 487, + 505, + 586 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 591, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 106, + 592, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 505, + 603 + ], + "score": 1.0, + "content": "Since the reward terms are mainly calculated according to joint positions and velocities, it may seem", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 602, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 106, + 602, + 505, + 614 + ], + "score": 1.0, + "content": "that it is inherently biased in favour of PD and Vel. However, the real challenges for the control", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "spans": [ + { + "bbox": [ + 106, + 614, + 505, + 625 + ], + "score": 1.0, + "content": "policies lie elsewhere, such as learning to compensate for gravity and ground-reaction forces, and", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "learning foot-placement strategies that are needed to maintain balance for the locomotion gaits. The", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 647 + ], + "score": 1.0, + "content": "reference pose terms provide little information on how to achieve these hidden aspects of motion", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 106, + 646, + 505, + 658 + ], + "score": 1.0, + "content": "control that will ultimately determine the success of the locomotion policy. While we have yet to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 505, + 668 + ], + "score": 1.0, + "content": "provide a concrete answer for the generalization of our results to different reward functions, we", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 668, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 505, + 680 + ], + "score": 1.0, + "content": "believe that the choice of action parameterization is a design decision that deserves greater attention", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 678, + 281, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 281, + 691 + ], + "score": 1.0, + "content": "regardless of the choice of reward function.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 46, + "bbox_fs": [ + 105, + 592, + 505, + 691 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 696, + 504, + 730 + ], + "lines": [ + { + "bbox": [ + 105, + 695, + 505, + 709 + ], + "spans": [ + { + "bbox": [ + 105, + 695, + 505, + 709 + ], + "score": 1.0, + "content": "Finally, it is reasonable to expect that evolutionary processes would result in the effective co-design", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 707, + 505, + 720 + ], + "spans": [ + { + "bbox": [ + 106, + 707, + 505, + 720 + ], + "score": 1.0, + "content": "of actuation mechanics and control capabilities. Developing optimization and learning algorithms", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 718, + 416, + 730 + ], + "spans": [ + { + "bbox": [ + 105, + 718, + 416, + 730 + ], + "score": 1.0, + "content": "to allow for this kind of co-design is a fascinating possibility for future work.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 52, + "bbox_fs": [ + 105, + 695, + 505, + 730 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 82, + 176, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 176, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 176, + 95 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 108, + 99, + 504, + 133 + ], + "lines": [ + { + "bbox": [ + 106, + 99, + 505, + 111 + ], + "spans": [ + { + "bbox": [ + 106, + 99, + 505, + 111 + ], + "score": 1.0, + "content": "Reinhard Blickhan, Andre Seyfarth, Hartmut Geyer, Sten Grimmer, Heiko Wagner, and Michael", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 116, + 111, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 116, + 111, + 505, + 123 + ], + "score": 1.0, + "content": "Gunther. 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ParameterValueDescription
Y0.9cumulative reward discount factor
απ0.001actor learning rate
αv0.01critic learning rate
momentum0.9stochastic gradient descent momentum
weight decay0L2 regularizer for critic parameters
θ weight decay0.0005L2 regularizer for actor parameters
minibatch size32tuples per stochastic gradient descent step
replay memory size500000number of the most recent tuples stored for future updates
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ParameterValueDescription
Y0.9cumulative reward discount factor
απ0.001actor learning rate
αv0.01critic learning rate
momentum0.9stochastic gradient descent momentum
weight decay0L2 regularizer for critic parameters
θ weight decay0.0005L2 regularizer for actor parameters
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replay memory size500000number of the most recent tuples stored for future updates
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Character+ Actuation ModelState ParametersAction ParametersActuator Parameters
Biped + TorBiped + VelBiped + PDBiped +MTU5858587460612114
6616
Raptor + TorRaptor + VelRaptor + PDRaptor + MTU1541541541941801836258
18
1840
Dog + TorDog + VelDog + PDDog + MTU1701701702142002040282
20
20
44
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Character+ActuationMotionPerformance (NCR)Learning Speed (AUC)
Biped + Tor Biped + Vel Biped + PDWalk Walk Walk0.7662 ± 0.3117 0.9520 ± 0.0034 0.9524 ± 0.0034 0.9584 ± 0.00650.4788 0.6308 0.6997 0.7165
Biped + MTU Biped +Tor Biped + Vel Biped +PDWalk March March March March0.9353 ±0.0072 0.9784 ± 0.0018 0.9767 士 0.0068 0.9484 ±0.00210.7478 0.9035 0.9136 0.5587
Biped + MTU Biped + Tor Biped + Vel Biped + PDRun Run Run0.9032± 0.0102 0.9070 ± 0.0106 0.9057 ± 0.00560.6938 0.7301 0.7880
Biped +MTU Raptor+ Tor Raptor + VelRun Run (Sim) Run (Sim)0.8988 ± 0.0094 0.7265 士 0.0037 0.9612 土 0.00550.5360 0.5061 0.8118
Raptor + PDRun (Sim)0.9863 ± 0.00170.9282
Raptor + MTURun (Sim)0.9708 ± 0.00230.6330
Raptor +TorRun
0.6141 ± 0.00910.3814
Raptor + VelRun0.8732 ±0.00370.7008
Raptor + PDRun0.9548 ± 0.00100.8372
Raptor + MTURun0.9533 ± 0.0015
Dog + Tor0.7258
Bound (Sim)0.7888 ± 0.00460.4895
Dog + VelBound (Sim)0.9788 ±0.00440.7862
Dog +PDBound (Sim)0.9797 ± 0.00120.9280
Dog+ MTUBound (Sim)0.9033 ±0.00290.6825
Dog + TorRear-Up0.8151 士 0.01130.5550
Dog + VelRear-Up0.7364 ± 0.27070.7454
Dog + PDRear-Up0.9565 ± 0.00580.8701
Dog +MTURear-Up0.8744 ± 0.25660.7932
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Character+ Actuation ModelState ParametersAction ParametersActuator Parameters
Biped + TorBiped + VelBiped + PDBiped +MTU5858587460612114
6616
Raptor + TorRaptor + VelRaptor + PDRaptor + MTU1541541541941801836258
18
1840
Dog + TorDog + VelDog + PDDog + MTU1701701702142002040282
20
20
44
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Character+ActuationMotionPerformance (NCR)Learning Speed (AUC)
Biped + Tor Biped + Vel Biped + PDWalk Walk Walk0.7662 ± 0.3117 0.9520 ± 0.0034 0.9524 ± 0.0034 0.9584 ± 0.00650.4788 0.6308 0.6997 0.7165
Biped + MTU Biped +Tor Biped + Vel Biped +PDWalk March March March March0.9353 ±0.0072 0.9784 ± 0.0018 0.9767 士 0.0068 0.9484 ±0.00210.7478 0.9035 0.9136 0.5587
Biped + MTU Biped + Tor Biped + Vel Biped + PDRun Run Run0.9032± 0.0102 0.9070 ± 0.0106 0.9057 ± 0.00560.6938 0.7301 0.7880
Biped +MTU Raptor+ Tor Raptor + VelRun Run (Sim) Run (Sim)0.8988 ± 0.0094 0.7265 士 0.0037 0.9612 土 0.00550.5360 0.5061 0.8118
Raptor + PDRun (Sim)0.9863 ± 0.00170.9282
Raptor + MTURun (Sim)0.9708 ± 0.00230.6330
Raptor +TorRun
0.6141 ± 0.00910.3814
Raptor + VelRun0.8732 ±0.00370.7008
Raptor + PDRun0.9548 ± 0.00100.8372
Raptor + MTURun0.9533 ± 0.0015
Dog + Tor0.7258
Bound (Sim)0.7888 ± 0.00460.4895
Dog + VelBound (Sim)0.9788 ±0.00440.7862
Dog +PDBound (Sim)0.9797 ± 0.00120.9280
Dog+ MTUBound (Sim)0.9033 ±0.00290.6825
Dog + TorRear-Up0.8151 士 0.01130.5550
Dog + VelRear-Up0.7364 ± 0.27070.7454
Dog + PDRear-Up0.9565 ± 0.00580.8701
Dog +MTURear-Up0.8744 ± 0.25660.7932
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ActuationModelActuatorParameters
Target Joint Angles (PD) proportional gains kp, derivative gains kd
Target Joint Velocities(Vel)derivative gains kd
Torques (Tor)none
Muscle Activations (MTU)optimal contractile element length, serial elastic element rest length,maximum isometric force, pennation,moment arm,maximum moment arm joint orientation, rest joint orientation.
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1:0←random weights
2:Φ ← random weights
3:while not done do
4:for step = 1.,..., .m do
5:s← start state
6:入←Ber(∈t)
7:a ← μθ(s)+λN(0,∑)
8:Apply α and simulate forward 1 step
9:s′← end state
10:r←reward
11:T ← (s,a,r,s',λ)
12:store T in replay memory
13:if episode terminated then
14:Sample so from po(s)
15:Reinitialize state s to so
16:end if
17:end for
18:Update critic:
19:Sample minibatch of n tuples {Ti = (si, ai,ri, Xi, s')} from replay memory
20:for each Ti do
21:δ←ri+γV(s)-V(si)
22:←Φ+avδiV(si)
23:end for
25:Update actor: Sample minibatch of n tuples {Tj = (sj,αj,rj,入j,s'j)} from replay memory where 入j = 1
26:for each Tj do
27:δj←rj+γV(sj)-V(sj)
28:
if δj >O then
29:∀aj←aj-μ(sj)
30:Vaj ← BoundActionGradient(Vaj, μe(sj))
31:θ←θ+αππ∀θμ(sj）∑-1∀aj
32:end if
33:end for
34: end while
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ParameterValueDescription
Y0.9cumulative reward discount factor
απ0.001actor learning rate
αv0.01critic learning rate
momentum0.9stochastic gradient descent momentum
weight decay0L2 regularizer for critic parameters
θ weight decay0.0005L2 regularizer for actor parameters
minibatch size32tuples per stochastic gradient descent step
replay memory size500000number of the most recent tuples stored for future updates
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Character+ActuationMotionPerformance (NCR)Learning Speed (AUC)
Biped + Tor Biped + Vel Biped + PDWalk Walk Walk0.7662 ± 0.3117 0.9520 ± 0.0034 0.9524 ± 0.0034 0.9584 ± 0.00650.4788 0.6308 0.6997 0.7165
Biped + MTU Biped +Tor Biped + Vel Biped +PDWalk March March March March0.9353 ±0.0072 0.9784 ± 0.0018 0.9767 士 0.0068 0.9484 ±0.00210.7478 0.9035 0.9136 0.5587
Biped + MTU Biped + Tor Biped + Vel Biped + PDRun Run Run0.9032± 0.0102 0.9070 ± 0.0106 0.9057 ± 0.00560.6938 0.7301 0.7880
Biped +MTU Raptor+ Tor Raptor + VelRun Run (Sim) Run (Sim)0.8988 ± 0.0094 0.7265 士 0.0037 0.9612 土 0.00550.5360 0.5061 0.8118
Raptor + PDRun (Sim)0.9863 ± 0.00170.9282
Raptor + MTURun (Sim)0.9708 ± 0.00230.6330
Raptor +TorRun
0.6141 ± 0.00910.3814
Raptor + VelRun0.8732 ±0.00370.7008
Raptor + PDRun0.9548 ± 0.00100.8372
Raptor + MTURun0.9533 ± 0.0015
Dog + Tor0.7258
Bound (Sim)0.7888 ± 0.00460.4895
Dog + VelBound (Sim)0.9788 ±0.00440.7862
Dog +PDBound (Sim)0.9797 ± 0.00120.9280
Dog+ MTUBound (Sim)0.9033 ±0.00290.6825
Dog + TorRear-Up0.8151 士 0.01130.5550
Dog + VelRear-Up0.7364 ± 0.27070.7454
Dog + PDRear-Up0.9565 ± 0.00580.8701
Dog +MTURear-Up0.8744 ± 0.25660.7932
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Character+ Actuation ModelState ParametersAction ParametersActuator Parameters
Biped + TorBiped + VelBiped + PDBiped +MTU5858587460612114
6616
Raptor + TorRaptor + VelRaptor + PDRaptor + MTU1541541541941801836258
18
1840
Dog + TorDog + VelDog + PDDog + MTU1701701702142002040282
20
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This comes with several immediate problems:", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 240, + 470, + 251 + ], + "spans": [ + { + "bbox": [ + 141, + 240, + 470, + 251 + ], + "score": 1.0, + "content": "For example, MI is notoriously hard to estimate, and using it as an objective for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 251, + 470, + 263 + ], + "spans": [ + { + "bbox": [ + 141, + 251, + 470, + 263 + ], + "score": 1.0, + "content": "representation learning may lead to highly entangled representations due to its", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 262, + 469, + 273 + ], + "spans": [ + { + "bbox": [ + 141, + 262, + 469, + 273 + ], + "score": 1.0, + "content": "invariance under arbitrary invertible transformations. 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In this paper we argue, and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 284, + 470, + 294 + ], + "spans": [ + { + "bbox": [ + 141, + 284, + 470, + 294 + ], + "score": 1.0, + "content": "provide empirical evidence, that the success of these methods cannot be attributed", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 142, + 295, + 469, + 305 + ], + "spans": [ + { + "bbox": [ + 142, + 295, + 469, + 305 + ], + "score": 1.0, + "content": "to the properties of MI alone, and that they strongly depend on the inductive bias", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 306, + 469, + 317 + ], + "spans": [ + { + "bbox": [ + 141, + 306, + 469, + 317 + ], + "score": 1.0, + "content": "in both the choice of feature extractor architectures and the parametrization of the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 315, + 470, + 330 + ], + "spans": [ + { + "bbox": [ + 141, + 315, + 470, + 330 + ], + "score": 1.0, + "content": "employed MI estimators. Finally, we establish a connection to deep metric learning", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 327, + 469, + 339 + ], + "spans": [ + { + "bbox": [ + 141, + 327, + 469, + 339 + ], + "score": 1.0, + "content": "and argue that this interpretation may be a plausible explanation for the success of", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 142, + 338, + 275, + 351 + ], + "spans": [ + { + "bbox": [ + 142, + 338, + 275, + 351 + ], + "score": 1.0, + "content": "the recently introduced methods.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 11, + "bbox_fs": [ + 141, + 205, + 470, + 351 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 371, + 205, + 384 + ], + "lines": [ + { + "bbox": [ + 105, + 370, + 208, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 370, + 208, + 387 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 396, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 106, + 396, + 505, + 410 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 410 + ], + "score": 1.0, + "content": "Unsupervised representation learning is a fundamental problem in machine learning. 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Even though the task seems ill-posed as there is no natural objective one should", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "optimize, by leveraging domain knowledge this approach can be successfully applied to a variety of", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 463, + 505, + 474 + ], + "score": 1.0, + "content": "problem areas, including image (Kolesnikov et al., 2019; van den Oord et al., 2018; Hénaff et al.,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 474, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 485 + ], + "score": 1.0, + "content": "2019; Tian et al., 2019; Hjelm et al., 2019; Bachman et al., 2019) and video classification (Wang and", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 484, + 505, + 497 + ], + "score": 1.0, + "content": "Gupta, 2015; Sun et al., 2019), and natural language understanding (van den Oord et al., 2018; Peters", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 496, + 238, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 496, + 238, + 507 + ], + "score": 1.0, + "content": "et al., 2018; Devlin et al., 2019).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 23.5, + "bbox_fs": [ + 105, + 396, + 506, + 507 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 512, + 505, + 579 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 506, + 524 + ], + "score": 1.0, + "content": "Recently, there has been a revival of approaches inspired by the InfoMax principle (Linsker, 1988):", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 207, + 536 + ], + "score": 1.0, + "content": "Choose a representation", + "type": "text" + }, + { + "bbox": [ + 208, + 523, + 227, + 535 + ], + "score": 0.92, + "content": "g ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "maximizing the mutual information (MI) between the input and its", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "representation, possibly subject to some structural constraints. 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Secondly, estimating MI in high-dimensional spaces is a notoriously difficult", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "task, and in practice one often maximizes a tractable lower bound on this quantity (Poole et al., 2019).", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 617, + 506, + 672 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "Nonetheless, any distribution-free high-confidence lower bound on entropy requires a sample size", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 376, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 376, + 106 + ], + "score": 1.0, + "content": "exponential in the size of the bound (McAllester and Statos, 2018).", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 110, + 505, + 187 + ], + "lines": [ + { + "bbox": [ + 106, + 111, + 504, + 122 + ], + "spans": [ + { + "bbox": [ + 106, + 111, + 504, + 122 + ], + "score": 1.0, + "content": "Despite these fundamental challenges, several recent works have demonstrated promising empirical", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 122, + 505, + 133 + ], + "spans": [ + { + "bbox": [ + 105, + 122, + 505, + 133 + ], + "score": 1.0, + "content": "results in representation learning using MI maximization (van den Oord et al., 2018; Hénaff et al.,", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "2019; Tian et al., 2019; Hjelm et al., 2019; Bachman et al., 2019; Sun et al., 2019). In this work we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "argue, and provide empirical evidence, that the success of these methods cannot be attributed to the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "properties of MI alone. In fact, we show that maximizing tighter bounds on MI can result in worse", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "representations. 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In this work we", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "argue, and provide empirical evidence, that the success of these methods cannot be attributed to the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 506, + 167 + ], + "score": 1.0, + "content": "properties of MI alone. In fact, we show that maximizing tighter bounds on MI can result in worse", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 177 + ], + "score": 1.0, + "content": "representations. 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Contrastive multiview coding (CMC) (Tian et al., 2019) generalizes the objective in (2) to", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 125, + 531, + 507, + 548 + ], + "spans": [ + { + "bbox": [ + 125, + 531, + 223, + 548 + ], + "score": 1.0, + "content": "consider multiple views", + "type": "text" + }, + { + "bbox": [ + 223, + 532, + 243, + 544 + ], + "score": 0.9, + "content": "X ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 531, + 294, + 548 + ], + "score": 1.0, + "content": ", where each", + "type": "text" + }, + { + "bbox": [ + 294, + 532, + 314, + 543 + ], + "score": 0.91, + "content": "X ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 531, + 507, + 548 + ], + "score": 1.0, + "content": "corresponds to a different image modality (e.g.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 126, + 545, + 391, + 557 + ], + "spans": [ + { + "bbox": [ + 126, + 545, + 391, + 557 + ], + "score": 1.0, + "content": "different color channels, or the image and its segmentation mask).", + "type": "text" + } + ], + "index": 35, + "is_list_end_line": true + }, + { + "bbox": [ + 114, + 556, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 114, + 556, + 505, + 570 + ], + "score": 1.0, + "content": "3. Contrastive predictive coding (CPC) (van den Oord et al., 2018; Hénaff et al., 2019) incorporates", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 125, + 568, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 125, + 568, + 506, + 582 + ], + "score": 1.0, + "content": "a sequential component of the data. 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Given", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 670, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 506, + 682 + ], + "score": 1.0, + "content": "the fundamental limitations of MI estimation (McAllester and Statos, 2018), recent work has focused", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 681, + 507, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 507, + 694 + ], + "score": 1.0, + "content": "on deriving lower bounds on MI (Barber and Agakov, 2003; Belghazi et al., 2018; Poole et al.,", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 45, + "bbox_fs": [ + 105, + 657, + 507, + 694 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "2019). Intuitively, these bounds are based on the following idea: If a classifier can accurately", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 317, + 106 + ], + "score": 1.0, + "content": "distinguish between samples drawn from the joint", + "type": "text" + }, + { + "bbox": [ + 317, + 93, + 347, + 106 + ], + "score": 0.92, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "and those drawn from the product of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 313, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 148, + 116 + ], + "score": 1.0, + "content": "marginals", + "type": "text" + }, + { + "bbox": [ + 148, + 104, + 185, + 117 + ], + "score": 0.93, + "content": "p ( x ) p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 105, + 209, + 116 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 209, + 105, + 219, + 114 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 105, + 237, + 116 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 105, + 247, + 114 + ], + "score": 0.76, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 105, + 313, + 116 + ], + "score": 1.0, + "content": "have a high MI.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 104, + 121, + 504, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 506, + 135 + ], + "score": 1.0, + "content": "We will focus on two such estimators, which are most commonly used in the representation learning", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 452, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 452, + 144 + ], + "score": 1.0, + "content": "literature. The first of them, termed InfoNCE (van den Oord et al., 2018), is defined as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 147, + 430, + 182 + ], + "lines": [ + { + "bbox": [ + 180, + 147, + 430, + 182 + ], + "spans": [ + { + "bbox": [ + 180, + 147, + 430, + 182 + ], + "score": 0.94, + "content": "I ( X ; Y ) \\geq \\mathbb { E } \\left[ \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) } } \\right] \\triangleq I _ { \\tt N C E } ( X ; Y ) ,", + "type": "interline_equation", + "image_path": "243e78ea4391040da2daea342569b96683de91841d9344da31ddf2c2eff5e20a.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 180, + 147, + 430, + 158.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 180, + 158.66666666666666, + 430, + 170.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 180, + 170.33333333333331, + 430, + 181.99999999999997 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 104, + 185, + 507, + 202 + ], + "spans": [ + { + "bbox": [ + 104, + 185, + 234, + 202 + ], + "score": 1.0, + "content": "where the expectation is over", + "type": "text" + }, + { + "bbox": [ + 234, + 187, + 245, + 197 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 185, + 337, + 202 + ], + "score": 1.0, + "content": "independent samples", + "type": "text" + }, + { + "bbox": [ + 337, + 186, + 390, + 199 + ], + "score": 0.93, + "content": "\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 185, + 507, + 202 + ], + "score": 1.0, + "content": "from the joint distribution", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 107, + 198, + 136, + 210 + ], + "score": 0.91, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "(Poole et al., 2019). In practice we estimate (3) using Monte Carlo estimation by averaging", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 363, + 221 + ], + "score": 1.0, + "content": "over multiple batches of samples. Intuitively, the critic function", + "type": "text" + }, + { + "bbox": [ + 363, + 210, + 370, + 221 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 209, + 466, + 221 + ], + "score": 1.0, + "content": "tries to predict for each", + "type": "text" + }, + { + "bbox": [ + 467, + 210, + 477, + 220 + ], + "score": 0.82, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 132, + 234 + ], + "score": 1.0, + "content": "of the", + "type": "text" + }, + { + "bbox": [ + 132, + 221, + 142, + 230 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 219, + 177, + 234 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 178, + 221, + 220, + 232 + ], + "score": 0.9, + "content": "y _ { 1 } , \\ldots , y _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 219, + 506, + 234 + ], + "score": 1.0, + "content": "it was jointly drawn with, by assigning high values to the jointly drawn", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "pair, and low values to all other pairs. The second estimator is based on the variational form of the KL", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "divergence due to Nguyen, Wainwright, and Jordan (NWJ) (Nguyen et al., 2010) and takes the form", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 257, + 445, + 273 + ], + "lines": [ + { + "bbox": [ + 165, + 257, + 445, + 273 + ], + "spans": [ + { + "bbox": [ + 165, + 257, + 445, + 273 + ], + "score": 0.91, + "content": "I ( X ; Y ) \\geq \\mathbb { E } _ { p ( x , y ) } [ f ( x , y ) ] - e ^ { - 1 } \\mathbb { E } _ { p ( x ) } [ \\mathbb { E } _ { p ( y ) } e ^ { f ( x , y ) } ] \\triangleq I _ { \\mathbb { N } \\mathbb { N } } ( X ; Y ) .", + "type": "interline_equation", + "image_path": "6b3d9b83561e96c5b3c906e840480bf764ed97f34d3354022f149b3be25fbdd1.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 165, + 257, + 445, + 273 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "For detailed derivations we refer the reader to (Ruderman et al., 2012; Poole et al., 2019). Note", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 253, + 300 + ], + "score": 1.0, + "content": "that these bounds hold for any critic", + "type": "text" + }, + { + "bbox": [ + 253, + 288, + 261, + 299 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "and when used in (2) one in practice jointly maximizes over", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 298, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 131, + 310 + ], + "score": 0.86, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 298, + 149, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 149, + 299, + 156, + 310 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 298, + 384, + 312 + ], + "score": 1.0, + "content": ". Furthermore, it can be shown that (3) is maximized by", + "type": "text" + }, + { + "bbox": [ + 384, + 298, + 473, + 311 + ], + "score": 0.92, + "content": "f ^ { * } ( x , \\bar { y } ) = \\log p ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 298, + 506, + 312 + ], + "score": 1.0, + "content": "and (4)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 120, + 322 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 310, + 227, + 321 + ], + "score": 0.87, + "content": "f ^ { * } ( x , y ) \\mathop { = } 1 + \\log p ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 309, + 402, + 322 + ], + "score": 1.0, + "content": "(Poole et al., 2019). Common choices for", + "type": "text" + }, + { + "bbox": [ + 403, + 311, + 410, + 321 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "include bilinear critics", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 107, + 321, + 182, + 332 + ], + "score": 0.86, + "content": "f ( x , y ) = x ^ { \\top } W y", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "(van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019), separable", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 330, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 133, + 345 + ], + "score": 1.0, + "content": "critics", + "type": "text" + }, + { + "bbox": [ + 134, + 331, + 231, + 344 + ], + "score": 0.92, + "content": "f ( x , y ) = \\phi _ { 1 } ( x ) ^ { \\top } \\phi _ { 2 } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 330, + 427, + 345 + ], + "score": 1.0, + "content": "(Bachman et al., 2019), and concatenated critics", + "type": "text" + }, + { + "bbox": [ + 427, + 331, + 505, + 344 + ], + "score": 0.92, + "content": "f ( x , y ) = \\phi ( [ x , y ] )", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 342, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 213, + 356 + ], + "score": 1.0, + "content": "(Hjelm et al., 2019) (here", + "type": "text" + }, + { + "bbox": [ + 213, + 343, + 249, + 354 + ], + "score": 0.92, + "content": "\\phi , \\phi _ { 1 } , \\phi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 342, + 506, + 356 + ], + "score": 1.0, + "content": "are typically shallow multi-layer perceptrons (MLPs)). When", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 449, + 366 + ], + "score": 1.0, + "content": "applying these estimators to solve (2), the line between the critic and the encoders", + "type": "text" + }, + { + "bbox": [ + 450, + 355, + 474, + 365 + ], + "score": 0.89, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 360, + 376 + ], + "score": 1.0, + "content": "blurry. For example, one can train with an inner product critic", + "type": "text" + }, + { + "bbox": [ + 360, + 364, + 421, + 376 + ], + "score": 0.93, + "content": "f ( x , y ) = x ^ { \\top } y", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 364, + 505, + 376 + ], + "score": 1.0, + "content": ", but extract features", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 228, + 388 + ], + "score": 1.0, + "content": "from an intermediate layer of", + "type": "text" + }, + { + "bbox": [ + 229, + 376, + 253, + 387 + ], + "score": 0.88, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 374, + 384, + 388 + ], + "score": 1.0, + "content": ", in which case the top layers of", + "type": "text" + }, + { + "bbox": [ + 385, + 378, + 408, + 387 + ], + "score": 0.9, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "form a separable critic.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "Nevertheless, this boundary is crucial for the interplay between MI estimation and the interpretation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 396, + 229, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 229, + 410 + ], + "score": 1.0, + "content": "of the learned representations.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 108, + 425, + 411, + 437 + ], + "lines": [ + { + "bbox": [ + 104, + 424, + 413, + 439 + ], + "spans": [ + { + "bbox": [ + 104, + 424, + 413, + 439 + ], + "score": 1.0, + "content": "3 BIASES IN APPROXIMATE INFORMATION MAXIMIZATION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "It is folklore knowledge that maximizing MI does not necessarily lead to useful representations.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "Already Linsker (1988) talks in his seminal work about constraints, while a manifestation of the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "problem in clustering approaches using MI criteria has been brought up by Bridle et al. (1992) and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "subsequently addressed using regularization by Krause et al. (2010). To what can we then attribute", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 493, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 506, + 505 + ], + "score": 1.0, + "content": "the recent success of methods building on the principles of MI maximization? We will argue that their", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "connection to the InfoMax principle might be very loose. Namely, we will show that they behave", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "counter-intuitively if one equates them with MI maximization, and that the performance of these", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "methods depends strongly on the bias that is encoded not only in the encoders, but also on the actual", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 536, + 220, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 220, + 549 + ], + "score": 1.0, + "content": "form of the used estimators.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 114, + 555, + 506, + 692 + ], + "lines": [ + { + "bbox": [ + 114, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 114, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "1. We first consider encoders which are bijective by design. Even though the true MI is maximized", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 125, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 125, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "for any choice of model parameters, the representation quality (measured by downstream linear", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 126, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 126, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "classification accuracy) improves during training. Furthermore, there exist invertible encoders", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 125, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 125, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "for which the representation quality is worse than using raw pixels, despite also maximizing MI.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 114, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 114, + 602, + 124, + 611 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 125, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "We next consider encoders that can model both invertible and non-invertible functions. When", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 126, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 126, + 612, + 390, + 623 + ], + "score": 1.0, + "content": "the encoder can be non-invertible, but is initialized to be invertible,", + "type": "text" + }, + { + "bbox": [ + 390, + 612, + 409, + 623 + ], + "score": 0.92, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "still biases the encoders", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 126, + 622, + 304, + 635 + ], + "spans": [ + { + "bbox": [ + 126, + 622, + 304, + 635 + ], + "score": 1.0, + "content": "to be very ill-conditioned and hard to invert.", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 112, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 112, + 634, + 143, + 648 + ], + "score": 1.0, + "content": "3. For", + "type": "text" + }, + { + "bbox": [ + 144, + 635, + 162, + 646 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 634, + 181, + 648 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 182, + 635, + 200, + 646 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 634, + 506, + 648 + ], + "score": 1.0, + "content": ", higher-capacity critics admit tighter bounds on MI. We demonstrate that", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 125, + 645, + 507, + 660 + ], + "spans": [ + { + "bbox": [ + 125, + 645, + 507, + 660 + ], + "score": 1.0, + "content": "simple critics yielding loose bounds can lead to better representations than high-capacity critics.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 113, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 113, + 658, + 505, + 671 + ], + "score": 1.0, + "content": "4. Finally, we optimize the estimators to the same MI lower-bound value with different encoder", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 126, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 126, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "architectures and show that the representation quality can be impacted more by the choice of the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 126, + 681, + 254, + 693 + ], + "spans": [ + { + "bbox": [ + 126, + 681, + 254, + 693 + ], + "score": 1.0, + "content": "architecture, than the estimator.", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 42.5 + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "As a consequence, we argue that the success of these methods and the way they are instantiated in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "practice is only loosely connected to MI. Then, in Section 4 we provide an alternative explanation for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 719, + 493, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 493, + 734 + ], + "score": 1.0, + "content": "the success of recent methods through a connection to classic triplet losses from metric learning.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 117 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 95 + ], + "score": 1.0, + "content": "2019). Intuitively, these bounds are based on the following idea: If a classifier can accurately", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 317, + 106 + ], + "score": 1.0, + "content": "distinguish between samples drawn from the joint", + "type": "text" + }, + { + "bbox": [ + 317, + 93, + 347, + 106 + ], + "score": 0.92, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 93, + 506, + 106 + ], + "score": 1.0, + "content": "and those drawn from the product of", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 313, + 117 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 148, + 116 + ], + "score": 1.0, + "content": "marginals", + "type": "text" + }, + { + "bbox": [ + 148, + 104, + 185, + 117 + ], + "score": 0.93, + "content": "p ( x ) p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 105, + 209, + 116 + ], + "score": 1.0, + "content": ", then", + "type": "text" + }, + { + "bbox": [ + 209, + 105, + 219, + 114 + ], + "score": 0.84, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 105, + 237, + 116 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 237, + 105, + 247, + 114 + ], + "score": 0.76, + "content": "Y", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 105, + 313, + 116 + ], + "score": 1.0, + "content": "have a high MI.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 81, + 506, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 121, + 504, + 144 + ], + "lines": [ + { + "bbox": [ + 105, + 119, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 119, + 506, + 135 + ], + "score": 1.0, + "content": "We will focus on two such estimators, which are most commonly used in the representation learning", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 452, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 452, + 144 + ], + "score": 1.0, + "content": "literature. The first of them, termed InfoNCE (van den Oord et al., 2018), is defined as", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 119, + 506, + 144 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 147, + 430, + 182 + ], + "lines": [ + { + "bbox": [ + 180, + 147, + 430, + 182 + ], + "spans": [ + { + "bbox": [ + 180, + 147, + 430, + 182 + ], + "score": 0.94, + "content": "I ( X ; Y ) \\geq \\mathbb { E } \\left[ \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) } } \\right] \\triangleq I _ { \\tt N C E } ( X ; Y ) ,", + "type": "interline_equation", + "image_path": "243e78ea4391040da2daea342569b96683de91841d9344da31ddf2c2eff5e20a.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 180, + 147, + 430, + 158.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 180, + 158.66666666666666, + 430, + 170.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 180, + 170.33333333333331, + 430, + 181.99999999999997 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 186, + 505, + 254 + ], + "lines": [ + { + "bbox": [ + 104, + 185, + 507, + 202 + ], + "spans": [ + { + "bbox": [ + 104, + 185, + 234, + 202 + ], + "score": 1.0, + "content": "where the expectation is over", + "type": "text" + }, + { + "bbox": [ + 234, + 187, + 245, + 197 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 185, + 337, + 202 + ], + "score": 1.0, + "content": "independent samples", + "type": "text" + }, + { + "bbox": [ + 337, + 186, + 390, + 199 + ], + "score": 0.93, + "content": "\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 185, + 507, + 202 + ], + "score": 1.0, + "content": "from the joint distribution", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 197, + 506, + 212 + ], + "spans": [ + { + "bbox": [ + 107, + 198, + 136, + 210 + ], + "score": 0.91, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 197, + 506, + 212 + ], + "score": 1.0, + "content": "(Poole et al., 2019). In practice we estimate (3) using Monte Carlo estimation by averaging", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 363, + 221 + ], + "score": 1.0, + "content": "over multiple batches of samples. Intuitively, the critic function", + "type": "text" + }, + { + "bbox": [ + 363, + 210, + 370, + 221 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 209, + 466, + 221 + ], + "score": 1.0, + "content": "tries to predict for each", + "type": "text" + }, + { + "bbox": [ + 467, + 210, + 477, + 220 + ], + "score": 0.82, + "content": "x _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "which", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 132, + 234 + ], + "score": 1.0, + "content": "of the", + "type": "text" + }, + { + "bbox": [ + 132, + 221, + 142, + 230 + ], + "score": 0.82, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 219, + 177, + 234 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 178, + 221, + 220, + 232 + ], + "score": 0.9, + "content": "y _ { 1 } , \\ldots , y _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 219, + 506, + 234 + ], + "score": 1.0, + "content": "it was jointly drawn with, by assigning high values to the jointly drawn", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 506, + 243 + ], + "score": 1.0, + "content": "pair, and low values to all other pairs. The second estimator is based on the variational form of the KL", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "divergence due to Nguyen, Wainwright, and Jordan (NWJ) (Nguyen et al., 2010) and takes the form", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5, + "bbox_fs": [ + 104, + 185, + 507, + 254 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 165, + 257, + 445, + 273 + ], + "lines": [ + { + "bbox": [ + 165, + 257, + 445, + 273 + ], + "spans": [ + { + "bbox": [ + 165, + 257, + 445, + 273 + ], + "score": 0.91, + "content": "I ( X ; Y ) \\geq \\mathbb { E } _ { p ( x , y ) } [ f ( x , y ) ] - e ^ { - 1 } \\mathbb { E } _ { p ( x ) } [ \\mathbb { E } _ { p ( y ) } e ^ { f ( x , y ) } ] \\triangleq I _ { \\mathbb { N } \\mathbb { N } } ( X ; Y ) .", + "type": "interline_equation", + "image_path": "6b3d9b83561e96c5b3c906e840480bf764ed97f34d3354022f149b3be25fbdd1.jpg" + } + ] + } + ], + "index": 14, + "virtual_lines": [ + { + "bbox": [ + 165, + 257, + 445, + 273 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 276, + 505, + 408 + ], + "lines": [ + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 505, + 288 + ], + "score": 1.0, + "content": "For detailed derivations we refer the reader to (Ruderman et al., 2012; Poole et al., 2019). Note", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 287, + 506, + 300 + ], + "spans": [ + { + "bbox": [ + 105, + 287, + 253, + 300 + ], + "score": 1.0, + "content": "that these bounds hold for any critic", + "type": "text" + }, + { + "bbox": [ + 253, + 288, + 261, + 299 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 287, + 506, + 300 + ], + "score": 1.0, + "content": "and when used in (2) one in practice jointly maximizes over", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 107, + 298, + 506, + 312 + ], + "spans": [ + { + "bbox": [ + 107, + 300, + 131, + 310 + ], + "score": 0.86, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 298, + 149, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 149, + 299, + 156, + 310 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 298, + 384, + 312 + ], + "score": 1.0, + "content": ". Furthermore, it can be shown that (3) is maximized by", + "type": "text" + }, + { + "bbox": [ + 384, + 298, + 473, + 311 + ], + "score": 0.92, + "content": "f ^ { * } ( x , \\bar { y } ) = \\log p ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 298, + 506, + 312 + ], + "score": 1.0, + "content": "and (4)", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 309, + 506, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 309, + 120, + 322 + ], + "score": 1.0, + "content": "by", + "type": "text" + }, + { + "bbox": [ + 120, + 310, + 227, + 321 + ], + "score": 0.87, + "content": "f ^ { * } ( x , y ) \\mathop { = } 1 + \\log p ( y | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 309, + 402, + 322 + ], + "score": 1.0, + "content": "(Poole et al., 2019). Common choices for", + "type": "text" + }, + { + "bbox": [ + 403, + 311, + 410, + 321 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 410, + 309, + 506, + 322 + ], + "score": 1.0, + "content": "include bilinear critics", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 320, + 506, + 333 + ], + "spans": [ + { + "bbox": [ + 107, + 321, + 182, + 332 + ], + "score": 0.86, + "content": "f ( x , y ) = x ^ { \\top } W y", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 320, + 506, + 333 + ], + "score": 1.0, + "content": "(van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019), separable", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 330, + 505, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 133, + 345 + ], + "score": 1.0, + "content": "critics", + "type": "text" + }, + { + "bbox": [ + 134, + 331, + 231, + 344 + ], + "score": 0.92, + "content": "f ( x , y ) = \\phi _ { 1 } ( x ) ^ { \\top } \\phi _ { 2 } ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 330, + 427, + 345 + ], + "score": 1.0, + "content": "(Bachman et al., 2019), and concatenated critics", + "type": "text" + }, + { + "bbox": [ + 427, + 331, + 505, + 344 + ], + "score": 0.92, + "content": "f ( x , y ) = \\phi ( [ x , y ] )", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 342, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 213, + 356 + ], + "score": 1.0, + "content": "(Hjelm et al., 2019) (here", + "type": "text" + }, + { + "bbox": [ + 213, + 343, + 249, + 354 + ], + "score": 0.92, + "content": "\\phi , \\phi _ { 1 } , \\phi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 342, + 506, + 356 + ], + "score": 1.0, + "content": "are typically shallow multi-layer perceptrons (MLPs)). When", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 353, + 506, + 366 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 449, + 366 + ], + "score": 1.0, + "content": "applying these estimators to solve (2), the line between the critic and the encoders", + "type": "text" + }, + { + "bbox": [ + 450, + 355, + 474, + 365 + ], + "score": 0.89, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 353, + 506, + 366 + ], + "score": 1.0, + "content": "can be", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 360, + 376 + ], + "score": 1.0, + "content": "blurry. For example, one can train with an inner product critic", + "type": "text" + }, + { + "bbox": [ + 360, + 364, + 421, + 376 + ], + "score": 0.93, + "content": "f ( x , y ) = x ^ { \\top } y", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 364, + 505, + 376 + ], + "score": 1.0, + "content": ", but extract features", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 374, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 104, + 374, + 228, + 388 + ], + "score": 1.0, + "content": "from an intermediate layer of", + "type": "text" + }, + { + "bbox": [ + 229, + 376, + 253, + 387 + ], + "score": 0.88, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 374, + 384, + 388 + ], + "score": 1.0, + "content": ", in which case the top layers of", + "type": "text" + }, + { + "bbox": [ + 385, + 378, + 408, + 387 + ], + "score": 0.9, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 374, + 506, + 388 + ], + "score": 1.0, + "content": "form a separable critic.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 386, + 505, + 399 + ], + "score": 1.0, + "content": "Nevertheless, this boundary is crucial for the interplay between MI estimation and the interpretation", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 396, + 229, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 229, + 410 + ], + "score": 1.0, + "content": "of the learned representations.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 20.5, + "bbox_fs": [ + 104, + 276, + 506, + 410 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 425, + 411, + 437 + ], + "lines": [ + { + "bbox": [ + 104, + 424, + 413, + 439 + ], + "spans": [ + { + "bbox": [ + 104, + 424, + 413, + 439 + ], + "score": 1.0, + "content": "3 BIASES IN APPROXIMATE INFORMATION MAXIMIZATION", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 448, + 505, + 548 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "It is folklore knowledge that maximizing MI does not necessarily lead to useful representations.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "Already Linsker (1988) talks in his seminal work about constraints, while a manifestation of the", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 471, + 506, + 484 + ], + "score": 1.0, + "content": "problem in clustering approaches using MI criteria has been brought up by Bridle et al. (1992) and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "spans": [ + { + "bbox": [ + 106, + 482, + 505, + 494 + ], + "score": 1.0, + "content": "subsequently addressed using regularization by Krause et al. (2010). To what can we then attribute", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 493, + 506, + 505 + ], + "spans": [ + { + "bbox": [ + 106, + 493, + 506, + 505 + ], + "score": 1.0, + "content": "the recent success of methods building on the principles of MI maximization? We will argue that their", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 517 + ], + "score": 1.0, + "content": "connection to the InfoMax principle might be very loose. Namely, we will show that they behave", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "counter-intuitively if one equates them with MI maximization, and that the performance of these", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "spans": [ + { + "bbox": [ + 106, + 526, + 505, + 538 + ], + "score": 1.0, + "content": "methods depends strongly on the bias that is encoded not only in the encoders, but also on the actual", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 536, + 220, + 549 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 220, + 549 + ], + "score": 1.0, + "content": "form of the used estimators.", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 448, + 506, + 549 + ] + }, + { + "type": "list", + "bbox": [ + 114, + 555, + 506, + 692 + ], + "lines": [ + { + "bbox": [ + 114, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 114, + 555, + 505, + 567 + ], + "score": 1.0, + "content": "1. We first consider encoders which are bijective by design. Even though the true MI is maximized", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 125, + 565, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 125, + 565, + 505, + 579 + ], + "score": 1.0, + "content": "for any choice of model parameters, the representation quality (measured by downstream linear", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 126, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 126, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "classification accuracy) improves during training. Furthermore, there exist invertible encoders", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 125, + 587, + 506, + 600 + ], + "spans": [ + { + "bbox": [ + 125, + 587, + 506, + 600 + ], + "score": 1.0, + "content": "for which the representation quality is worse than using raw pixels, despite also maximizing MI.", + "type": "text" + } + ], + "index": 40, + "is_list_end_line": true + }, + { + "bbox": [ + 114, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 114, + 602, + 124, + 611 + ], + "score": 1.0, + "content": "2.", + "type": "text" + }, + { + "bbox": [ + 125, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "We next consider encoders that can model both invertible and non-invertible functions. When", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 126, + 612, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 126, + 612, + 390, + 623 + ], + "score": 1.0, + "content": "the encoder can be non-invertible, but is initialized to be invertible,", + "type": "text" + }, + { + "bbox": [ + 390, + 612, + 409, + 623 + ], + "score": 0.92, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 409, + 612, + 505, + 623 + ], + "score": 1.0, + "content": "still biases the encoders", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 126, + 622, + 304, + 635 + ], + "spans": [ + { + "bbox": [ + 126, + 622, + 304, + 635 + ], + "score": 1.0, + "content": "to be very ill-conditioned and hard to invert.", + "type": "text" + } + ], + "index": 43, + "is_list_end_line": true + }, + { + "bbox": [ + 112, + 634, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 112, + 634, + 143, + 648 + ], + "score": 1.0, + "content": "3. For", + "type": "text" + }, + { + "bbox": [ + 144, + 635, + 162, + 646 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 163, + 634, + 181, + 648 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 182, + 635, + 200, + 646 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 634, + 506, + 648 + ], + "score": 1.0, + "content": ", higher-capacity critics admit tighter bounds on MI. We demonstrate that", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 125, + 645, + 507, + 660 + ], + "spans": [ + { + "bbox": [ + 125, + 645, + 507, + 660 + ], + "score": 1.0, + "content": "simple critics yielding loose bounds can lead to better representations than high-capacity critics.", + "type": "text" + } + ], + "index": 45, + "is_list_end_line": true + }, + { + "bbox": [ + 113, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 113, + 658, + 505, + 671 + ], + "score": 1.0, + "content": "4. Finally, we optimize the estimators to the same MI lower-bound value with different encoder", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 126, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 126, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "architectures and show that the representation quality can be impacted more by the choice of the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 126, + 681, + 254, + 693 + ], + "spans": [ + { + "bbox": [ + 126, + 681, + 254, + 693 + ], + "score": 1.0, + "content": "architecture, than the estimator.", + "type": "text" + } + ], + "index": 48, + "is_list_end_line": true + } + ], + "index": 42.5, + "bbox_fs": [ + 112, + 555, + 507, + 693 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 699, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "As a consequence, we argue that the success of these methods and the way they are instantiated in", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "practice is only loosely connected to MI. Then, in Section 4 we provide an alternative explanation for", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 719, + 493, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 719, + 493, + 734 + ], + "score": 1.0, + "content": "the success of recent methods through a connection to classic triplet losses from metric learning.", + "type": "text" + } + ], + "index": 51 + } + ], + "index": 50, + "bbox_fs": [ + 106, + 699, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 77, + 500, + 184 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 77, + 500, + 184 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 77, + 500, + 184 + ], + "spans": [ + { + "bbox": [ + 111, + 77, + 500, + 184 + ], + "score": 0.968, + "type": "image", + "image_path": "a84c5f10c5703c8bb062be5139fa94f689d1924b0e49aabbcdefbabec169b769.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 77, + 500, + 112.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 112.66666666666666, + 500, + 148.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 148.33333333333331, + 500, + 183.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 195, + 505, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 219, + 208 + ], + "score": 1.0, + "content": "Figure 1: (a, b) Maximizing", + "type": "text" + }, + { + "bbox": [ + 220, + 195, + 239, + 206 + ], + "score": 0.9, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 194, + 505, + 208 + ], + "score": 1.0, + "content": "over a family of invertible models. We can see that during training", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 392, + 219 + ], + "score": 1.0, + "content": "the downstream classification performance improves (and the testing", + "type": "text" + }, + { + "bbox": [ + 392, + 206, + 411, + 217 + ], + "score": 0.91, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 205, + 505, + 219 + ], + "score": 1.0, + "content": "value increases), even", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 229 + ], + "score": 1.0, + "content": "though the true MI remains constant throughout. (c) Downstream classification accuracy of a different", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 228, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 505, + 240 + ], + "score": 1.0, + "content": "invertible encoder (with the same architecture) trained to have poor performance. This demonstrates", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 238, + 479, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 479, + 252 + ], + "score": 1.0, + "content": "the existence of encoders that provably maximize MI yet have bad downstream performance.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 280 + ], + "score": 1.0, + "content": "Setup Our goal is to provide a minimal set of easily reproducible empirical experiments to under-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "stand the role of MI estimators, critic and encoder architectures when learning representations via the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "objective (2). To this end, we consider a simple setup of learning a representation of the top half of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "MNIST handwritten digit images (we present results for the experiments from Sections 3.2 and 3.3", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "on CIFAR10 in Appendix G; the conclusions are analogous). This setup has been used in the context", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "of deep canonical correlation analysis (Andrew et al., 2013), where the target is to maximize the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "correlation between the representations. Following the widely adopted downstream linear evaluation", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 104, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "protocol (Kolesnikov et al., 2019; van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019;", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "Hjelm et al., 2019; Bachman et al., 2019), we train a linear classifier4 for digit classification on the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "learned representation using all available training labels (other evaluation protocols are discussed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 376, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 505, + 391 + ], + "score": 1.0, + "content": "in Section 5). To learn the representation we instantiate (2) and split each input MNIST image", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 388, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 107, + 389, + 159, + 402 + ], + "score": 0.93, + "content": "x \\in [ 0 , 1 ] ^ { 7 8 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 388, + 327, + 405 + ], + "score": 1.0, + "content": "into two parts, the top part of the image", + "type": "text" + }, + { + "bbox": [ + 327, + 389, + 389, + 402 + ], + "score": 0.93, + "content": "x _ { \\mathrm { t o p } } \\in [ 0 , 1 ] ^ { 3 9 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 388, + 462, + 405 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 463, + 389, + 483, + 400 + ], + "score": 0.9, + "content": "X ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 388, + 506, + 405 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 399, + 504, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 176, + 417 + ], + "score": 1.0, + "content": "the bottom part,", + "type": "text" + }, + { + "bbox": [ + 176, + 402, + 250, + 414 + ], + "score": 0.92, + "content": "x _ { \\mathrm { b o t t o m } } \\in [ 0 , 1 ] ^ { 3 9 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 399, + 327, + 417 + ], + "score": 1.0, + "content": ", corresponding to", + "type": "text" + }, + { + "bbox": [ + 328, + 402, + 348, + 413 + ], + "score": 0.9, + "content": "X ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 399, + 448, + 417 + ], + "score": 1.0, + "content": ", respectively. We train", + "type": "text" + }, + { + "bbox": [ + 448, + 404, + 474, + 415 + ], + "score": 0.31, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 399, + 496, + 417 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 496, + 403, + 504, + 414 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 353, + 427 + ], + "score": 1.0, + "content": "using the Adam optimizer (Kingma and Ba, 2015), and use", + "type": "text" + }, + { + "bbox": [ + 353, + 414, + 386, + 426 + ], + "score": 0.92, + "content": "g _ { 1 } \\big ( x _ { \\mathrm { t o p } } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 412, + 506, + 427 + ], + "score": 1.0, + "content": "as the representation for the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 369, + 439 + ], + "score": 1.0, + "content": "linear evaluation. Unless stated otherwise, we use a bilinear critic", + "type": "text" + }, + { + "bbox": [ + 370, + 425, + 442, + 438 + ], + "score": 0.93, + "content": "f ( x , y ) = x ^ { \\top } W y", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "(we investigate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 475, + 450 + ], + "score": 1.0, + "content": "its effect in a separate ablation study), set the batch size to 128 and the learning rate to", + "type": "text" + }, + { + "bbox": [ + 475, + 437, + 497, + 447 + ], + "score": 0.87, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 435, + 506, + 450 + ], + "score": 1.0, + "content": ". 5", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 159, + 460 + ], + "score": 1.0, + "content": "Throughout,", + "type": "text" + }, + { + "bbox": [ + 160, + 448, + 178, + 459 + ], + "score": 0.9, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "values and downstream classification accuracies are averaged over 20 runs and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "reported on the testing set (we did not observe large gaps between the training and testing values of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 470, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 126, + 481 + ], + "score": 0.87, + "content": "I _ { \\mathrm { E S T } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 470, + 418, + 482 + ], + "score": 1.0, + "content": "). As a common baseline, we rely on a linear classifier in pixel space on", + "type": "text" + }, + { + "bbox": [ + 419, + 471, + 435, + 482 + ], + "score": 0.87, + "content": "x _ { \\mathrm { t o p } }", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 470, + 506, + 482 + ], + "score": 1.0, + "content": ", which obtains a", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 481, + 505, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 481, + 212, + 493 + ], + "score": 1.0, + "content": "testing accuracy of about", + "type": "text" + }, + { + "bbox": [ + 212, + 481, + 231, + 492 + ], + "score": 0.88, + "content": "8 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 481, + 505, + 493 + ], + "score": 1.0, + "content": ". For comparison, a simple MLP or ConvNet architecture achieves", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 491, + 266, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 131, + 504 + ], + "score": 1.0, + "content": "about", + "type": "text" + }, + { + "bbox": [ + 131, + 492, + 151, + 502 + ], + "score": 0.87, + "content": "5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 491, + 266, + 504 + ], + "score": 1.0, + "content": "(see Section 3.3 for details).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 523, + 407, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 408, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 408, + 534 + ], + "score": 1.0, + "content": "3.1 LARGE MI IS NOT PREDICTIVE OF DOWNSTREAM PERFORMANCE", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 506, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 283, + 558 + ], + "score": 1.0, + "content": "We start by investigating the behavior of", + "type": "text" + }, + { + "bbox": [ + 284, + 545, + 302, + 557 + ], + "score": 0.92, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 544, + 323, + 558 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 546, + 343, + 557 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N W } } { \\boldsymbol { \\mathbf { \\mathit { J } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 544, + 371, + 558 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 371, + 547, + 382, + 557 + ], + "score": 0.84, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 544, + 403, + 558 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 403, + 547, + 414, + 557 + ], + "score": 0.85, + "content": "g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "are parameterized to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "be always invertible. Hence, for any choice of the encoder parameters, the MI is constant, i.e.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 565, + 507, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 268, + 580 + ], + "score": 0.93, + "content": "I ( g _ { 1 } ( X ^ { \\acute { ( 1 ) } } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) ) = I ( X ^ { ( 1 ) } ; X ^ { ( 2 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 565, + 296, + 582 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 296, + 569, + 320, + 580 + ], + "score": 0.89, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 565, + 507, + 582 + ], + "score": 1.0, + "content": ". 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We experimented", + "type": "text" + } + ] + }, + { + "bbox": [ + 106, + 691, + 444, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 444, + 702 + ], + "score": 1.0, + "content": "with batch sizes up to 512 and obtained consistent results aligned with the stated conclusions.", + "type": "text" + } + ] + }, + { + "bbox": [ + 119, + 699, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 119, + 700, + 132, + 711 + ], + "score": 0.61, + "content": "^ 6 \\mathrm { I n }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 699, + 415, + 713 + ], + "score": 1.0, + "content": "the context of continuous distributions and invertible representation functions", + "type": "text" + }, + { + "bbox": [ + 415, + 703, + 421, + 712 + ], + "score": 0.81, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 699, + 505, + 713 + ], + "score": 1.0, + "content": "the InfoMax objective", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 711, + 506, + 723 + ], + "score": 1.0, + "content": "might be infinite. Bell and Sejnowski (1995) suggest to instead maximize the entropy of the representation. In", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 505, + 732 + ], + "score": 1.0, + "content": "our case the MI between the two views is finite as the two halves are not deterministic functions of each another.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 111, + 77, + 500, + 184 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 111, + 77, + 500, + 184 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 111, + 77, + 500, + 184 + ], + "spans": [ + { + "bbox": [ + 111, + 77, + 500, + 184 + ], + "score": 0.968, + "type": "image", + "image_path": "a84c5f10c5703c8bb062be5139fa94f689d1924b0e49aabbcdefbabec169b769.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 111, + 77, + 500, + 112.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 111, + 112.66666666666666, + 500, + 148.33333333333331 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 111, + 148.33333333333331, + 500, + 183.99999999999997 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 195, + 505, + 250 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 219, + 208 + ], + "score": 1.0, + "content": "Figure 1: (a, b) Maximizing", + "type": "text" + }, + { + "bbox": [ + 220, + 195, + 239, + 206 + ], + "score": 0.9, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 194, + 505, + 208 + ], + "score": 1.0, + "content": "over a family of invertible models. We can see that during training", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 205, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 392, + 219 + ], + "score": 1.0, + "content": "the downstream classification performance improves (and the testing", + "type": "text" + }, + { + "bbox": [ + 392, + 206, + 411, + 217 + ], + "score": 0.91, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 411, + 205, + 505, + 219 + ], + "score": 1.0, + "content": "value increases), even", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 216, + 505, + 229 + ], + "score": 1.0, + "content": "though the true MI remains constant throughout. 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This demonstrates", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 238, + 479, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 479, + 252 + ], + "score": 1.0, + "content": "the existence of encoders that provably maximize MI yet have bad downstream performance.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 106, + 268, + 505, + 503 + ], + "lines": [ + { + "bbox": [ + 105, + 268, + 506, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 280 + ], + "score": 1.0, + "content": "Setup Our goal is to provide a minimal set of easily reproducible empirical experiments to under-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "spans": [ + { + "bbox": [ + 105, + 279, + 505, + 291 + ], + "score": 1.0, + "content": "stand the role of MI estimators, critic and encoder architectures when learning representations via the", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 506, + 302 + ], + "score": 1.0, + "content": "objective (2). To this end, we consider a simple setup of learning a representation of the top half of", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 300, + 506, + 313 + ], + "score": 1.0, + "content": "MNIST handwritten digit images (we present results for the experiments from Sections 3.2 and 3.3", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 312, + 506, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 506, + 324 + ], + "score": 1.0, + "content": "on CIFAR10 in Appendix G; the conclusions are analogous). This setup has been used in the context", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 106, + 323, + 505, + 335 + ], + "score": 1.0, + "content": "of deep canonical correlation analysis (Andrew et al., 2013), where the target is to maximize the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "correlation between the representations. Following the widely adopted downstream linear evaluation", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 104, + 344, + 506, + 357 + ], + "spans": [ + { + "bbox": [ + 104, + 344, + 506, + 357 + ], + "score": 1.0, + "content": "protocol (Kolesnikov et al., 2019; van den Oord et al., 2018; Hénaff et al., 2019; Tian et al., 2019;", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 505, + 368 + ], + "score": 1.0, + "content": "Hjelm et al., 2019; Bachman et al., 2019), we train a linear classifier4 for digit classification on the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 366, + 506, + 380 + ], + "score": 1.0, + "content": "learned representation using all available training labels (other evaluation protocols are discussed", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 376, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 104, + 376, + 505, + 391 + ], + "score": 1.0, + "content": "in Section 5). To learn the representation we instantiate (2) and split each input MNIST image", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 388, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 107, + 389, + 159, + 402 + ], + "score": 0.93, + "content": "x \\in [ 0 , 1 ] ^ { 7 8 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 388, + 327, + 405 + ], + "score": 1.0, + "content": "into two parts, the top part of the image", + "type": "text" + }, + { + "bbox": [ + 327, + 389, + 389, + 402 + ], + "score": 0.93, + "content": "x _ { \\mathrm { t o p } } \\in [ 0 , 1 ] ^ { 3 9 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 388, + 462, + 405 + ], + "score": 1.0, + "content": "corresponding to", + "type": "text" + }, + { + "bbox": [ + 463, + 389, + 483, + 400 + ], + "score": 0.9, + "content": "X ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 388, + 506, + 405 + ], + "score": 1.0, + "content": ", and", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 399, + 504, + 417 + ], + "spans": [ + { + "bbox": [ + 104, + 399, + 176, + 417 + ], + "score": 1.0, + "content": "the bottom part,", + "type": "text" + }, + { + "bbox": [ + 176, + 402, + 250, + 414 + ], + "score": 0.92, + "content": "x _ { \\mathrm { b o t t o m } } \\in [ 0 , 1 ] ^ { 3 9 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 399, + 327, + 417 + ], + "score": 1.0, + "content": ", corresponding to", + "type": "text" + }, + { + "bbox": [ + 328, + 402, + 348, + 413 + ], + "score": 0.9, + "content": "X ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 349, + 399, + 448, + 417 + ], + "score": 1.0, + "content": ", respectively. We train", + "type": "text" + }, + { + "bbox": [ + 448, + 404, + 474, + 415 + ], + "score": 0.31, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 399, + 496, + 417 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 496, + 403, + 504, + 414 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 412, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 353, + 427 + ], + "score": 1.0, + "content": "using the Adam optimizer (Kingma and Ba, 2015), and use", + "type": "text" + }, + { + "bbox": [ + 353, + 414, + 386, + 426 + ], + "score": 0.92, + "content": "g _ { 1 } \\big ( x _ { \\mathrm { t o p } } \\big )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 412, + 506, + 427 + ], + "score": 1.0, + "content": "as the representation for the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 369, + 439 + ], + "score": 1.0, + "content": "linear evaluation. Unless stated otherwise, we use a bilinear critic", + "type": "text" + }, + { + "bbox": [ + 370, + 425, + 442, + 438 + ], + "score": 0.93, + "content": "f ( x , y ) = x ^ { \\top } W y", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "(we investigate", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 104, + 435, + 506, + 450 + ], + "spans": [ + { + "bbox": [ + 104, + 435, + 475, + 450 + ], + "score": 1.0, + "content": "its effect in a separate ablation study), set the batch size to 128 and the learning rate to", + "type": "text" + }, + { + "bbox": [ + 475, + 437, + 497, + 447 + ], + "score": 0.87, + "content": "1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 497, + 435, + 506, + 450 + ], + "score": 1.0, + "content": ". 5", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 448, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 159, + 460 + ], + "score": 1.0, + "content": "Throughout,", + "type": "text" + }, + { + "bbox": [ + 160, + 448, + 178, + 459 + ], + "score": 0.9, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 448, + 505, + 460 + ], + "score": 1.0, + "content": "values and downstream classification accuracies are averaged over 20 runs and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 460, + 505, + 471 + ], + "score": 1.0, + "content": "reported on the testing set (we did not observe large gaps between the training and testing values of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 470, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 106, + 470, + 126, + 481 + ], + "score": 0.87, + "content": "I _ { \\mathrm { E S T } } ,", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 470, + 418, + 482 + ], + "score": 1.0, + "content": "). 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For comparison, a simple MLP or ConvNet architecture achieves", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 491, + 266, + 504 + ], + "spans": [ + { + "bbox": [ + 106, + 491, + 131, + 504 + ], + "score": 1.0, + "content": "about", + "type": "text" + }, + { + "bbox": [ + 131, + 492, + 151, + 502 + ], + "score": 0.87, + "content": "5 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 491, + 266, + 504 + ], + "score": 1.0, + "content": "(see Section 3.3 for details).", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 18, + "bbox_fs": [ + 104, + 268, + 506, + 504 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 523, + 407, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 522, + 408, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 522, + 408, + 534 + ], + "score": 1.0, + "content": "3.1 LARGE MI IS NOT PREDICTIVE OF DOWNSTREAM PERFORMANCE", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 506, + 612 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 283, + 558 + ], + "score": 1.0, + "content": "We start by investigating the behavior of", + "type": "text" + }, + { + "bbox": [ + 284, + 545, + 302, + 557 + ], + "score": 0.92, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 544, + 323, + 558 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 324, + 546, + 343, + 557 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N W } } { \\boldsymbol { \\mathbf { \\mathit { J } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 544, + 371, + 558 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 371, + 547, + 382, + 557 + ], + "score": 0.84, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 544, + 403, + 558 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 403, + 547, + 414, + 557 + ], + "score": 0.85, + "content": "g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "are parameterized to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 506, + 569 + ], + "score": 1.0, + "content": "be always invertible. Hence, for any choice of the encoder parameters, the MI is constant, i.e.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 565, + 507, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 567, + 268, + 580 + ], + "score": 0.93, + "content": "I ( g _ { 1 } ( X ^ { \\acute { ( 1 ) } } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) ) = I ( X ^ { ( 1 ) } ; X ^ { ( 2 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 565, + 296, + 582 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 296, + 569, + 320, + 580 + ], + "score": 0.89, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 565, + 507, + 582 + ], + "score": 1.0, + "content": ". 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This confirms our hypothesis that the estimator biases the encoders", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 320, + 411, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 411, + 331 + ], + "score": 1.0, + "content": "towards solutions suitable to solve the downstream linear classification task.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 504, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 504, + 348 + ], + "score": 1.0, + "content": "The previous experiment demonstrated that among many invertible encoders, all of which are globally", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "optimal MI maximizers, some give rise to improved linear classification performance over raw", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 202, + 371 + ], + "score": 1.0, + "content": "pixels, and maximizing", + "type": "text" + }, + { + "bbox": [ + 203, + 359, + 221, + 370 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 358, + 239, + 371 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 359, + 258, + 370 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N W } } { \\boldsymbol { \\mathbf { \\mathit { J } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "yields such encoders. Next we demonstrate that for the same", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "score": 1.0, + "content": "invertible encoder architecture there are model parameters for which linear classification performance", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 380, + 491, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 491, + 393 + ], + "score": 1.0, + "content": "is significantly worse than using raw pixels, despite also being globally optimal MI maximizers.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "Maximized MI and worsened downstream performance The goal is to learn a (bijective) rep-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 409, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 420 + ], + "score": 1.0, + "content": "resentation maximizing MI such that the optimal linear classifier performs poorly; we achieve this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "by jointly training a representation and classifier in an adversarial fashion (a separate classifier is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 431, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 442 + ], + "score": 1.0, + "content": "trained for the evaluation), without using a MI estimator. Intuitively, we will train the encoder to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "make the classification task for the linear layer as hard as possible. The experimental details are", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "presented in Appendix B. Figure 1c shows the result of one such training run, displaying the loss", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "of a separately trained classifier on top of the frozen representation. At the beginning of training", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "the network is initialized to be close to the identity mapping, and as such achieves the baseline", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "classification accuracy corresponding to raw pixels. All points beyond this correspond to invertible", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 496, + 495, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 495, + 509 + ], + "score": 1.0, + "content": "feature maps with worse classification performance, despite still achieving globally maximal MI.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "Alternatively, the following thought experiment would yield the same conclusion: Using a loss-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 293, + 539 + ], + "score": 1.0, + "content": "less compression algorithm (e.g. PNG) for", + "type": "text" + }, + { + "bbox": [ + 293, + 527, + 304, + 537 + ], + "score": 0.85, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 522, + 326, + 539 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 326, + 526, + 337, + 537 + ], + "score": 0.85, + "content": "g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 522, + 397, + 539 + ], + "score": 1.0, + "content": "also satisfies", + "type": "text" + }, + { + "bbox": [ + 398, + 524, + 505, + 537 + ], + "score": 0.89, + "content": "I ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) ) \\ =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 534, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 536, + 164, + 549 + ], + "score": 0.92, + "content": "I ( X ^ { ( 1 ) } ; X ^ { ( 2 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 534, + 466, + 551 + ], + "score": 1.0, + "content": ". Yet, performing linear classification on the raw compressed bit stream", + "type": "text" + }, + { + "bbox": [ + 467, + 537, + 505, + 549 + ], + "score": 0.9, + "content": "g _ { 1 } ( X ^ { ( 1 ) } )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "will likely lead to worse performance than the baseline in pixel space. The information content alone", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 558, + 406, + 573 + ], + "spans": [ + { + "bbox": [ + 104, + 558, + 406, + 573 + ], + "score": 1.0, + "content": "is not sufficient to guarantee a useful geometry in the representation space.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "We next investigate the behavior of the model if we use a network architecture that can model both", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 407, + 599 + ], + "score": 1.0, + "content": "invertible and non-invertible functions. We would like to understand whether", + "type": "text" + }, + { + "bbox": [ + 408, + 587, + 426, + 598 + ], + "score": 0.91, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "prefers the network", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "to remain bijective, thus maximizing the true MI, or to ignore part of the input signal, which can be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 608, + 257, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 257, + 623 + ], + "score": 1.0, + "content": "beneficial for representation learning.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 625, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 505, + 638 + ], + "score": 1.0, + "content": "Bias towards hard-to-invert encoders We use an MLP architecture with 4 hidden layers of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "score": 1.0, + "content": "same dimension as the input, and with a skip connection added to each layer (hence by setting all", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 647, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 506, + 660 + ], + "score": 1.0, + "content": "weights to 0 the network becomes the identity function). As quantifying invertibility is hard, we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "analyze the condition number, i.e. the ratio between the largest and the smallest singular value, of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 172, + 682 + ], + "score": 1.0, + "content": "the Jacobian of", + "type": "text" + }, + { + "bbox": [ + 172, + 671, + 182, + 681 + ], + "score": 0.84, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 669, + 506, + 682 + ], + "score": 1.0, + "content": ": By the implicit function theorem, the function is invertible if the Jacobian is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 679, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 506, + 694 + ], + "score": 1.0, + "content": "non-singular.7 However, the data itself might lie on a low-dimensional manifold, so that having", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "score": 1.0, + "content": "a singular Jacobian is not necessarily indicative of losing invertibility on the support of the data", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 711, + 504, + 731 + ], + "lines": [ + { + "bbox": [ + 118, + 710, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 118, + 710, + 159, + 723 + ], + "score": 1.0, + "content": "7Formally,", + "type": "text" + }, + { + "bbox": [ + 159, + 713, + 169, + 722 + ], + "score": 0.79, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 710, + 506, + 723 + ], + "score": 1.0, + "content": "is invertible as long as the condition number of the Jacobian is finite. 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Despite the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 285, + 302, + 299 + ], + "score": 1.0, + "content": "fact that MI is maximized by any instantiation of", + "type": "text" + }, + { + "bbox": [ + 303, + 288, + 313, + 298 + ], + "score": 0.86, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 285, + 331, + 299 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 331, + 288, + 341, + 298 + ], + "score": 0.77, + "content": "g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 285, + 345, + 299 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 345, + 287, + 364, + 298 + ], + "score": 0.86, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "and downstream accuracy increase", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 106, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "during training, implying that the estimators provide gradient feedback leading to a representation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "useful for linear classification. This confirms our hypothesis that the estimator biases the encoders", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 320, + 411, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 320, + 411, + 331 + ], + "score": 1.0, + "content": "towards solutions suitable to solve the downstream linear classification task.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 276, + 505, + 331 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 336, + 505, + 392 + ], + "lines": [ + { + "bbox": [ + 106, + 336, + 504, + 348 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 504, + 348 + ], + "score": 1.0, + "content": "The previous experiment demonstrated that among many invertible encoders, all of which are globally", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 347, + 506, + 361 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 506, + 361 + ], + "score": 1.0, + "content": "optimal MI maximizers, some give rise to improved linear classification performance over raw", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 202, + 371 + ], + "score": 1.0, + "content": "pixels, and maximizing", + "type": "text" + }, + { + "bbox": [ + 203, + 359, + 221, + 370 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 358, + 239, + 371 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 239, + 359, + 258, + 370 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N W } } { \\boldsymbol { \\mathbf { \\mathit { J } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "yields such encoders. Next we demonstrate that for the same", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 505, + 383 + ], + "score": 1.0, + "content": "invertible encoder architecture there are model parameters for which linear classification performance", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 380, + 491, + 393 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 491, + 393 + ], + "score": 1.0, + "content": "is significantly worse than using raw pixels, despite also being globally optimal MI maximizers.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 336, + 506, + 393 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 397, + 505, + 507 + ], + "lines": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 410 + ], + "score": 1.0, + "content": "Maximized MI and worsened downstream performance The goal is to learn a (bijective) rep-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 409, + 505, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 409, + 505, + 420 + ], + "score": 1.0, + "content": "resentation maximizing MI such that the optimal linear classifier performs poorly; we achieve this", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 106, + 419, + 506, + 432 + ], + "score": 1.0, + "content": "by jointly training a representation and classifier in an adversarial fashion (a separate classifier is", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 431, + 505, + 442 + ], + "spans": [ + { + "bbox": [ + 106, + 431, + 505, + 442 + ], + "score": 1.0, + "content": "trained for the evaluation), without using a MI estimator. Intuitively, we will train the encoder to", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 505, + 453 + ], + "score": 1.0, + "content": "make the classification task for the linear layer as hard as possible. The experimental details are", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 505, + 465 + ], + "score": 1.0, + "content": "presented in Appendix B. Figure 1c shows the result of one such training run, displaying the loss", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 462, + 506, + 477 + ], + "spans": [ + { + "bbox": [ + 104, + 462, + 506, + 477 + ], + "score": 1.0, + "content": "of a separately trained classifier on top of the frozen representation. At the beginning of training", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 487 + ], + "score": 1.0, + "content": "the network is initialized to be close to the identity mapping, and as such achieves the baseline", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 506, + 497 + ], + "score": 1.0, + "content": "classification accuracy corresponding to raw pixels. All points beyond this correspond to invertible", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 496, + 495, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 495, + 509 + ], + "score": 1.0, + "content": "feature maps with worse classification performance, despite still achieving globally maximal MI.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 396, + 506, + 509 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 513, + 505, + 570 + ], + "lines": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 512, + 506, + 525 + ], + "score": 1.0, + "content": "Alternatively, the following thought experiment would yield the same conclusion: Using a loss-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 522, + 505, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 293, + 539 + ], + "score": 1.0, + "content": "less compression algorithm (e.g. PNG) for", + "type": "text" + }, + { + "bbox": [ + 293, + 527, + 304, + 537 + ], + "score": 0.85, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 522, + 326, + 539 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 326, + 526, + 337, + 537 + ], + "score": 0.85, + "content": "g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 522, + 397, + 539 + ], + "score": 1.0, + "content": "also satisfies", + "type": "text" + }, + { + "bbox": [ + 398, + 524, + 505, + 537 + ], + "score": 0.89, + "content": "I ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) ) \\ =", + "type": "inline_equation" + } + ], + "index": 31 + }, + { + "bbox": [ + 107, + 534, + 505, + 551 + ], + "spans": [ + { + "bbox": [ + 107, + 536, + 164, + 549 + ], + "score": 0.92, + "content": "I ( X ^ { ( 1 ) } ; X ^ { ( 2 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 534, + 466, + 551 + ], + "score": 1.0, + "content": ". Yet, performing linear classification on the raw compressed bit stream", + "type": "text" + }, + { + "bbox": [ + 467, + 537, + 505, + 549 + ], + "score": 0.9, + "content": "g _ { 1 } ( X ^ { ( 1 ) } )", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 506, + 561 + ], + "score": 1.0, + "content": "will likely lead to worse performance than the baseline in pixel space. The information content alone", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 558, + 406, + 573 + ], + "spans": [ + { + "bbox": [ + 104, + 558, + 406, + 573 + ], + "score": 1.0, + "content": "is not sufficient to guarantee a useful geometry in the representation space.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 512, + 506, + 573 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 576, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 106, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "We next investigate the behavior of the model if we use a network architecture that can model both", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 586, + 505, + 599 + ], + "spans": [ + { + "bbox": [ + 105, + 586, + 407, + 599 + ], + "score": 1.0, + "content": "invertible and non-invertible functions. We would like to understand whether", + "type": "text" + }, + { + "bbox": [ + 408, + 587, + 426, + 598 + ], + "score": 0.91, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 427, + 586, + 505, + 599 + ], + "score": 1.0, + "content": "prefers the network", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 505, + 611 + ], + "score": 1.0, + "content": "to remain bijective, thus maximizing the true MI, or to ignore part of the input signal, which can be", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 608, + 257, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 608, + 257, + 623 + ], + "score": 1.0, + "content": "beneficial for representation learning.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36.5, + "bbox_fs": [ + 105, + 576, + 505, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 626, + 505, + 703 + ], + "lines": [ + { + "bbox": [ + 106, + 625, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 625, + 505, + 638 + ], + "score": 1.0, + "content": "Bias towards hard-to-invert encoders We use an MLP architecture with 4 hidden layers of the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 505, + 650 + ], + "score": 1.0, + "content": "same dimension as the input, and with a skip connection added to each layer (hence by setting all", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 647, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 647, + 506, + 660 + ], + "score": 1.0, + "content": "weights to 0 the network becomes the identity function). As quantifying invertibility is hard, we", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 671 + ], + "score": 1.0, + "content": "analyze the condition number, i.e. the ratio between the largest and the smallest singular value, of", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 669, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 172, + 682 + ], + "score": 1.0, + "content": "the Jacobian of", + "type": "text" + }, + { + "bbox": [ + 172, + 671, + 182, + 681 + ], + "score": 0.84, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 182, + 669, + 506, + 682 + ], + "score": 1.0, + "content": ": By the implicit function theorem, the function is invertible if the Jacobian is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 679, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 679, + 506, + 694 + ], + "score": 1.0, + "content": "non-singular.7 However, the data itself might lie on a low-dimensional manifold, so that having", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "spans": [ + { + "bbox": [ + 105, + 692, + 505, + 704 + ], + "score": 1.0, + "content": "a singular Jacobian is not necessarily indicative of losing invertibility on the support of the data", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 218, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 232 + ], + "score": 1.0, + "content": "distribution. To ensure the support of the data distribution covers the complete input space, we corrupt", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 228, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 107, + 230, + 127, + 241 + ], + "score": 0.89, + "content": "X ^ { ( 1 ) }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 128, + 228, + 145, + 245 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 146, + 230, + 166, + 241 + ], + "score": 0.91, + "content": "X ^ { ( 2 ) }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 166, + 228, + 506, + 245 + ], + "score": 1.0, + "content": "in a coupled way by adding to each the same 392-dimensional random vector, whose", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 241, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 279, + 256 + ], + "score": 1.0, + "content": "coordinates are sampled (independently of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 280, + 242, + 327, + 254 + ], + "score": 0.9, + "content": "X ^ { ( 1 ) } , X ^ { ( 2 ) } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 328, + 241, + 506, + 256 + ], + "score": 1.0, + "content": "from a normal with standard deviation 0.05", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 253, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 466, + 268 + ], + "score": 1.0, + "content": "(the standard deviation of the pixels themselves is 0.3). Hence, non-invertible encoders", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 466, + 255, + 491, + 266 + ], + "score": 0.89, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 491, + 253, + 506, + 268 + ], + "score": 1.0, + "content": "do", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 263, + 507, + 280 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 163, + 280 + ], + "score": 1.0, + "content": "not maximize", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 164, + 265, + 254, + 278 + ], + "score": 0.91, + "content": "I ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 255, + 263, + 507, + 280 + ], + "score": 1.0, + "content": ". 8 As a reference point, the linear classification accuracy from", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 277, + 307, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 192, + 290 + ], + "score": 1.0, + "content": "pixels drops to about", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 192, + 278, + 212, + 287 + ], + "score": 0.84, + "content": "84 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 212, + 277, + 307, + 290 + ], + "score": 1.0, + "content": "due to the added noise.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 625, + 506, + 704 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 115, + 77, + 496, + 168 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 115, + 77, + 496, + 168 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 115, + 77, + 496, + 168 + ], + "spans": [ + { + "bbox": [ + 115, + 77, + 496, + 168 + ], + "score": 0.965, + "type": "image", + "image_path": "8aa32e9c03a77e8657ee9a3fc96b1e03d12ad15997f3d7f40349ab8ce7ec0d00.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 115, + 77, + 496, + 107.33333333333333 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 115, + 107.33333333333333, + 496, + 137.66666666666666 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 115, + 137.66666666666666, + 496, + 168.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 107, + 175, + 505, + 209 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 175, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 280, + 189 + ], + "score": 1.0, + "content": "Figure 3: Downstream testing accuracy for", + "type": "text" + }, + { + "bbox": [ + 281, + 176, + 299, + 186 + ], + "score": 0.94, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 175, + 317, + 189 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 317, + 176, + 336, + 186 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 336, + 175, + 385, + 189 + ], + "score": 1.0, + "content": ", and testing", + "type": "text" + }, + { + "bbox": [ + 386, + 176, + 405, + 187 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 175, + 505, + 189 + ], + "score": 1.0, + "content": "value for MLP encoders", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 131, + 198 + ], + "score": 0.88, + "content": "g _ { 1 } , g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 131, + 186, + 307, + 200 + ], + "score": 1.0, + "content": "and different critic architectures (the testing", + "type": "text" + }, + { + "bbox": [ + 308, + 187, + 326, + 198 + ], + "score": 0.89, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "curve can be found in Appendix F). Bilinear", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 196, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 482, + 212 + ], + "score": 1.0, + "content": "and separable critics lead to higher downstream accuracy than MLP critics, while reaching lower", + "type": "text" + }, + { + "bbox": [ + 483, + 198, + 501, + 209 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 196, + 505, + 212 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 219, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 218, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 105, + 218, + 505, + 232 + ], + "score": 1.0, + "content": "distribution. To ensure the support of the data distribution covers the complete input space, we corrupt", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 107, + 228, + 506, + 245 + ], + "spans": [ + { + "bbox": [ + 107, + 230, + 127, + 241 + ], + "score": 0.89, + "content": "X ^ { ( 1 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 228, + 145, + 245 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 146, + 230, + 166, + 241 + ], + "score": 0.91, + "content": "X ^ { ( 2 ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 228, + 506, + 245 + ], + "score": 1.0, + "content": "in a coupled way by adding to each the same 392-dimensional random vector, whose", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 241, + 506, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 279, + 256 + ], + "score": 1.0, + "content": "coordinates are sampled (independently of", + "type": "text" + }, + { + "bbox": [ + 280, + 242, + 327, + 254 + ], + "score": 0.9, + "content": "X ^ { ( 1 ) } , X ^ { ( 2 ) } )", + "type": "inline_equation" + }, + { + "bbox": [ + 328, + 241, + 506, + 256 + ], + "score": 1.0, + "content": "from a normal with standard deviation 0.05", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 253, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 466, + 268 + ], + "score": 1.0, + "content": "(the standard deviation of the pixels themselves is 0.3). Hence, non-invertible encoders", + "type": "text" + }, + { + "bbox": [ + 466, + 255, + 491, + 266 + ], + "score": 0.89, + "content": "g _ { 1 } , g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 491, + 253, + 506, + 268 + ], + "score": 1.0, + "content": "do", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 263, + 507, + 280 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 163, + 280 + ], + "score": 1.0, + "content": "not maximize", + "type": "text" + }, + { + "bbox": [ + 164, + 265, + 254, + 278 + ], + "score": 0.91, + "content": "I ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 2 } ( X ^ { ( 2 ) } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 263, + 507, + 280 + ], + "score": 1.0, + "content": ". 8 As a reference point, the linear classification accuracy from", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 277, + 307, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 192, + 290 + ], + "score": 1.0, + "content": "pixels drops to about", + "type": "text" + }, + { + "bbox": [ + 192, + 278, + 212, + 287 + ], + "score": 0.84, + "content": "84 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 277, + 307, + 290 + ], + "score": 1.0, + "content": "due to the added noise.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 107, + 294, + 505, + 382 + ], + "lines": [ + { + "bbox": [ + 105, + 293, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 243, + 307 + ], + "score": 1.0, + "content": "In Figure 2 we can see that the", + "type": "text" + }, + { + "bbox": [ + 243, + 294, + 262, + 305 + ], + "score": 0.91, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 293, + 506, + 307 + ], + "score": 1.0, + "content": "value and the downstream accuracy both increase during", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 305, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 305, + 277, + 317 + ], + "score": 1.0, + "content": "training, as before. Moreover, even though", + "type": "text" + }, + { + "bbox": [ + 277, + 307, + 288, + 316 + ], + "score": 0.86, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "is initialized very close to the identity function (which", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "maximizes the true MI), the condition number of its Jacobian evaluated at inputs randomly sampled", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "from the data-distribution steadily deteriorates over time, suggesting that in practice (i.e. numerically)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "score": 1.0, + "content": "inverting the model becomes increasingly hard. It therefore seems that the bounds we consider favor", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 349, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 360 + ], + "score": 1.0, + "content": "hard-to-invert encoders, which heavily attenuate part of the noise (as the support of the noise is the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "entire input space), over well conditioned encoders (such as the identity function at initialization),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 371, + 364, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 364, + 383 + ], + "score": 1.0, + "content": "which preserve the noise and hence the entropy of the data well.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "title", + "bbox": [ + 108, + 399, + 472, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 474, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 474, + 411 + ], + "score": 1.0, + "content": "3.2 HIGHER CAPACITY CRITICS CAN LEAD TO WORSE DOWNSTREAM PERFORMANCE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "In the previous section we have established that MI and downstream performance are only loosely", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "connected. Clearly, maximizing MI is not sufficient to learn good representations and there is a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "non-trivial interplay between the architectures of the encoder, critic, and the underlying estimators.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "In this section, we will focus on how one of these factors, namely the critic architecture, impacts", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 485, + 479 + ], + "score": 1.0, + "content": "the quality of the learned representation. Recall that it determines how the estimators such as", + "type": "text" + }, + { + "bbox": [ + 485, + 465, + 504, + 475 + ], + "score": 0.89, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 124, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 476, + 144, + 487 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N W } } { \\boldsymbol { \\mathbf { \\mathit { I } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 475, + 378, + 488 + ], + "score": 1.0, + "content": "distinguish between samples from the joint distribution", + "type": "text" + }, + { + "bbox": [ + 378, + 475, + 407, + 487 + ], + "score": 0.93, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "and the product of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 149, + 500 + ], + "score": 1.0, + "content": "marginals", + "type": "text" + }, + { + "bbox": [ + 149, + 486, + 187, + 498 + ], + "score": 0.93, + "content": "p ( x ) p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 486, + 505, + 500 + ], + "score": 1.0, + "content": ", and thereby determines the tightness on the lower bound. A higher capacity", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "critic should allow for a tighter lower-bound on MI (Belghazi et al., 2018). Furthermore, in the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 275, + 520 + ], + "score": 1.0, + "content": "context of representation learning where", + "type": "text" + }, + { + "bbox": [ + 275, + 509, + 282, + 520 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "is instantiated as a neural network, the critic provides", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 519, + 417, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 190, + 532 + ], + "score": 1.0, + "content": "gradient feedback to", + "type": "text" + }, + { + "bbox": [ + 190, + 520, + 201, + 531 + ], + "score": 0.84, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 201, + 519, + 218, + 532 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 219, + 520, + 229, + 531 + ], + "score": 0.85, + "content": "g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 230, + 519, + 417, + 532 + ], + "score": 1.0, + "content": "and thereby shapes the learned representation.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 106, + 536, + 505, + 613 + ], + "lines": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 535, + 505, + 549 + ], + "score": 1.0, + "content": "Looser bounds with simpler critics can lead to better representations We compare three critic", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 546, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 547, + 303, + 560 + ], + "score": 1.0, + "content": "architectures, a bilinear critic, a separable critic", + "type": "text" + }, + { + "bbox": [ + 303, + 546, + 433, + 559 + ], + "score": 0.9, + "content": "f ( x , y ) \\stackrel { \\mathrm { ~ } } { = } \\phi _ { 1 } ( x ) ^ { \\top } \\phi _ { 2 } ( y ) ( \\phi _ { 1 } , \\phi _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 547, + 506, + 560 + ], + "score": 1.0, + "content": "are MLPs with a", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "single hidden layer with 100 units and ReLU activations, followed by a linear layer with 100 units;", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 569, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 106, + 569, + 153, + 582 + ], + "score": 1.0, + "content": "comprising", + "type": "text" + }, + { + "bbox": [ + 154, + 569, + 170, + 580 + ], + "score": 0.68, + "content": "4 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 569, + 505, + 582 + ], + "score": 1.0, + "content": "parameters in total) and an MLP critic with a single hidden layer with 200 units and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 579, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 105, + 579, + 320, + 593 + ], + "score": 1.0, + "content": "ReLU activations, applied to the concatenated input", + "type": "text" + }, + { + "bbox": [ + 320, + 580, + 342, + 592 + ], + "score": 0.91, + "content": "[ x , y ]", + "type": "inline_equation" + }, + { + "bbox": [ + 342, + 579, + 505, + 593 + ], + "score": 1.0, + "content": "(40k trainable parameters). Further, we", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 591, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 247, + 604 + ], + "score": 1.0, + "content": "use identical MLP architectures for", + "type": "text" + }, + { + "bbox": [ + 248, + 593, + 258, + 603 + ], + "score": 0.83, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 591, + 275, + 604 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 276, + 592, + 286, + 603 + ], + "score": 0.85, + "content": "g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 591, + 505, + 604 + ], + "score": 1.0, + "content": "with two hidden layers comprising 300 units each, and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 602, + 363, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 363, + 615 + ], + "score": 1.0, + "content": "a third linear layer mapping to a 100-dimensional feature space.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 106, + 619, + 505, + 696 + ], + "lines": [ + { + "bbox": [ + 106, + 619, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 106, + 619, + 375, + 631 + ], + "score": 1.0, + "content": "Figure 3 shows the downstream testing accuracy and the testing", + "type": "text" + }, + { + "bbox": [ + 375, + 619, + 394, + 630 + ], + "score": 0.9, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 394, + 619, + 505, + 631 + ], + "score": 1.0, + "content": "value as a function of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 630, + 505, + 642 + ], + "score": 1.0, + "content": "iteration (see Appendix G for the corresponding results on CIFAR10). It can be seen that for both", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 106, + 641, + 505, + 653 + ], + "score": 1.0, + "content": "lower bounds, representations trained with the MLP critic barely outperform the baseline on pixel", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 651, + 505, + 664 + ], + "score": 1.0, + "content": "space, whereas the same lower bounds with bilinear and separable critics clearly lead to a higher", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 663, + 505, + 675 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 298, + 675 + ], + "score": 1.0, + "content": "accuracy than the baseline. 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Moreover, even though", + "type": "text" + }, + { + "bbox": [ + 277, + 307, + 288, + 316 + ], + "score": 0.86, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 305, + 505, + 317 + ], + "score": 1.0, + "content": "is initialized very close to the identity function (which", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 505, + 328 + ], + "score": 1.0, + "content": "maximizes the true MI), the condition number of its Jacobian evaluated at inputs randomly sampled", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 506, + 339 + ], + "score": 1.0, + "content": "from the data-distribution steadily deteriorates over time, suggesting that in practice (i.e. numerically)", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 106, + 338, + 505, + 349 + ], + "score": 1.0, + "content": "inverting the model becomes increasingly hard. It therefore seems that the bounds we consider favor", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 349, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 505, + 360 + ], + "score": 1.0, + "content": "hard-to-invert encoders, which heavily attenuate part of the noise (as the support of the noise is the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 106, + 360, + 506, + 372 + ], + "score": 1.0, + "content": "entire input space), over well conditioned encoders (such as the identity function at initialization),", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 371, + 364, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 364, + 383 + ], + "score": 1.0, + "content": "which preserve the noise and hence the entropy of the data well.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 293, + 506, + 383 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 399, + 472, + 410 + ], + "lines": [ + { + "bbox": [ + 105, + 399, + 474, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 474, + 411 + ], + "score": 1.0, + "content": "3.2 HIGHER CAPACITY CRITICS CAN LEAD TO WORSE DOWNSTREAM PERFORMANCE", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 106, + 420, + 505, + 531 + ], + "lines": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 420, + 505, + 433 + ], + "score": 1.0, + "content": "In the previous section we have established that MI and downstream performance are only loosely", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 506, + 444 + ], + "score": 1.0, + "content": "connected. Clearly, maximizing MI is not sufficient to learn good representations and there is a", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 106, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "non-trivial interplay between the architectures of the encoder, critic, and the underlying estimators.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 105, + 453, + 505, + 466 + ], + "score": 1.0, + "content": "In this section, we will focus on how one of these factors, namely the critic architecture, impacts", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 462, + 504, + 479 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 485, + 479 + ], + "score": 1.0, + "content": "the quality of the learned representation. Recall that it determines how the estimators such as", + "type": "text" + }, + { + "bbox": [ + 485, + 465, + 504, + 475 + ], + "score": 0.89, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 475, + 505, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 124, + 488 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 476, + 144, + 487 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N W } } { \\boldsymbol { \\mathbf { \\mathit { I } } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 475, + 378, + 488 + ], + "score": 1.0, + "content": "distinguish between samples from the joint distribution", + "type": "text" + }, + { + "bbox": [ + 378, + 475, + 407, + 487 + ], + "score": 0.93, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 475, + 505, + 488 + ], + "score": 1.0, + "content": "and the product of the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 486, + 505, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 486, + 149, + 500 + ], + "score": 1.0, + "content": "marginals", + "type": "text" + }, + { + "bbox": [ + 149, + 486, + 187, + 498 + ], + "score": 0.93, + "content": "p ( x ) p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 486, + 505, + 500 + ], + "score": 1.0, + "content": ", and thereby determines the tightness on the lower bound. A higher capacity", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "critic should allow for a tighter lower-bound on MI (Belghazi et al., 2018). 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If negative samples", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 246, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 210, + 261 + ], + "score": 1.0, + "content": "are not drawn i.i.d., both", + "type": "text" + }, + { + "bbox": [ + 210, + 248, + 229, + 259 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 246, + 247, + 261 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 248, + 248, + 267, + 259 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N } \\bar { W } \\bar { J } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 246, + 506, + 261 + ], + "score": 1.0, + "content": "estimators can be greater than the true MI. Despite being", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 259, + 499, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 286, + 271 + ], + "score": 1.0, + "content": "commonly justified as a lower bound on MI,", + "type": "text" + }, + { + "bbox": [ + 286, + 259, + 304, + 270 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 259, + 499, + 271 + ], + "score": 1.0, + "content": "is often used in the non-i.i.d. setting in practice.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 503, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 126, + 297 + ], + "score": 0.88, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "testing value, the simple bilinear critic leads to a better downstream performance than the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 297, + 280, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 280, + 309 + ], + "score": 1.0, + "content": "higher-capacity separable and MLP critics.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "A related phenomenon was observed in the context of variational autoencoders (VAEs) (Kingma and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "Welling, 2014), where one maximizes a lower bound on the data likelihood: Looser bounds often", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 335, + 426, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 426, + 348 + ], + "score": 1.0, + "content": "yield better inference models, i.e. latent representations (Rainforth et al., 2018).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "title", + "bbox": [ + 106, + 365, + 494, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 496, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 496, + 376 + ], + "score": 1.0, + "content": "3.3 ENCODER ARCHITECTURE CAN BE MORE IMPORTANT THAN THE SPECIFIC ESTIMATOR", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "We will now show that the encoder architecture is a critical design choice and we will investigate its", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "effect on the learned representation. We consider the same MLP architecture (238k parameters) as", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 407, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 480, + 420 + ], + "score": 1.0, + "content": "in Section 3.2, as well as a ConvNet architecture comprising two convolution layers (with a", + "type": "text" + }, + { + "bbox": [ + 480, + 408, + 504, + 419 + ], + "score": 0.89, + "content": "5 \\times 5", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 429, + 432 + ], + "score": 1.0, + "content": "kernel, stride of 2, ReLU activations, and 64 and 128 channels, respectively;", + "type": "text" + }, + { + "bbox": [ + 430, + 419, + 452, + 429 + ], + "score": 0.28, + "content": "2 2 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "parameters),", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "followed by spatial average pooling and a fully connected layer. Before the average pooling operation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 462, + 454 + ], + "score": 1.0, + "content": "we apply layer normalization (Ba et al., 2016) which greatly reduces the variance of", + "type": "text" + }, + { + "bbox": [ + 462, + 441, + 481, + 452 + ], + "score": 0.89, + "content": "\\bar { I } _ { \\mathrm { N W } } ^ { \\mathrm { ~ ~ } } \\bar { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 440, + 488, + 454 + ], + "score": 1.0, + "content": ". 9", + "type": "text" + }, + { + "bbox": [ + 491, + 442, + 505, + 452 + ], + "score": 1.0, + "content": "To", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 385, + 464 + ], + "score": 1.0, + "content": "ensure that both network architectures achieve the same lower bound", + "type": "text" + }, + { + "bbox": [ + 386, + 452, + 405, + 463 + ], + "score": 0.91, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "on the MI, we minimize", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 462, + 503, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 286, + 475 + ], + "score": 0.91, + "content": "L _ { t } ( g _ { 1 } , g _ { 2 } ) = | I _ { \\mathrm { E S T } } ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 1 } ( X ^ { ( 2 ) } ) ) - t |", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 462, + 470, + 475 + ], + "score": 1.0, + "content": "instead of solving (2), for two different values", + "type": "text" + }, + { + "bbox": [ + 470, + 463, + 503, + 474 + ], + "score": 0.91, + "content": "t = 2 , 4", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 506, + 492 + ], + "score": 1.0, + "content": "Figure 4 shows the downstream testing accuracy as a function of the training iteration (see Appendix G", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 490, + 507, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 507, + 503 + ], + "score": 1.0, + "content": "for the corresponding results on CIFAR10). It can be seen in the testing loss curves in Appendix F", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "that for both architectures and estimators the objective value after 7k iterations matches the target", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 112, + 523 + ], + "score": 0.69, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 513, + 134, + 525 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 134, + 512, + 198, + 524 + ], + "score": 0.92, + "content": "L _ { t } ( g _ { 1 } , g _ { 2 } ) \\approx 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "which implies that they achieve the same lower-bound on the MI. Despite", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "matching lower bounds, ConvNet encoders lead to clearly superior classification accuracy, for both", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 125, + 546 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 534, + 145, + 547 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 145, + 534, + 164, + 545 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 534, + 506, + 547 + ], + "score": 1.0, + "content": ". Note that, in contrast, the MLP and ConvNet architectures trained end-to-end in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 545, + 444, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 420, + 558 + ], + "score": 1.0, + "content": "supervised fashion both achieve essentially the same testing accuracy of about", + "type": "text" + }, + { + "bbox": [ + 420, + 545, + 440, + 556 + ], + "score": 0.89, + "content": "9 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 545, + 444, + 558 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 562, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "score": 1.0, + "content": "In the context of VAEs, Alemi et al. (2018) similarly observed that models achieving the same", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "evidence lower bound value can lead to vastly different representations depending on the employed", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 584, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 506, + 597 + ], + "score": 1.0, + "content": "encoder architecture, and do not necessarily capture useful information about the data (Tschannen", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 595, + 263, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 263, + 608 + ], + "score": 1.0, + "content": "et al., 2018; Blau and Michaeli, 2019).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5 + }, + { + "type": "title", + "bbox": [ + 108, + 626, + 452, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 455, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 455, + 641 + ], + "score": 1.0, + "content": "4 CONNECTION TO DEEP METRIC LEARNING AND TRIPLET LOSSES", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "score": 1.0, + "content": "In the previous section we empirically demonstrated that there is a disconnect between approximate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 663, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 484, + 677 + ], + "score": 1.0, + "content": "MI maximization and representation quality. However, many recent works have applied the", + "type": "text" + }, + { + "bbox": [ + 485, + 664, + 504, + 675 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "estimator to obtain state-of-the-art results in practice. 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For a given estimator and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 107, + 227, + 111, + 235 + ], + "score": 0.67, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 111, + 225, + 506, + 239 + ], + "score": 1.0, + "content": ", ConvNet encoders clearly outperform MLP encoders in terms of downstream testing accuracy. (c)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "Estimating MI from i.i.d. and non-i.i.d. samples in a synthetic setting (Section 4). If negative samples", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 246, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 210, + 261 + ], + "score": 1.0, + "content": "are not drawn i.i.d., both", + "type": "text" + }, + { + "bbox": [ + 210, + 248, + 229, + 259 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 246, + 247, + 261 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 248, + 248, + 267, + 259 + ], + "score": 0.91, + "content": "I _ { \\mathrm { N } \\bar { W } \\bar { J } }", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 246, + 506, + 261 + ], + "score": 1.0, + "content": "estimators can be greater than the true MI. Despite being", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 259, + 499, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 286, + 271 + ], + "score": 1.0, + "content": "commonly justified as a lower bound on MI,", + "type": "text" + }, + { + "bbox": [ + 286, + 259, + 304, + 270 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 259, + 499, + 271 + ], + "score": 1.0, + "content": "is often used in the non-i.i.d. setting in practice.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 286, + 503, + 308 + ], + "lines": [ + { + "bbox": [ + 106, + 285, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 126, + 297 + ], + "score": 0.88, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 285, + 505, + 299 + ], + "score": 1.0, + "content": "testing value, the simple bilinear critic leads to a better downstream performance than the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 297, + 280, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 280, + 309 + ], + "score": 1.0, + "content": "higher-capacity separable and MLP critics.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5, + "bbox_fs": [ + 105, + 285, + 505, + 309 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 314, + 505, + 347 + ], + "lines": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 506, + 326 + ], + "score": 1.0, + "content": "A related phenomenon was observed in the context of variational autoencoders (VAEs) (Kingma and", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 506, + 337 + ], + "score": 1.0, + "content": "Welling, 2014), where one maximizes a lower bound on the data likelihood: Looser bounds often", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 335, + 426, + 348 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 426, + 348 + ], + "score": 1.0, + "content": "yield better inference models, i.e. latent representations (Rainforth et al., 2018).", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 313, + 506, + 348 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 365, + 494, + 375 + ], + "lines": [ + { + "bbox": [ + 105, + 363, + 496, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 496, + 376 + ], + "score": 1.0, + "content": "3.3 ENCODER ARCHITECTURE CAN BE MORE IMPORTANT THAN THE SPECIFIC ESTIMATOR", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 475 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 505, + 398 + ], + "score": 1.0, + "content": "We will now show that the encoder architecture is a critical design choice and we will investigate its", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "spans": [ + { + "bbox": [ + 105, + 396, + 506, + 409 + ], + "score": 1.0, + "content": "effect on the learned representation. We consider the same MLP architecture (238k parameters) as", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 407, + 504, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 407, + 480, + 420 + ], + "score": 1.0, + "content": "in Section 3.2, as well as a ConvNet architecture comprising two convolution layers (with a", + "type": "text" + }, + { + "bbox": [ + 480, + 408, + 504, + 419 + ], + "score": 0.89, + "content": "5 \\times 5", + "type": "inline_equation" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 418, + 506, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 418, + 429, + 432 + ], + "score": 1.0, + "content": "kernel, stride of 2, ReLU activations, and 64 and 128 channels, respectively;", + "type": "text" + }, + { + "bbox": [ + 430, + 419, + 452, + 429 + ], + "score": 0.28, + "content": "2 2 0 \\mathrm { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 418, + 506, + 432 + ], + "score": 1.0, + "content": "parameters),", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "followed by spatial average pooling and a fully connected layer. Before the average pooling operation", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 440, + 505, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 462, + 454 + ], + "score": 1.0, + "content": "we apply layer normalization (Ba et al., 2016) which greatly reduces the variance of", + "type": "text" + }, + { + "bbox": [ + 462, + 441, + 481, + 452 + ], + "score": 0.89, + "content": "\\bar { I } _ { \\mathrm { N W } } ^ { \\mathrm { ~ ~ } } \\bar { J }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 440, + 488, + 454 + ], + "score": 1.0, + "content": ". 9", + "type": "text" + }, + { + "bbox": [ + 491, + 442, + 505, + 452 + ], + "score": 1.0, + "content": "To", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 452, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 452, + 385, + 464 + ], + "score": 1.0, + "content": "ensure that both network architectures achieve the same lower bound", + "type": "text" + }, + { + "bbox": [ + 386, + 452, + 405, + 463 + ], + "score": 0.91, + "content": "I _ { \\mathrm { E S T } }", + "type": "inline_equation" + }, + { + "bbox": [ + 405, + 452, + 506, + 464 + ], + "score": 1.0, + "content": "on the MI, we minimize", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 462, + 503, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 286, + 475 + ], + "score": 0.91, + "content": "L _ { t } ( g _ { 1 } , g _ { 2 } ) = | I _ { \\mathrm { E S T } } ( g _ { 1 } ( X ^ { ( 1 ) } ) ; g _ { 1 } ( X ^ { ( 2 ) } ) ) - t |", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 462, + 470, + 475 + ], + "score": 1.0, + "content": "instead of solving (2), for two different values", + "type": "text" + }, + { + "bbox": [ + 470, + 463, + 503, + 474 + ], + "score": 0.91, + "content": "t = 2 , 4", + "type": "inline_equation" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 105, + 386, + 506, + 475 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 479, + 505, + 557 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 506, + 492 + ], + "score": 1.0, + "content": "Figure 4 shows the downstream testing accuracy as a function of the training iteration (see Appendix G", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 490, + 507, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 490, + 507, + 503 + ], + "score": 1.0, + "content": "for the corresponding results on CIFAR10). It can be seen in the testing loss curves in Appendix F", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 514 + ], + "score": 1.0, + "content": "that for both architectures and estimators the objective value after 7k iterations matches the target", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 512, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 112, + 523 + ], + "score": 0.69, + "content": "t", + "type": "inline_equation" + }, + { + "bbox": [ + 112, + 513, + 134, + 525 + ], + "score": 1.0, + "content": "(i.e.,", + "type": "text" + }, + { + "bbox": [ + 134, + 512, + 198, + 524 + ], + "score": 0.92, + "content": "L _ { t } ( g _ { 1 } , g _ { 2 } ) \\approx 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "which implies that they achieve the same lower-bound on the MI. Despite", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "matching lower bounds, ConvNet encoders lead to clearly superior classification accuracy, for both", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 534, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 534, + 125, + 546 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 126, + 534, + 145, + 547 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 145, + 534, + 164, + 545 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 534, + 506, + 547 + ], + "score": 1.0, + "content": ". Note that, in contrast, the MLP and ConvNet architectures trained end-to-end in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 545, + 444, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 420, + 558 + ], + "score": 1.0, + "content": "supervised fashion both achieve essentially the same testing accuracy of about", + "type": "text" + }, + { + "bbox": [ + 420, + 545, + 440, + 556 + ], + "score": 0.89, + "content": "9 4 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 545, + 444, + 558 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 479, + 507, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 562, + 505, + 606 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 575 + ], + "score": 1.0, + "content": "In the context of VAEs, Alemi et al. (2018) similarly observed that models achieving the same", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "evidence lower bound value can lead to vastly different representations depending on the employed", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 584, + 506, + 597 + ], + "spans": [ + { + "bbox": [ + 105, + 584, + 506, + 597 + ], + "score": 1.0, + "content": "encoder architecture, and do not necessarily capture useful information about the data (Tschannen", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 595, + 263, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 263, + 608 + ], + "score": 1.0, + "content": "et al., 2018; Blau and Michaeli, 2019).", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 561, + 506, + 608 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 626, + 452, + 639 + ], + "lines": [ + { + "bbox": [ + 105, + 625, + 455, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 625, + 455, + 641 + ], + "score": 1.0, + "content": "4 CONNECTION TO DEEP METRIC LEARNING AND TRIPLET LOSSES", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "text", + "bbox": [ + 107, + 653, + 505, + 698 + ], + "lines": [ + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "spans": [ + { + "bbox": [ + 105, + 652, + 505, + 666 + ], + "score": 1.0, + "content": "In the previous section we empirically demonstrated that there is a disconnect between approximate", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 663, + 504, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 663, + 484, + 677 + ], + "score": 1.0, + "content": "MI maximization and representation quality. However, many recent works have applied the", + "type": "text" + }, + { + "bbox": [ + 485, + 664, + 504, + 675 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "spans": [ + { + "bbox": [ + 105, + 675, + 505, + 687 + ], + "score": 1.0, + "content": "estimator to obtain state-of-the-art results in practice. We provide some insight on this conundrum by", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 686, + 488, + 698 + ], + "spans": [ + { + "bbox": [ + 106, + 686, + 153, + 698 + ], + "score": 1.0, + "content": "connecting", + "type": "text" + }, + { + "bbox": [ + 153, + 686, + 171, + 697 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 686, + 252, + 698 + ], + "score": 1.0, + "content": "to a popular triplet", + "type": "text" + }, + { + "bbox": [ + 253, + 687, + 259, + 696 + ], + "score": 0.66, + "content": "k", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 686, + 488, + 698 + ], + "score": 1.0, + "content": "-plet) loss known in the deep metric learning community.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 652, + 505, + 698 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 506, + 204 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 415, + 96 + ], + "score": 1.0, + "content": "The metric learning view Given sets of triplets, namely an anchor point", + "type": "text" + }, + { + "bbox": [ + 416, + 85, + 423, + 93 + ], + "score": 0.68, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 82, + 506, + 96 + ], + "score": 1.0, + "content": ", a positive instance", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 94, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 96, + 113, + 105 + ], + "score": 0.78, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 113, + 94, + 213, + 106 + ], + "score": 1.0, + "content": ", and a negative instance", + "type": "text" + }, + { + "bbox": [ + 214, + 96, + 220, + 104 + ], + "score": 0.68, + "content": "z", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 94, + 370, + 106 + ], + "score": 1.0, + "content": ", the goal is to learn a representation", + "type": "text" + }, + { + "bbox": [ + 370, + 94, + 389, + 105 + ], + "score": 0.89, + "content": "g ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 390, + 94, + 506, + 106 + ], + "score": 1.0, + "content": "such that the distances (i.e.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 104, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 117, + 115 + ], + "score": 0.74, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 104, + 157, + 118 + ], + "score": 1.0, + "content": ") between", + "type": "text" + }, + { + "bbox": [ + 158, + 104, + 177, + 116 + ], + "score": 0.91, + "content": "g ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 104, + 196, + 118 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 196, + 104, + 215, + 116 + ], + "score": 0.91, + "content": "g ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 104, + 366, + 118 + ], + "score": 1.0, + "content": "is smaller than the distance between", + "type": "text" + }, + { + "bbox": [ + 366, + 105, + 386, + 116 + ], + "score": 0.9, + "content": "g ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 387, + 104, + 405, + 118 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 405, + 104, + 424, + 117 + ], + "score": 0.91, + "content": "g ( z )", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 104, + 506, + 118 + ], + "score": 1.0, + "content": ", for each triplet. In", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "the supervised setting, the positive instances are usually sampled from the same class, while the", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 104, + 126, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 104, + 126, + 506, + 140 + ], + "score": 1.0, + "content": "negative instances are sampled from any other class. A major focus in deep metric learning is how", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "spans": [ + { + "bbox": [ + 105, + 136, + 506, + 151 + ], + "score": 1.0, + "content": "to perform (semi-)hard positive mining — we want to present non-trivial triplets to the learning", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 506, + 161 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 294, + 161 + ], + "score": 1.0, + "content": "algorithm which become more challenging as", + "type": "text" + }, + { + "bbox": [ + 294, + 150, + 301, + 160 + ], + "score": 0.78, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 149, + 506, + 161 + ], + "score": 1.0, + "content": "improves. Natural extensions to the unsupervised", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 158, + 507, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 158, + 507, + 173 + ], + "score": 1.0, + "content": "setting can be obtained by exploiting the structure present in the input data, namely spatial (e.g.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 506, + 183 + ], + "score": 1.0, + "content": "patches from the same image should be closer than patches from different images) and temporal", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 506, + 194 + ], + "score": 1.0, + "content": "information (temporally close video frames should be encoded closer than the ones which are further", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 192, + 268, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 268, + 205 + ], + "score": 1.0, + "content": "away in time) (Hoffer and Ailon, 2015).", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 108, + 208, + 424, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 208, + 425, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 208, + 425, + 222 + ], + "score": 1.0, + "content": "Connection to InfoNCE The InfoNCE objective can be rewritten as follows:", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 224, + 507, + 264 + ], + "lines": [ + { + "bbox": [ + 111, + 224, + 507, + 264 + ], + "spans": [ + { + "bbox": [ + 111, + 224, + 507, + 264 + ], + "score": 0.93, + "content": "\\tilde { \\mathrm { \\Sigma } } _ { \\mathrm { N C E } } = \\mathbb { E } \\left[ \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { i } , y _ { j } ) } } \\right] = \\log K - \\mathbb { E } \\left[ \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\log \\left( 1 + \\sum _ { j \\ne i } e ^ { f ( x _ { i } , y _ { j } ) - f ( x _ { i } , y _ { i } ) } \\right) \\right] .", + "type": "interline_equation", + "image_path": "4865ee3219d7d938dfd53613ca8157fbc691b587965551a9b9a6bd525099ed70.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 111, + 224, + 507, + 237.33333333333334 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 111, + 237.33333333333334, + 507, + 250.66666666666669 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 111, + 250.66666666666669, + 507, + 264.0 + ], + "spans": [], + "index": 14 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 266, + 505, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 266, + 505, + 279 + ], + "spans": [ + { + "bbox": [ + 105, + 266, + 398, + 279 + ], + "score": 1.0, + "content": "The derivation is presented in Appendix C. In the particular case that", + "type": "text" + }, + { + "bbox": [ + 398, + 269, + 405, + 277 + ], + "score": 0.76, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 266, + 425, + 279 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 425, + 269, + 432, + 279 + ], + "score": 0.79, + "content": "y", + "type": "inline_equation" + }, + { + "bbox": [ + 432, + 266, + 505, + 279 + ], + "score": 1.0, + "content": "take value in the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 172, + 290 + ], + "score": 1.0, + "content": "same space and", + "type": "text" + }, + { + "bbox": [ + 172, + 278, + 179, + 289 + ], + "score": 0.86, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 180, + 277, + 309, + 290 + ], + "score": 1.0, + "content": "is constrained to be of the form", + "type": "text" + }, + { + "bbox": [ + 310, + 277, + 398, + 290 + ], + "score": 0.93, + "content": "f ( x , y ) = \\phi ( x ) ^ { \\top } \\phi ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 277, + 477, + 290 + ], + "score": 1.0, + "content": ", for some function", + "type": "text" + }, + { + "bbox": [ + 477, + 279, + 484, + 289 + ], + "score": 0.84, + "content": "\\phi", + "type": "inline_equation" + }, + { + "bbox": [ + 484, + 277, + 505, + 290 + ], + "score": 1.0, + "content": ", this", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 106, + 289, + 456, + 301 + ], + "score": 1.0, + "content": "coincides (up to constants and change of sign) with the expectation of the multi-class", + "type": "text" + }, + { + "bbox": [ + 457, + 290, + 465, + 299 + ], + "score": 0.76, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "-pair loss", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 299, + 252, + 312 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 252, + 312 + ], + "score": 1.0, + "content": "proposed in (Sohn, 2016, Eqn. (7)):", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16.5 + }, + { + "type": "interline_equation", + "bbox": [ + 144, + 315, + 467, + 354 + ], + "lines": [ + { + "bbox": [ + 144, + 315, + 467, + 354 + ], + "spans": [ + { + "bbox": [ + 144, + 315, + 467, + 354 + ], + "score": 0.95, + "content": "L _ { \\mathrm { K - p a i r r o c } } \\left( \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { K } , \\phi \\right) = \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\log \\left( 1 + \\sum _ { j \\ne i } e ^ { \\phi ( x _ { i } ) ^ { \\top } \\phi ( y _ { j } ) - \\phi ( x _ { i } ) ^ { \\top } \\phi ( y _ { i } ) } \\right) .", + "type": "interline_equation", + "image_path": "9ce25da6e970470c3dc44ec565657cfb0753a9c0cdc951af563afa335e874c47.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 144, + 315, + 467, + 328.0 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 144, + 328.0, + 467, + 341.0 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 144, + 341.0, + 467, + 354.0 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 358, + 505, + 425 + ], + "lines": [ + { + "bbox": [ + 105, + 358, + 505, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 262, + 372 + ], + "score": 1.0, + "content": "Representation learning by maximizing", + "type": "text" + }, + { + "bbox": [ + 262, + 359, + 281, + 370 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 358, + 416, + 372 + ], + "score": 1.0, + "content": "using a symmetric separable critic", + "type": "text" + }, + { + "bbox": [ + 417, + 358, + 505, + 371 + ], + "score": 0.93, + "content": "f ( x , y ) = \\phi ( x ) ^ { \\top } \\phi ( y )", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 370, + 506, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 370, + 170, + 383 + ], + "score": 1.0, + "content": "and an encoder", + "type": "text" + }, + { + "bbox": [ + 170, + 371, + 221, + 381 + ], + "score": 0.91, + "content": "g = g _ { 1 } = g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 370, + 506, + 383 + ], + "score": 1.0, + "content": "shared across views is thus equivalent to metric learning based on (5).", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 381, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 506, + 393 + ], + "score": 1.0, + "content": "When using different encoders for different views and asymmetric critics as employed by CPC,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 390, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 104, + 390, + 506, + 405 + ], + "score": 1.0, + "content": "DeepInfoMax, and CMC one recovers asymmetric variants of (5), see, e.g. (Yu et al., 2017; Zhang", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "et al., 2019). As a result, one can view (5) as learning encoders with a parameter-less inner product", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 414, + 353, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 353, + 426 + ], + "score": 1.0, + "content": "critic, for which the MI lower-bound is very weak in general.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "There are (at least) two immediate benefits of viewing recent representation learning methods based", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 442, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 506, + 454 + ], + "score": 1.0, + "content": "on MI estimators through the lens of metric learning. Firstly, in the MI view, using inner product", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "or bilinear critic functions is sub-optimal since the critic should ideally be as flexible as possible in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "score": 1.0, + "content": "order to reduce the gap between the lower bound and the true MI. In the metric learning view, the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 475, + 504, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 504, + 486 + ], + "score": 1.0, + "content": "inner product critic corresponds to a simple metric on the embedding space. The metric learning view", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 485, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 506, + 497 + ], + "score": 1.0, + "content": "seems hence in better accordance with the observations from Section 3.2 than the MI view. Secondly,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "it elucidates the importance of appropriately choosing the negative samples, which is indeed a critical", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 508, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 519 + ], + "score": 1.0, + "content": "component in deep metric learning based on triplet losses (Norouzi et al., 2012; Schroff et al., 2015).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "InfoNCE and the importance of negative sampling The negative sample mining issue also", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "manifests itself in MI-based contrastive losses. In fact, while InfoNCE is a lower bound on MI if the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 546, + 456, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 456, + 559 + ], + "score": 1.0, + "content": "negative samples are drawn from the true marginal distribution (Poole et al., 2019), i.e.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37 + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 561, + 437, + 596 + ], + "lines": [ + { + "bbox": [ + 173, + 561, + 437, + 596 + ], + "spans": [ + { + "bbox": [ + 173, + 561, + 437, + 596 + ], + "score": 0.94, + "content": "I ( X , Y ) \\geq \\mathbb { E } _ { \\prod _ { k } p ( x _ { k } , y _ { k } ) } { \\frac { 1 } { K } } \\sum _ { i = 1 } ^ { K } \\left[ \\log { \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { { \\frac { 1 } { K } } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } } \\right] \\triangleq I _ { \\mathbb { N } \\mathbb { C } } ,", + "type": "interline_equation", + "image_path": "52e91f01da6694fcddd8df01621988aa6ac5cf229695ad2c31dfba41037daae4.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 173, + 561, + 437, + 572.6666666666666 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 173, + 572.6666666666666, + 437, + 584.3333333333333 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 173, + 584.3333333333333, + 437, + 595.9999999999999 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 474, + 614 + ], + "score": 1.0, + "content": "we show that if the negative samples are drawn in a dependent fashion (corresponding to the", + "type": "text" + }, + { + "bbox": [ + 475, + 600, + 505, + 612 + ], + "score": 0.91, + "content": "( x _ { i } , y _ { i } )", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 308, + 624 + ], + "score": 1.0, + "content": "being drawn identically but not independently), the", + "type": "text" + }, + { + "bbox": [ + 308, + 612, + 327, + 622 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "estimator is in general neither a lower nor an", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 219, + 635 + ], + "score": 1.0, + "content": "upper bound on the true MI", + "type": "text" + }, + { + "bbox": [ + 220, + 622, + 254, + 633 + ], + "score": 0.92, + "content": "I ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 622, + 506, + 635 + ], + "score": 1.0, + "content": ". We prove this in Appendix D and present empirical evidence", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 632, + 507, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 145, + 646 + ], + "score": 1.0, + "content": "here. Let", + "type": "text" + }, + { + "bbox": [ + 145, + 633, + 212, + 645 + ], + "score": 0.92, + "content": "( X , Y ) = Z + \\epsilon .", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 632, + 241, + 646 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 242, + 633, + 303, + 644 + ], + "score": 0.91, + "content": "\\bar { Z } \\sim \\mathcal { N } ( 0 , \\Sigma _ { Z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 632, + 322, + 646 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 322, + 633, + 378, + 645 + ], + "score": 0.93, + "content": "\\epsilon \\sim \\mathcal { N } ( 0 , \\Sigma _ { \\epsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 632, + 507, + 646 + ], + "score": 1.0, + "content": "are two-dimensional Gaussians.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 225, + 657 + ], + "score": 1.0, + "content": "We generate batches of data", + "type": "text" + }, + { + "bbox": [ + 225, + 644, + 301, + 656 + ], + "score": 0.9, + "content": "( X _ { i } , Y _ { i } ) = Z + \\epsilon _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 643, + 352, + 657 + ], + "score": 1.0, + "content": "where each", + "type": "text" + }, + { + "bbox": [ + 352, + 646, + 361, + 655 + ], + "score": 0.86, + "content": "\\epsilon _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "is sampled independently for each", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 206, + 667 + ], + "score": 1.0, + "content": "element of the batch, but", + "type": "text" + }, + { + "bbox": [ + 207, + 655, + 215, + 665 + ], + "score": 0.82, + "content": "Z", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 654, + 380, + 667 + ], + "score": 1.0, + "content": "is sampled only once per batch. As such,", + "type": "text" + }, + { + "bbox": [ + 380, + 655, + 414, + 667 + ], + "score": 0.92, + "content": "( X _ { i } , Y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "has the same marginal", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 194, + 678 + ], + "score": 1.0, + "content": "distribution for each", + "type": "text" + }, + { + "bbox": [ + 194, + 667, + 198, + 676 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 665, + 506, + 678 + ], + "score": 1.0, + "content": ", but the elements of the batch are not independent. 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We observe in Figure", + "type": "text" + }, + { + "bbox": [ + 405, + 688, + 416, + 698 + ], + "score": 0.29, + "content": "_ \\mathrm { 4 c }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "that when using non-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 196, + 712 + ], + "score": 1.0, + "content": "i.i.d. samples both the", + "type": "text" + }, + { + "bbox": [ + 197, + 699, + 215, + 710 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 698, + 233, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 699, + 252, + 710 + ], + "score": 0.93, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "values are larger than the true MI, and that when i.i.d. samples", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "are used, both are lower bounds on the true MI. 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As a result, one can view (5) as learning encoders with a parameter-less inner product", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 414, + 353, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 414, + 353, + 426 + ], + "score": 1.0, + "content": "critic, for which the MI lower-bound is very weak in general.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 358, + 506, + 426 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 430, + 505, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 505, + 443 + ], + "score": 1.0, + "content": "There are (at least) two immediate benefits of viewing recent representation learning methods based", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 442, + 506, + 454 + ], + "spans": [ + { + "bbox": [ + 106, + 442, + 506, + 454 + ], + "score": 1.0, + "content": "on MI estimators through the lens of metric learning. Firstly, in the MI view, using inner product", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 465 + ], + "score": 1.0, + "content": "or bilinear critic functions is sub-optimal since the critic should ideally be as flexible as possible in", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 505, + 475 + ], + "score": 1.0, + "content": "order to reduce the gap between the lower bound and the true MI. In the metric learning view, the", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 475, + 504, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 475, + 504, + 486 + ], + "score": 1.0, + "content": "inner product critic corresponds to a simple metric on the embedding space. The metric learning view", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 485, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 106, + 485, + 506, + 497 + ], + "score": 1.0, + "content": "seems hence in better accordance with the observations from Section 3.2 than the MI view. Secondly,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 506, + 509 + ], + "score": 1.0, + "content": "it elucidates the importance of appropriately choosing the negative samples, which is indeed a critical", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 508, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 508, + 506, + 519 + ], + "score": 1.0, + "content": "component in deep metric learning based on triplet losses (Norouzi et al., 2012; Schroff et al., 2015).", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 430, + 506, + 519 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 524, + 505, + 558 + ], + "lines": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "spans": [ + { + "bbox": [ + 105, + 524, + 505, + 537 + ], + "score": 1.0, + "content": "InfoNCE and the importance of negative sampling The negative sample mining issue also", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 106, + 536, + 505, + 547 + ], + "score": 1.0, + "content": "manifests itself in MI-based contrastive losses. In fact, while InfoNCE is a lower bound on MI if the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 546, + 456, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 456, + 559 + ], + "score": 1.0, + "content": "negative samples are drawn from the true marginal distribution (Poole et al., 2019), i.e.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 37, + "bbox_fs": [ + 105, + 524, + 505, + 559 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 561, + 437, + 596 + ], + "lines": [ + { + "bbox": [ + 173, + 561, + 437, + 596 + ], + "spans": [ + { + "bbox": [ + 173, + 561, + 437, + 596 + ], + "score": 0.94, + "content": "I ( X , Y ) \\geq \\mathbb { E } _ { \\prod _ { k } p ( x _ { k } , y _ { k } ) } { \\frac { 1 } { K } } \\sum _ { i = 1 } ^ { K } \\left[ \\log { \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { { \\frac { 1 } { K } } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } } \\right] \\triangleq I _ { \\mathbb { N } \\mathbb { C } } ,", + "type": "interline_equation", + "image_path": "52e91f01da6694fcddd8df01621988aa6ac5cf229695ad2c31dfba41037daae4.jpg" + } + ] + } + ], + "index": 40, + "virtual_lines": [ + { + "bbox": [ + 173, + 561, + 437, + 572.6666666666666 + ], + "spans": [], + "index": 39 + }, + { + "bbox": [ + 173, + 572.6666666666666, + 437, + 584.3333333333333 + ], + "spans": [], + "index": 40 + }, + { + "bbox": [ + 173, + 584.3333333333333, + 437, + 595.9999999999999 + ], + "spans": [], + "index": 41 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 599, + 506, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 474, + 614 + ], + "score": 1.0, + "content": "we show that if the negative samples are drawn in a dependent fashion (corresponding to the", + "type": "text" + }, + { + "bbox": [ + 475, + 600, + 505, + 612 + ], + "score": 0.91, + "content": "( x _ { i } , y _ { i } )", + "type": "inline_equation" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 611, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 308, + 624 + ], + "score": 1.0, + "content": "being drawn identically but not independently), the", + "type": "text" + }, + { + "bbox": [ + 308, + 612, + 327, + 622 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 327, + 611, + 506, + 624 + ], + "score": 1.0, + "content": "estimator is in general neither a lower nor an", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 622, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 622, + 219, + 635 + ], + "score": 1.0, + "content": "upper bound on the true MI", + "type": "text" + }, + { + "bbox": [ + 220, + 622, + 254, + 633 + ], + "score": 0.92, + "content": "I ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 622, + 506, + 635 + ], + "score": 1.0, + "content": ". We prove this in Appendix D and present empirical evidence", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 632, + 507, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 145, + 646 + ], + "score": 1.0, + "content": "here. Let", + "type": "text" + }, + { + "bbox": [ + 145, + 633, + 212, + 645 + ], + "score": 0.92, + "content": "( X , Y ) = Z + \\epsilon .", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 632, + 241, + 646 + ], + "score": 1.0, + "content": ", where", + "type": "text" + }, + { + "bbox": [ + 242, + 633, + 303, + 644 + ], + "score": 0.91, + "content": "\\bar { Z } \\sim \\mathcal { N } ( 0 , \\Sigma _ { Z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 632, + 322, + 646 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 322, + 633, + 378, + 645 + ], + "score": 0.93, + "content": "\\epsilon \\sim \\mathcal { N } ( 0 , \\Sigma _ { \\epsilon } )", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 632, + 507, + 646 + ], + "score": 1.0, + "content": "are two-dimensional Gaussians.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 643, + 506, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 225, + 657 + ], + "score": 1.0, + "content": "We generate batches of data", + "type": "text" + }, + { + "bbox": [ + 225, + 644, + 301, + 656 + ], + "score": 0.9, + "content": "( X _ { i } , Y _ { i } ) = Z + \\epsilon _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 643, + 352, + 657 + ], + "score": 1.0, + "content": "where each", + "type": "text" + }, + { + "bbox": [ + 352, + 646, + 361, + 655 + ], + "score": 0.86, + "content": "\\epsilon _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 643, + 506, + 657 + ], + "score": 1.0, + "content": "is sampled independently for each", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 206, + 667 + ], + "score": 1.0, + "content": "element of the batch, but", + "type": "text" + }, + { + "bbox": [ + 207, + 655, + 215, + 665 + ], + "score": 0.82, + "content": "Z", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 654, + 380, + 667 + ], + "score": 1.0, + "content": "is sampled only once per batch. As such,", + "type": "text" + }, + { + "bbox": [ + 380, + 655, + 414, + 667 + ], + "score": 0.92, + "content": "( X _ { i } , Y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "has the same marginal", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 665, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 194, + 678 + ], + "score": 1.0, + "content": "distribution for each", + "type": "text" + }, + { + "bbox": [ + 194, + 667, + 198, + 676 + ], + "score": 0.7, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 199, + 665, + 506, + 678 + ], + "score": 1.0, + "content": ", but the elements of the batch are not independent. Although we do not", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 423, + 689 + ], + "score": 1.0, + "content": "treat it theoretically, we also display results of the same experiment using the", + "type": "text" + }, + { + "bbox": [ + 423, + 677, + 442, + 688 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "estimator. The", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 688, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 405, + 700 + ], + "score": 1.0, + "content": "experimental details are presented in Appendix E. We observe in Figure", + "type": "text" + }, + { + "bbox": [ + 405, + 688, + 416, + 698 + ], + "score": 0.29, + "content": "_ \\mathrm { 4 c }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 688, + 506, + 700 + ], + "score": 1.0, + "content": "that when using non-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 196, + 712 + ], + "score": 1.0, + "content": "i.i.d. samples both the", + "type": "text" + }, + { + "bbox": [ + 197, + 699, + 215, + 710 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 698, + 233, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 234, + 699, + 252, + 710 + ], + "score": 0.93, + "content": "I _ { \\mathrm { N } \\bar { w } \\bar { s } }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "values are larger than the true MI, and that when i.i.d. samples", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "are used, both are lower bounds on the true MI. Hence, the connection to MI under improper negative", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 720, + 336, + 734 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 336, + 734 + ], + "score": 1.0, + "content": "sampling is no longer clear and might vanish completely.", + "type": "text" + } + ], + "index": 53 + } + ], + "index": 47.5, + "bbox_fs": [ + 104, + 599, + 507, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Notwithstanding this fundamental problem, the negative sampling strategy is often treated as a design", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "choice. In Hénaff et al. (2019), CPC is applied to images by partitioning the input image into patches.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 119 + ], + "score": 1.0, + "content": "Then, MI (estimated by InfoNCE) between representations of patches and a context summarizing", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "several patches that are vertically above or below in the same image is minimized. Negative samples", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "are obtained by patches from different images as well as patches from the same image, violating", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 504, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 504, + 149 + ], + "score": 1.0, + "content": "the independence assumption. Similarly, van den Oord et al. (2018) learn representations of speech", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "using samples from a variety of speakers. It was found that using utterances from the same speaker", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "as negative samples is more effective, whereas the “proper” negative samples should be drawn from", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 327, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 327, + 183 + ], + "score": 1.0, + "content": "an appropriate mixture of utterances from all speakers.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "A common observation is that increasing the number of negative examples helps in practice (Hjelm", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 504, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 485, + 212 + ], + "score": 1.0, + "content": "et al., 2019; Tian et al., 2019; Bachman et al., 2019). Indeed, Ma and Collins (2018) show that", + "type": "text" + }, + { + "bbox": [ + 485, + 199, + 504, + 209 + ], + "score": 0.88, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "is consistent for any number of negative samples (under technical conditions), and Poole et al. (2019)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "show that the signal-to-noise ratio increases with the number of negative samples. On the other", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "hand, (Arora et al., 2019) have demonstrated, both theoretically and empirically, that increasing the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 504, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 504, + 253 + ], + "score": 1.0, + "content": "number of negative samples does not necessarily help, and can even deteriorate the performance. The", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 394, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 394, + 266 + ], + "score": 1.0, + "content": "intricacies of negative sampling hence remain a key research challenge.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 108, + 293, + 195, + 305 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 197, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 197, + 308 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "Is MI maximization a good objective for learning good representations in an unsupervised fashion?", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "Possibly, but it is clearly not sufficient. In this work we have demonstrated that, under the common", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "linear evaluation protocol, maximizing lower bounds on MI as done in modern incarnations of the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "InfoMax principle can result in bad representations. We have revealed that the commonly used", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 369, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 381 + ], + "score": 1.0, + "content": "estimators have strong inductive biases and—perhaps surprisingly—looser bounds can lead to better", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 104, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "representations. Furthermore, we have demonstrated that the connection of recent approaches to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "MI maximization might vanish if negative samples are not drawn independently (as done by some", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "approaches in the literature). As a result, it is unclear whether the connection to MI is a sufficient (or", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "necessary) component for designing powerful unsupervised representation learning algorithms. We", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "score": 1.0, + "content": "propose that the success of these recent methods could be explained through the view of triplet-based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "metric learning and that leveraging advances in that domain might lead to further improvements. We", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 428, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 428, + 459 + ], + "score": 1.0, + "content": "have several suggestions for future work, which we summarize in the following.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "Alternative measures of information We believe that the question of developing new notions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 474, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 485 + ], + "score": 1.0, + "content": "of information suitable for representation learning should receive more attention. While MI has", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 507, + 497 + ], + "score": 1.0, + "content": "appealing theoretical properties, it is clearly not sufficient for this task—it is hard to estimate,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "invariant to bijections and can result in suboptimal representations which do not correlate with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "downstream performance. Therefore, a new notion of information should account for both the amount", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "of information stored in a representation and the geometry of the induced space necessary for good", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 529, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 540 + ], + "score": 1.0, + "content": "performance on downstream tasks. One possible avenue is to consider extensions to MI which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "explicitly account for the modeling power and computational constraints of the observer, such as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 204, + 563 + ], + "score": 1.0, + "content": "the recently introduced", + "type": "text" + }, + { + "bbox": [ + 205, + 551, + 214, + 560 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 550, + 267, + 563 + ], + "score": 1.0, + "content": "-information", + "type": "text" + }, + { + "bbox": [ + 267, + 550, + 282, + 561 + ], + "score": 0.36, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "et al. (2020). Alternatively, one can investigate other", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 560, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 338, + 575 + ], + "score": 1.0, + "content": "statistical divergences to measure the discrepancy between", + "type": "text" + }, + { + "bbox": [ + 338, + 561, + 367, + 573 + ], + "score": 0.93, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 560, + 385, + 575 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 561, + 423, + 573 + ], + "score": 0.92, + "content": "p ( x ) p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 560, + 505, + 575 + ], + "score": 1.0, + "content": ". For example, using", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "the Wasserstein distance leads to promising results in representation learning as it naturally enforces", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 583, + 298, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 298, + 595 + ], + "score": 1.0, + "content": "smoothness in the encoders (Ozair et al., 2019).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34.5 + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "A holistic view We believe that any theory on measuring information for representation learning", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "built on critics should explicitly take into account the function families one uses (e.g. that of the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "critic and estimator). Most importantly, we would expect some natural trade-offs between the amount", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "of information that can be stored against how hard it is to extract it in the downstream tasks as a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 645, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 655 + ], + "score": 1.0, + "content": "function of the architectural choices. While the distribution of downstream tasks is typically assumed", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "unknown in representation learning, it might be possible to rely on weaker assumptions such as a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "family of invariances relevant for the downstream tasks. Moreover, it seems that in the literature", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 505, + 690 + ], + "score": 1.0, + "content": "(i) the critics that are used to measure the information, (ii) the encoders, and (iii) the downstream", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "models/evaluation protocol are all mostly chosen independently of each other. Our empirical results", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "show that the downstream performance depends on the intricate balance between these choices and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "we believe that one should co-design them. This holistic view is currently under-explored and due to", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 721, + 381, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 381, + 733 + ], + "score": 1.0, + "content": "the lack of any theory or extensive studies to guide the practitioners.", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 46.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 108, + 27, + 293, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 294, + 38 + ], + "score": 1.0, + "content": "Published as a conference paper at ICLR 2020", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Notwithstanding this fundamental problem, the negative sampling strategy is often treated as a design", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 107 + ], + "score": 1.0, + "content": "choice. In Hénaff et al. (2019), CPC is applied to images by partitioning the input image into patches.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 102, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 102, + 506, + 119 + ], + "score": 1.0, + "content": "Then, MI (estimated by InfoNCE) between representations of patches and a context summarizing", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 128 + ], + "score": 1.0, + "content": "several patches that are vertically above or below in the same image is minimized. Negative samples", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 505, + 140 + ], + "score": 1.0, + "content": "are obtained by patches from different images as well as patches from the same image, violating", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 504, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 504, + 149 + ], + "score": 1.0, + "content": "the independence assumption. Similarly, van den Oord et al. (2018) learn representations of speech", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 161 + ], + "score": 1.0, + "content": "using samples from a variety of speakers. It was found that using utterances from the same speaker", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "as negative samples is more effective, whereas the “proper” negative samples should be drawn from", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 171, + 327, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 327, + 183 + ], + "score": 1.0, + "content": "an appropriate mixture of utterances from all speakers.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 82, + 506, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 264 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 199 + ], + "score": 1.0, + "content": "A common observation is that increasing the number of negative examples helps in practice (Hjelm", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 196, + 504, + 212 + ], + "spans": [ + { + "bbox": [ + 105, + 196, + 485, + 212 + ], + "score": 1.0, + "content": "et al., 2019; Tian et al., 2019; Bachman et al., 2019). Indeed, Ma and Collins (2018) show that", + "type": "text" + }, + { + "bbox": [ + 485, + 199, + 504, + 209 + ], + "score": 0.88, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 505, + 221 + ], + "score": 1.0, + "content": "is consistent for any number of negative samples (under technical conditions), and Poole et al. (2019)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 505, + 232 + ], + "score": 1.0, + "content": "show that the signal-to-noise ratio increases with the number of negative samples. On the other", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 243 + ], + "score": 1.0, + "content": "hand, (Arora et al., 2019) have demonstrated, both theoretically and empirically, that increasing the", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 243, + 504, + 253 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 504, + 253 + ], + "score": 1.0, + "content": "number of negative samples does not necessarily help, and can even deteriorate the performance. The", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 253, + 394, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 253, + 394, + 266 + ], + "score": 1.0, + "content": "intricacies of negative sampling hence remain a key research challenge.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 186, + 505, + 266 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 293, + 195, + 305 + ], + "lines": [ + { + "bbox": [ + 105, + 291, + 197, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 197, + 308 + ], + "score": 1.0, + "content": "5 CONCLUSION", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 325, + 505, + 457 + ], + "lines": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 505, + 338 + ], + "score": 1.0, + "content": "Is MI maximization a good objective for learning good representations in an unsupervised fashion?", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 506, + 349 + ], + "score": 1.0, + "content": "Possibly, but it is clearly not sufficient. In this work we have demonstrated that, under the common", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 505, + 360 + ], + "score": 1.0, + "content": "linear evaluation protocol, maximizing lower bounds on MI as done in modern incarnations of the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 505, + 371 + ], + "score": 1.0, + "content": "InfoMax principle can result in bad representations. We have revealed that the commonly used", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 369, + 506, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 369, + 506, + 381 + ], + "score": 1.0, + "content": "estimators have strong inductive biases and—perhaps surprisingly—looser bounds can lead to better", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 380, + 506, + 393 + ], + "spans": [ + { + "bbox": [ + 104, + 380, + 506, + 393 + ], + "score": 1.0, + "content": "representations. Furthermore, we have demonstrated that the connection of recent approaches to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 404 + ], + "score": 1.0, + "content": "MI maximization might vanish if negative samples are not drawn independently (as done by some", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 415 + ], + "score": 1.0, + "content": "approaches in the literature). As a result, it is unclear whether the connection to MI is a sufficient (or", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 413, + 505, + 425 + ], + "score": 1.0, + "content": "necessary) component for designing powerful unsupervised representation learning algorithms. We", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 506, + 436 + ], + "score": 1.0, + "content": "propose that the success of these recent methods could be explained through the view of triplet-based", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "metric learning and that leveraging advances in that domain might lead to further improvements. We", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 444, + 428, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 428, + 459 + ], + "score": 1.0, + "content": "have several suggestions for future work, which we summarize in the following.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 325, + 506, + 459 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 462, + 505, + 594 + ], + "lines": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 505, + 475 + ], + "score": 1.0, + "content": "Alternative measures of information We believe that the question of developing new notions", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 474, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 505, + 485 + ], + "score": 1.0, + "content": "of information suitable for representation learning should receive more attention. While MI has", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 484, + 507, + 497 + ], + "spans": [ + { + "bbox": [ + 105, + 484, + 507, + 497 + ], + "score": 1.0, + "content": "appealing theoretical properties, it is clearly not sufficient for this task—it is hard to estimate,", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 505, + 507 + ], + "score": 1.0, + "content": "invariant to bijections and can result in suboptimal representations which do not correlate with", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 506, + 519 + ], + "score": 1.0, + "content": "downstream performance. Therefore, a new notion of information should account for both the amount", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 506, + 531 + ], + "score": 1.0, + "content": "of information stored in a representation and the geometry of the induced space necessary for good", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 529, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 506, + 540 + ], + "score": 1.0, + "content": "performance on downstream tasks. One possible avenue is to consider extensions to MI which", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "explicitly account for the modeling power and computational constraints of the observer, such as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 204, + 563 + ], + "score": 1.0, + "content": "the recently introduced", + "type": "text" + }, + { + "bbox": [ + 205, + 551, + 214, + 560 + ], + "score": 0.84, + "content": "\\mathcal { F }", + "type": "inline_equation" + }, + { + "bbox": [ + 214, + 550, + 267, + 563 + ], + "score": 1.0, + "content": "-information", + "type": "text" + }, + { + "bbox": [ + 267, + 550, + 282, + 561 + ], + "score": 0.36, + "content": "\\mathrm { X u }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "et al. (2020). Alternatively, one can investigate other", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 560, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 338, + 575 + ], + "score": 1.0, + "content": "statistical divergences to measure the discrepancy between", + "type": "text" + }, + { + "bbox": [ + 338, + 561, + 367, + 573 + ], + "score": 0.93, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 368, + 560, + 385, + 575 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 385, + 561, + 423, + 573 + ], + "score": 0.92, + "content": "p ( x ) p ( y )", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 560, + 505, + 575 + ], + "score": 1.0, + "content": ". For example, using", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 505, + 585 + ], + "score": 1.0, + "content": "the Wasserstein distance leads to promising results in representation learning as it naturally enforces", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 583, + 298, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 298, + 595 + ], + "score": 1.0, + "content": "smoothness in the encoders (Ozair et al., 2019).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 34.5, + "bbox_fs": [ + 105, + 462, + 507, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 600, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 104, + 599, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 505, + 614 + ], + "score": 1.0, + "content": "A holistic view We believe that any theory on measuring information for representation learning", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "built on critics should explicitly take into account the function families one uses (e.g. that of the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 635 + ], + "score": 1.0, + "content": "critic and estimator). Most importantly, we would expect some natural trade-offs between the amount", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "of information that can be stored against how hard it is to extract it in the downstream tasks as a", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 645, + 505, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 645, + 505, + 655 + ], + "score": 1.0, + "content": "function of the architectural choices. While the distribution of downstream tasks is typically assumed", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "score": 1.0, + "content": "unknown in representation learning, it might be possible to rely on weaker assumptions such as a", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "family of invariances relevant for the downstream tasks. 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Our empirical results", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "show that the downstream performance depends on the intricate balance between these choices and", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 506, + 722 + ], + "score": 1.0, + "content": "we believe that one should co-design them. 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(2018): They developed a multi-view representation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 230, + 505, + 244 + ], + "score": 1.0, + "content": "learning approach for video data similar to CMC, but without drawing negative samples independently", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 242, + 460, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 460, + 254 + ], + "score": 1.0, + "content": "and seemingly without relying on the MI mental model to motivate their design choices.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 11, + "bbox_fs": [ + 105, + 176, + 505, + 254 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 272, + 218, + 283 + ], + "lines": [ + { + "bbox": [ + 106, + 271, + 220, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 220, + 286 + ], + "score": 1.0, + "content": "ACKNOWLEDGMENTS", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 108, + 296, + 505, + 329 + ], + "lines": [ + { + "bbox": [ + 105, + 294, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 294, + 505, + 310 + ], + "score": 1.0, + "content": "We would like to thank Alex Alemi, Ben Poole, Olivier Bachem, and Alexey Dosovitskiy for inspiring", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "discussions and comments on the manuscript. 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Let", + "type": "text" + }, + { + "bbox": [ + 186, + 132, + 196, + 141 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 129, + 328, + 145 + ], + "score": 1.0, + "content": "be a random variable and define", + "type": "text" + }, + { + "bbox": [ + 329, + 131, + 381, + 144 + ], + "score": 0.93, + "content": "X _ { 1 } = g _ { 1 } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 129, + 400, + 145 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 400, + 131, + 453, + 144 + ], + "score": 0.92, + "content": "X _ { 2 } = g _ { 2 } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 129, + 505, + 145 + ], + "score": 1.0, + "content": "be arbitrary", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 142, + 320, + 155 + ], + "spans": [ + { + "bbox": [ + 104, + 142, + 156, + 155 + ], + "score": 1.0, + "content": "functions of", + "type": "text" + }, + { + "bbox": [ + 156, + 143, + 165, + 152 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 142, + 193, + 155 + ], + "score": 1.0, + "content": ". 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Follows by two applications of the data processing inequality, which states that for random", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 146, + 190 + ], + "score": 1.0, + "content": "variables", + "type": "text" + }, + { + "bbox": [ + 146, + 177, + 171, + 187 + ], + "score": 0.29, + "content": "X , Y", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 175, + 190, + 190 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 190, + 177, + 199, + 187 + ], + "score": 0.83, + "content": "Z", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 175, + 329, + 190 + ], + "score": 1.0, + "content": "satisfying the Markov relation", + "type": "text" + }, + { + "bbox": [ + 329, + 177, + 392, + 187 + ], + "score": 0.91, + "content": "X Y Z", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 175, + 456, + 190 + ], + "score": 1.0, + "content": ", the inequality", + "type": "text" + }, + { + "bbox": [ + 457, + 177, + 505, + 189 + ], + "score": 0.92, + "content": "I ( X ; Z ) \\ \\leq", + "type": "inline_equation" + } + ], + "index": 5 + }, + { + "bbox": [ + 107, + 187, + 171, + 200 + ], + "spans": [ + { + "bbox": [ + 107, + 187, + 142, + 200 + ], + "score": 0.92, + "content": "I ( X ; Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 187, + 171, + 200 + ], + "score": 1.0, + "content": "holds.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 204, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 105, + 204, + 505, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 240, + 217 + ], + "score": 1.0, + "content": "The first step is to observe that", + "type": "text" + }, + { + "bbox": [ + 241, + 205, + 250, + 215 + ], + "score": 0.67, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 204, + 255, + 217 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 256, + 205, + 270, + 216 + ], + "score": 0.78, + "content": "X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 270, + 204, + 290, + 217 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 291, + 205, + 304, + 216 + ], + "score": 0.89, + "content": "X _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 204, + 387, + 217 + ], + "score": 1.0, + "content": "satisfy the relation", + "type": "text" + }, + { + "bbox": [ + 388, + 205, + 462, + 216 + ], + "score": 0.91, + "content": "X _ { 1 } \\left. X \\right. 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The second step is to observe that", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 107, + 236, + 376, + 251 + ], + "spans": [ + { + "bbox": [ + 107, + 237, + 198, + 250 + ], + "score": 0.93, + "content": "X ( X _ { 1 } , X _ { 2 } ) X _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 236, + 254, + 251 + ], + "score": 1.0, + "content": "and therefore", + "type": "text" + }, + { + "bbox": [ + 255, + 238, + 371, + 250 + ], + "score": 0.88, + "content": "I ( X ; X _ { 1 } ) \\leq I ( X ; ( X _ { 1 } , X _ { 2 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 236, + 376, + 251 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 108, + 253, + 433, + 267 + ], + "lines": [ + { + "bbox": [ + 105, + 252, + 435, + 269 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 261, + 269 + ], + "score": 1.0, + "content": "Combining the two inequalities yields", + "type": "text" + }, + { + "bbox": [ + 261, + 254, + 381, + 267 + ], + "score": 0.91, + "content": "I ( X _ { 1 } ; X _ { 2 } ) \\leq I ( X ; ( X _ { 1 } , X _ { 2 } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 252, + 435, + 269 + ], + "score": 1.0, + "content": ", as required.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "title", + "bbox": [ + 107, + 281, + 440, + 308 + ], + "lines": [ + { + "bbox": [ + 105, + 281, + 441, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 441, + 295 + ], + "score": 1.0, + "content": "B EXPERIMENT DETAILS: ADVERSARIALLY TRAINED ENCODER", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 126, + 295, + 204, + 309 + ], + "spans": [ + { + "bbox": [ + 126, + 295, + 204, + 309 + ], + "score": 1.0, + "content": "(SECTION 3.1)", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 106, + 318, + 506, + 421 + ], + "lines": [ + { + "bbox": [ + 104, + 318, + 507, + 332 + ], + "spans": [ + { + "bbox": [ + 104, + 318, + 507, + 332 + ], + "score": 1.0, + "content": "In the following, we present the details for training the invertible model from Section 3.1 adversarially.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 147, + 343 + ], + "score": 1.0, + "content": "We model", + "type": "text" + }, + { + "bbox": [ + 148, + 331, + 158, + 342 + ], + "score": 0.82, + "content": "g _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 330, + 475, + 343 + ], + "score": 1.0, + "content": "with the same RealNVP architecture as in the first experiment, and do not model", + "type": "text" + }, + { + "bbox": [ + 476, + 332, + 486, + 342 + ], + "score": 0.84, + "content": "g _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 330, + 506, + 343 + ], + "score": 1.0, + "content": ". 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SAMPLING", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 658, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 671 + ], + "score": 1.0, + "content": "The proof that InfoNCE is a lower bound on MI presented in (Poole et al., 2019) makes crucial use of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 669, + 504, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 504, + 682 + ], + "score": 1.0, + "content": "the assumption that the negative samples are drawn from the true marginal distribution. We briefly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "review this proof to highlight the importance of the negative sampling distribution. 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Let", + "type": "text" + }, + { + "bbox": [ + 186, + 132, + 196, + 141 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 129, + 328, + 145 + ], + "score": 1.0, + "content": "be a random variable and define", + "type": "text" + }, + { + "bbox": [ + 329, + 131, + 381, + 144 + ], + "score": 0.93, + "content": "X _ { 1 } = g _ { 1 } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 382, + 129, + 400, + 145 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 400, + 131, + 453, + 144 + ], + "score": 0.92, + "content": "X _ { 2 } = g _ { 2 } ( X )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 129, + 505, + 145 + ], + "score": 1.0, + "content": "be arbitrary", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 104, + 142, + 320, + 155 + ], + "spans": [ + { + "bbox": [ + 104, + 142, + 156, + 155 + ], + "score": 1.0, + "content": "functions of", + "type": "text" + }, + { + "bbox": [ + 156, + 143, + 165, + 152 + ], + "score": 0.74, + "content": "X", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 142, + 193, + 155 + ], + "score": 1.0, + "content": ". 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SAMPLING", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 658, + 505, + 716 + ], + "lines": [ + { + "bbox": [ + 106, + 658, + 505, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 658, + 505, + 671 + ], + "score": 1.0, + "content": "The proof that InfoNCE is a lower bound on MI presented in (Poole et al., 2019) makes crucial use of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 669, + 504, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 669, + 504, + 682 + ], + "score": 1.0, + "content": "the assumption that the negative samples are drawn from the true marginal distribution. We briefly", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 505, + 693 + ], + "score": 1.0, + "content": "review this proof to highlight the importance of the negative sampling distribution. Their proof starts", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 691, + 506, + 706 + ], + "spans": [ + { + "bbox": [ + 105, + 691, + 417, + 706 + ], + "score": 1.0, + "content": "from the NWJ lower bound of the KL divergence, namely that for any function", + "type": "text" + }, + { + "bbox": [ + 418, + 692, + 425, + 705 + ], + "score": 0.85, + "content": "\\tilde { f }", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 691, + 506, + 706 + ], + "score": 1.0, + "content": "the following lower", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 703, + 335, + 716 + ], + "spans": [ + { + "bbox": [ + 106, + 703, + 335, + 716 + ], + "score": 1.0, + "content": "bound holds (Nguyen et al., 2010; Nowozin et al., 2016):", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 658, + 506, + 716 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 140, + 717, + 470, + 734 + ], + "lines": [ + { + "bbox": [ + 140, + 717, + 470, + 734 + ], + "spans": [ + { + "bbox": [ + 140, + 717, + 470, + 734 + ], + "score": 0.88, + "content": "I ( X ; Y ) = D _ { K L } ( p ( x , y ) | | p ( x ) p ( y ) ) \\geq \\mathbb { E } _ { p ( x , y ) } [ \\tilde { f } ( x , y ) ] - e ^ { - 1 } \\mathbb { E } _ { p ( x ) p ( y ) } [ e ^ { \\tilde { f } ( x , y ) } ] .", + "type": "interline_equation", + "image_path": "25acdbbb3fe5b324ed4038df7893a441aae4431216503c69fa6a1238d55372a6.jpg" + } + ] + } + ], + "index": 34, + "virtual_lines": [ + { + "bbox": [ + 140, + 717, + 470, + 734 + ], + "spans": [], + "index": 34 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 81, + 506, + 106 + ], + "lines": [ + { + "bbox": [ + 105, + 79, + 508, + 97 + ], + "spans": [ + { + "bbox": [ + 105, + 79, + 161, + 97 + ], + "score": 1.0, + "content": "Suppose that", + "type": "text" + }, + { + "bbox": [ + 161, + 82, + 208, + 95 + ], + "score": 0.93, + "content": "( X _ { i } , Y _ { i } ) _ { i = 1 } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 208, + 79, + 293, + 97 + ], + "score": 1.0, + "content": "are i.i.d. draws from", + "type": "text" + }, + { + "bbox": [ + 294, + 83, + 322, + 94 + ], + "score": 0.93, + "content": "p ( x , y )", + "type": "inline_equation" + }, + { + "bbox": [ + 323, + 79, + 364, + 97 + ], + "score": 1.0, + "content": "and write", + "type": "text" + }, + { + "bbox": [ + 364, + 82, + 477, + 95 + ], + "score": 0.92, + "content": "X _ { 1 : K } = ( X _ { 1 } , X _ { 2 } , \\dots , X _ { K } )", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 79, + 508, + 97 + ], + "score": 1.0, + "content": ". 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In particular, taking", + "type": "text" + }, + { + "bbox": [ + 284, + 160, + 440, + 181 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\tilde { f } ( x _ { 1 : K } , y _ { i } ) = 1 + \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 164, + 470, + 178 + ], + "score": 1.0, + "content": "ef(xj ,yi) yields", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 193, + 507, + 228 + ], + "lines": [ + { + "bbox": [ + 111, + 193, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 111, + 193, + 507, + 228 + ], + "score": 0.93, + "content": "\\textmd { ( X , Y ) \\geq 1 + } \\mathbb { E } _ { p ( x _ { i } , y _ { i } ) \\prod _ { k \\geqslant i } p ( x _ { k } ) } \\left[ \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } \\right] - \\mathbb { E } _ { p ( y _ { i } ) \\prod _ { k } p ( x _ { k } ) } \\left[ \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } \\right] .", + "type": "interline_equation", + "image_path": "827fc1bbe033cd21e55517b744938640b02166889ed26b97ecd5e68b374dac72.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 111, + 193, + 507, + 204.66666666666666 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 111, + 204.66666666666666, + 507, + 216.33333333333331 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 111, + 216.33333333333331, + 507, + 227.99999999999997 + ], + "spans": [], + "index": 7 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 249, + 506, + 284 + ], + "lines": [ + { + "bbox": [ + 106, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 231, + 262 + ], + "score": 1.0, + "content": "This is then averaged over the", + "type": "text" + }, + { + "bbox": [ + 231, + 251, + 242, + 260 + ], + "score": 0.83, + "content": "K", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 250, + 279, + 262 + ], + "score": 1.0, + "content": "samples", + "type": "text" + }, + { + "bbox": [ + 279, + 251, + 289, + 261 + ], + "score": 0.86, + "content": "Y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 250, + 505, + 262 + ], + "score": 1.0, + "content": ", in which case the third term above cancels with the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 227, + 273 + ], + "score": 1.0, + "content": "constant 1 (all occurences of", + "type": "text" + }, + { + "bbox": [ + 228, + 263, + 237, + 273 + ], + "score": 0.86, + "content": "y _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 261, + 418, + 273 + ], + "score": 1.0, + "content": "in the 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338.00000000000006 + ], + "spans": [], + "index": 13 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 347, + 506, + 404 + ], + "lines": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "The point in this proof that makes use of the i.i.d. assumption of the negative samples is in the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 358, + 506, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 141, + 372 + ], + "score": 1.0, + "content": "equality", + "type": "text" + }, + { + "bbox": [ + 142, + 359, + 243, + 371 + ], + "score": 0.92, + "content": "I ( X _ { i } , Y _ { i } ) = I ( X _ { 1 : K } , Y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 243, + 358, + 506, + 372 + ], + "score": 1.0, + "content": ", which allowed us to leverage multiple samples when estimating", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 370, + 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If instead the negative samples are drawn in a dependent fashion", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 380, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 380, + 194, + 394 + ], + "score": 1.0, + "content": "(corresponding to the", + "type": "text" + }, + { + "bbox": [ + 194, + 381, + 227, + 393 + ], + "score": 0.92, + "content": "( X _ { i } , Y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 380, + 454, + 394 + ], + "score": 1.0, + "content": "being drawn identically but not independently), we have", + "type": "text" + }, + { + "bbox": [ + 455, + 381, + 505, + 393 + ], + "score": 0.91, + "content": "I ( X _ { i } , Y _ { i } ) \\leq", + "type": "inline_equation" + } + ], + "index": 17 + }, + { + "bbox": [ + 107, + 391, + 389, + 405 + ], + "spans": [ + { + "bbox": [ + 107, + 392, + 156, + 404 + ], + "score": 0.92, + "content": "I ( X _ { 1 : K } , Y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 391, + 389, + 405 + ], + "score": 1.0, + "content": ", though the remainder of the proof still holds, resulting in", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 16, + "bbox_fs": [ + 105, + 348, + 506, + 405 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 142, + 423, + 468, + 458 + ], + "lines": [ + { + "bbox": [ + 142, + 423, + 468, + 458 + ], + "spans": [ + { + "bbox": [ + 142, + 423, + 468, + 458 + ], + "score": 0.94, + "content": "I ( X , Y ) \\leq \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } I ( X _ { 1 : K } ; Y _ { i } ) \\geq \\mathbb { E } _ { p ( x _ { 1 : K } , y _ { 1 : K } ) } \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\left[ \\log \\frac { e ^ { f ( x _ { i } , y _ { i } ) } } { \\frac { 1 } { K } \\sum _ { j = 1 } ^ { K } e ^ { f ( x _ { j } , y _ { i } ) } } \\right] .", + "type": "interline_equation", + "image_path": "ddf028218118e476dd360f213a65d0eaecbd9f3ba2f90ce6688d40ae085d2206.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 142, + 423, + 468, + 434.6666666666667 + ], + "spans": [], + "index": 19 + }, + { + "bbox": [ + 142, + 434.6666666666667, + 468, + 446.33333333333337 + ], + "spans": [], + "index": 20 + }, + { + "bbox": [ + 142, + 446.33333333333337, + 468, + 458.00000000000006 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 468, + 506, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 466, + 507, + 483 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 199, + 483 + ], + "score": 1.0, + "content": "Therefore the resulting", + "type": "text" + }, + { + "bbox": [ + 199, + 469, + 218, + 480 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 466, + 468, + 483 + ], + "score": 1.0, + "content": "estimator is neither a lower nor an upper bound on the true MI", + "type": "text" + }, + { + "bbox": [ + 468, + 468, + 502, + 480 + ], + "score": 0.92, + "content": "I ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 466, + 507, + 483 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 466, + 507, + 483 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 495, + 430, + 509 + ], + "lines": [ + { + "bbox": [ + 104, + 494, + 430, + 511 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 430, + 511 + ], + "score": 1.0, + "content": "E EXPERIMENT DETAILS: NON-I.I.D. 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The true MI", + "type": "text" + }, + { + "bbox": [ + 398, + 581, + 432, + 592 + ], + "score": 0.92, + "content": "I ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 580, + 506, + 594 + ], + "score": 1.0, + "content": "can be calculated", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 591, + 506, + 605 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 182, + 605 + ], + "score": 1.0, + "content": "analytically since", + "type": "text" + }, + { + "bbox": [ + 182, + 592, + 211, + 604 + ], + "score": 0.91, + "content": "( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 591, + 424, + 605 + ], + "score": 1.0, + "content": "is jointly Gaussian with known covariance matrix", + "type": "text" + }, + { + "bbox": [ + 424, + 592, + 463, + 603 + ], + "score": 0.91, + "content": "\\Sigma _ { Z } + \\Sigma _ { \\epsilon }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 591, + 506, + 605 + ], + "score": 1.0, + "content": ": For two", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 603, + 505, + 615 + ], + "spans": [ + { + "bbox": [ + 106, + 603, + 217, + 614 + ], + "score": 1.0, + "content": "univariate random variables", + "type": "text" + }, + { + "bbox": [ + 218, + 603, + 246, + 615 + ], + "score": 0.92, + "content": "( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 603, + 409, + 614 + ], + "score": 1.0, + "content": "that are jointly Gaussian with covariance", + "type": "text" + }, + { + "bbox": [ + 410, + 603, + 418, + 613 + ], + "score": 0.81, + "content": "\\Sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 418, + 603, + 505, + 614 + ], + "score": 1.0, + "content": "the MI can be written", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 615, + 118, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 118, + 626 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 569, + 506, + 626 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 235, + 621, + 376, + 646 + ], + "lines": [ + { + "bbox": [ + 235, + 621, + 376, + 646 + ], + "spans": [ + { + "bbox": [ + 235, + 621, + 376, + 646 + ], + "score": 0.94, + "content": "I ( X , Y ) = - \\frac { 1 } { 2 } \\log ( 1 - \\frac { \\Sigma _ { 1 2 } \\Sigma _ { 2 1 } } { \\Sigma _ { 1 1 } \\Sigma _ { 2 2 } } ) .", + "type": "interline_equation", + "image_path": "cd27ab7c0798506aa3961c144bebf1b6ef36dc50baae434841d0ee52b2c08ac8.jpg" + } + ] + } + ], + "index": 31, + "virtual_lines": [ + { + "bbox": [ + 235, + 621, + 376, + 646 + ], + "spans": [], + "index": 31 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 648, + 505, + 672 + ], + "lines": [ + { + "bbox": [ + 106, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 282, + 661 + ], + "score": 1.0, + "content": "This can be derived using the decomposition", + "type": "text" + }, + { + "bbox": [ + 282, + 649, + 440, + 661 + ], + "score": 0.93, + "content": "I ( X , Y ) = H ( X ) + H ( Y ) - H ( X , Y )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "and the analytic", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 660, + 284, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 213, + 673 + ], + "score": 1.0, + "content": "expression for the entropy", + "type": "text" + }, + { + "bbox": [ + 213, + 660, + 223, + 670 + ], + "score": 0.84, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 660, + 284, + 673 + ], + "score": 1.0, + "content": "of a Gaussian.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 648, + 505, + 673 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 676, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 676, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 106, + 676, + 352, + 690 + ], + "score": 1.0, + "content": "We compare the same setting trained using i.i.d. sampled pairs", + "type": "text" + }, + { + "bbox": [ + 352, + 677, + 385, + 689 + ], + "score": 0.92, + "content": "( X _ { i } , Y _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 676, + 506, + 690 + ], + "score": 1.0, + "content": "as a baseline. We parametrize", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "the critic as a MLP with 5 hidden layers, each with 10 units and ReLU activations, followed by a", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 216, + 712 + ], + "score": 1.0, + "content": "linear layer and maximize", + "type": "text" + }, + { + "bbox": [ + 217, + 699, + 235, + 710 + ], + "score": 0.9, + "content": "I _ { \\mathrm { N C E } }", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "using these non-i.i.d. samples with batch size 128. 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The key idea of our approach is to learn a rich latent space which effectively propagates program annotations from known questions to novel questions. We do this by formalizing prior work on VQA, called module networks (Andreas, 2016) as discrete, structured, latent variable models on the joint distribution over questions and answers given images, and devise a procedure to train the model effectively. Our results on a dataset of compositional questions about SHAPES (Andreas, 2016) show that our model generates more interpretable programs and obtains better accuracy on VQA in the low-data regime than prior work. + +# 1 INTRODUCTION + +Exciting progress has been made in recent years in deep representation learning (Hochreiter & Schmidhuber, 1997; LeCun et al., 1998), which has led to advances in artificial intelligence tasks such as image recognition (Krizhevsky et al., 2012; He et al., 2015), machine translation (Sutskever et al., 2014), visual question answering (Agrawal et al., 2015), visual dialog (Das et al., 2016), and reinforcement learning (Mnih et al., 2015), etc. While deep neural networks achieve such impressive performances, many aspects of human cognition such as compositional generalization and reasoning are harder to model with state-of-the art deep learning approaches (Battaglia et al., 2018; Barrett et al., 2018). + +Symbolic approaches (Newell & Simon, 1976) on the other hand provide strong compositional and reasoning capabilities which are challenging to model with neural networks (Lake et al., 2017). Consequently, a rich line of work in neuro-symbolic processing sought to build AI systems with both strong learning and reasoning capabilities (Valiant, 2003; Evans et al., 2018b; Bader & Hitzler, 2005). + +As we tackle higher-level tasks which involve reasoning, such as visual question answering (Agrawal et al., 2015), planning (Andreas et al., 2016b; Gregor et al., 2016) etc., it is natural to desire the ability to provide instructions to models to guide them. Symbolic instructions, by their very nature are easier to specify and more interpretable than specifying the parameters of a neural network. For such high-level tasks, a sensible approach is to specify “what” to do using symbols and learn how to do the task using modern representation learning techniques. + +For the example shown in Figure 1, given a question “Is a square to the left of a green shape?”, one can ask the model to reason about the answer by first applying a find[green] operator, then find[left], then And the result together with a find[square] operator in order to predict the answer, in other words specifying “what” are the computations we desire to be executed, in the form of a “program”. (Johnson et al., 2017; Hu et al., 2017; Andreas et al., 2016a). The network can then learn “how” to execute such a program from data using deep representation learning (Johnson et al., 2017; Hu et al., 2017; Andreas et al., 2016a). + +This paper addresses a very natural desiderata for such neuro-symbolic models, in the context of visual question answering – can we retain the intepretability of “what the network did” expressed in terms of the syntax and lexicon that we are interested in, while specifying minimal teaching examples of “what to do” given an input question? + +![](images/2aa1b4dd59dc5a8e445ac2166becf0f18b8fb3f6685e2df672b5e879925a26f6.jpg) +Figure 1: An illustration of neural module networks for visual question answering. Given a question, the program generator produces a program in prefix notation which is used to construct a neural network from the specified module networks. This network operates on the image and intermediate attention maps to answer the question. + +We propose to approach this problem with a treatment of programs $\mathbf { z }$ as a latent variable, embedded into a generative model of questions $\mathbf { x }$ and answers a given images i as visualized in Figure 2. This model class has certain desirable properties: first, a proper treatment of $\mathbf { z }$ as a stochastic latent variable is helpful especially in the presence of a limited number of expert human instructions, as we model the uncertainty associated with $\mathbf { z }$ explicitly, leading to more interepretable latent spaces (Section 5); second, the model makes intuitive independence assumptions with regards to the question answering task – conditioned on the program $\mathbf { z }$ , questions $\mathbf { x }$ are independent of the answers a for an image, meaning that the program must be a sufficient statistic for predicting the answer given a question1. Third, we parameterize the $p ( \mathbf { a } | \mathbf { i } , \mathbf { z } )$ term, which predicts an answer given an image and a program using a popular, related class of question answering models called neural module networks (NMNs) (Hu et al., 2017; Johnson et al., 2017; Andreas et al., 2016a) which previous work has shown leads to better fit between the program (or “what the model is doing”) and the execution (or “how it is doing what it is doing”). Given the close ties of our model to NMNs, we will refer to our model as variational neural module networks (V-NMN) for the rest of the paper. + +Apart from our modeling contribution of a probabilistic latent variable, neuro-symbolic model, our key technical contribution is to show how to train this model in the context of visual question answering. Optimizing such generative models with structured discrete latent variables is an intractable problem to solve in genereal – for this particular case, we formulate training into stages of grounding questions into programs (question coding), learning how to execute them and then with those initializations train the full variational objective. We demonstrate that this stage-wise optimization allows us to successfully learn a probabilistic neural-symbolic model that can answer compositional questions about shapes in an image (Andreas et al., 2016a). + +While the V-NMN model is instantiated for the specific problem of visual question answering, we believe the tools and techniques we develop are more generally applicable to probabilistic, structured, discrete, interpretable latent spaces for interpretable neuro-symbolic models for planning, instruction following, grounding referring expressions etc. + +We benchmark our model on the SHAPES (Andreas et al., 2016a) dataset with compositionally novel questions and show that our model is able to provide interpretable explanations of “what it is doing” (measured by program correctness) for unseen questions even in the setting where it is given as few as $\approx 2 0$ paired examples of programs and corresponding questions from the dataset, while also getting to high question answering accuracy. We find that the proposed approach outperforms a state of the art neural-symbolic model designed for VQA from Johnson et al. (2017), as well as deterministic ablations of our model. + +# 2 A PROBABILISTIC NEURAL-SYMBOLIC MODEL FOR VQA + +For a standard neural-symbolic NMN approach to question answering (Johnson et al., 2017), we are given an input image i and a question ${ \bf x } = ( x _ { 1 } , \cdots , x _ { N } )$ as a sequence of words $x _ { i }$ each encoded using a vocabulary $\mathcal { V } _ { \mathbf { x } }$ , and must produce answers a by planning and executing a series of modular computations $( { \bf z } = ( z _ { 1 } , \cdot \cdot \cdot , z _ { N } ) ) - i . e$ . a program of operations $( z _ { i } \in \mathcal { V } _ { \mathbf { z } } )$ ) to be executed in order to answer the question. Note that $\mathbf { z }$ here is the prefix serialization of the program and programs themselves often have tree-structured topologies. We denote the dataset of $( \mathbf { i } , \mathbf { x } , \mathbf { a } )$ triplets as $\mathcal { D }$ . Additionally, we are provided programs $\mathbf { z }$ for some input questions $( \mathbf { x } \in S )$ ), which specify the notion of grounding desired by an end user. Our goal is to provide faithful explanations on unseen questions (along with correct answers) consistent with the specified grounding. + +![](images/22b2bbae87156a9dbb5b0416e0b6f7d0dca7b729f9eb2243043afb4d763daba4.jpg) +Figure 2: An illustration of the graphical model used for neural module networks (NMNs). + +We accomplish this by modeling $\mathbf { z }$ as a latent variable that underlies both question and answer generation given an input image. Intuitively, we expect this will enable us to learn a more meaningful latent representation for questions, as modeling uncertainity should help better handle the potentially underspecified mapping between questions and programs provided by the user. As such, our model factorizes $p ( \mathbf { x } , \mathbf { z } , \bar { \mathbf { a } } | \bar { \mathbf { i } } ) \stackrel { - } { = } p ( \mathbf { x } | \mathbf { z } ) p ( \mathbf { a } | \mathbf { i } , \mathbf { z } ) p ( \mathbf { z } )$ as in the graphical model shown in Figure 2, where the prior $p ( \mathbf { z } )$ is learnt by training a neural sequence model on valid strings simulated from known syntax. Conceptually, this factorization captures the notion that many differently-worded questions are reducible to identical underlying semantics (captured by a program), making the program space, in a sense, a compression of the question space. As mentioned in Section 1 the model also captures the independence of questions from answers for an image given programs, which is also an intuitive desiderata for interpretable neural-symbolic models2. + +Given the proposed model shown in Figure 2, we must address two tasks to put the model to use for VQA – learning model parameters, and performing inference for a specific query. + +Learning. We will always condition on images i but may not observe underlying programs $\mathbf { z }$ , so our per-data point training objective is the log-marginal-conditional-likelihood: $\log p ( \mathbf { x } , \mathbf { a } \mid \mathbf { i } )$ . Since computing this objective involves an intractable summation over latent programs $\begin{array} { r } { ( \log \sum _ { \mathbf { z } } p ( \mathbf { x } _ { j } , \mathbf { z } , \mathbf { a } _ { j } \ | } \end{array}$ $\mathbf { i } _ { j }$ )), we use a variational approximation (Jordan et al., 1999). Specifically, let + +- $q _ { \phi } ( \mathbf { z } \mid \mathbf { x } )$ denote an amortized inference neural network parameterized by $\phi$ ,which instantiates a sequence to sequence LSTM (Hochreiter & Schmidhuber, 1997) model +- $p _ { \theta _ { \mathbf { z } } } ( \mathbf { a } \mid \mathbf { z } , \mathbf { i } )$ be a NMN parametrized by $\boldsymbol { \theta } _ { \mathbf { z } }$ run on image i with program $\mathbf { z }$ , +- $p _ { \theta } ( \mathbf { x } \mid \mathbf { z } )$ be a recurrent network (sequence to sequence model) parametrized by $\theta$ that maps programs to questions, and +- $\mathcal { D } ^ { ( z ) }$ be a prior over programs learned from data or based on program syntax directly , parameterized as an LSTM (Hochreiter & Schmidhuber, 1997) sequence model + +Then, for datapoints without program supervision, the objective is: + +$$ +\mathrm { e l b o } ( { \mathbf { a } } , { \mathbf { x } } | { \mathbf { i } } ) = E _ { z \sim q _ { \phi } ( { \mathbf { z } } | \mathbf { x } ) } [ \log p _ { \theta _ { \mathbf { z } } } ( { \mathbf { a } } | { \mathbf { z } } , { \mathbf { i } } ) + \log p _ { \theta } ( { \mathbf { x } } | { \mathbf { z } } ) ] - \beta { \mathrm { K L } } ( q _ { \phi } ( { \mathbf { z } } | \mathbf { x } ) , p ( \mathbf { z } ) ) +$$ + +It is easy to show that $\mathrm { e l b o } ( \mathbf { x } , \mathbf { a } | \mathbf { i } ) \leq \log p ( \mathbf { x } , \mathbf { a } | \mathbf { i } )$ for $\beta \geq 1$ ; as such, maximizing this objective is a form of approximate maximum likelihood training for distribution parameters $\theta _ { \mathbf { z } } , \theta$ , and $\phi$ . + +Additionally, for data points with annotations, we add the likelihood of the program given the question $\log r _ { \phi } ( \mathbf { z } | \mathbf { x } )$ to the objective. Note that $r$ shares parameters $\phi$ with the inference network $q$ . So our overall objective becomes: + +$$ +\mathrm { e l b o } ( \mathbf { a } , \mathbf { x } | \mathbf { i } ) + \alpha \log r ( \mathbf { z } | \mathbf { x } ) \cdot \left[ [ \mathbf { x } \in \mathcal { S } ] \right] +$$ + +where the indicator function $[ [ \mathbf { x } \in S ] ]$ is 1 if $\mathbf { x }$ is in the supervised set and 0 otherwise and $\alpha$ is a positive scaling factor. The goal is to learn the desired association of questions with programs from the second term, and to propagate them to other questions using the first term. + +Inference. At test time, we can answer questions for images by computing $p ( \mathbf { a } | \mathbf { i } , \mathbf { x } ) \ =$ $\begin{array} { r } { \sum _ { z } p ( \mathbf { a } | \mathbf { i } , \mathbf { z } ) p ( \mathbf { z } | \mathbf { x } ) } \end{array}$ using an approximation $p _ { q } ( \mathbf { \bar { a } } | \mathbf { i } , \mathbf { x } ) \approx E _ { \mathbf { z } _ { i } \sim q ( \mathbf { z } | \mathbf { x } ) } p ( \mathbf { a } | \mathbf { i } , \mathbf { z } _ { i } )$ . + +Challenges. While the elbo term in Equation (1) appears similar to a standard ELBO for joint latent variable models (Vedantam et al., 2018; Wang et al., 2016b; Suzuki et al., 2017), there are a few important departures: + +- Firstly, the latent variable $\mathbf { z }$ is not a Gaussian or categorical random variable (which admit straightforward reparameterizations (Maddison et al., 2017; Jang et al., 2016)) but rather a sequence of discrete tokens, which complicates the optimization problem in Equation (1). Secondly, the conditional log-likelihood for a is parameterized by a NMN that is dynamically assembled based on sample programs drawn from $q ( \mathbf { z } | \mathbf { x } )$ . Depending upon the uncertainty of the program posterior, program tokens may not be consistently grounded across examples, making learning the appropriate computation for the corresponding neural modules difficult. Lastly, our observations x are explained by powerful “autoregressive” sequence models. Learning meaningful latent variable models that are informative of observations for such models is difficult (c.f. (Bowman et al., 2016; Chen et al., 2016; Alemi et al., 2018)) as powerful decoder models can explain the data using stochasticity from sampling (referred to as autodecoding in Alemi et al. (2018)). Consequently, it also becomes hard to learn meaningful NMN parameters from the previous step. + +In the following sections, we present a training regime and adjustments to the objective in Equation (1) that make optimization more tractable and enable training NMNs with significantly reduced questionaligned program supervision. While instantiated for question answering, it is not difficult to conceive versions of this regime for other problems relevant to probabilistic neural-symbolic models. + +# 2.1 OPTIMIZING THE ELBO FOR V-NMN + +Rather than approaching the joint objective directly, we break ELBO training into three stages: + +1. Question coding, where we learn a latent ‘code’ for questions in the latent space of programs, 2. Module training, where we learn the ‘execution’ of the programs from the first stage of training, grounding tokens in programs into the operations they perform on images; and 3. Joint training, where we train both stages jointly given the ‘warm-starts’ from the other phases. + +Question Coding. The goal of this stage is to learn a code for questions in the latent space of programs $- i . e$ . an informative mapping from questions to programs. Given some set of question-aligned programs, we find this coding process effectively propagates these supervised groundings to unsupervised questions, which enable interpretable explanations $/$ groundings of queries to explanations of “what” the model is doing, which is our primary goal. + +During question coding, we drop the $p _ { \theta _ { z } } ( \mathbf { a } | \mathbf { z } , \mathbf { i } )$ term from Eq. 1, yielding the objective + +$$ +J ( \mathbf { x } ; \phi , \theta ) = E _ { z \sim q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } [ \log p _ { \theta } ( \mathbf { x } | \mathbf { z } ) ] - \beta \mathrm { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) , p ( \mathbf { z } ) ) + \alpha \log r _ { \phi } ( \mathbf { z } | \mathbf { x } ) \cdot [ [ \mathbf { x } \in \mathcal { S } ] +$$ + +Recent theory from Alemi et al. (2018) prescribes changes to the ELBO, setting $\beta < 1$ as a recipe to learn representations which avoid learning degenerate latent representations in the presence of powerful decoders. Essentially, Alemi et al. (2018) identify that upto a constant factor, the negative log-marignal likelihood term $D { = } { - } E _ { z { \sim } q _ { \phi } ( { \bf z } | { \bf x } ) } \big [ \log p _ { \theta } ( { \bf x } | { \bf z } ) \big ]$ , and the kl diveregence term $R = \mathrm { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) , p ( \mathbf { z } ) )$ bound the mutual information between the data $\mathbf { x }$ and the latent variable $\mathbf { z }$ as follows: + +$$ +H - D \leq I ( \mathbf { x } , \mathbf { z } ) \leq R +$$ + +where $H$ is the entropy of the data, which is a constant. One can immediately notice that the standard e $\mathrm { b o } = - D - R$ . This means that for the same value of the elbo one can get different models which make drastically different usage of the latent variable (based on the achieved values of $D$ and $R$ ). Thus, one way to achieve a desired behavior (of say high mutual information $I ( \mathbf { x } , \mathbf { z } )$ ), is to set $\beta$ to a lower value than the standard elbo (c.f. Eqn. 6 in Alemi et al. (2018).) This aligns with our goal in question coding, and so we use $\beta < 1$ . As this no longer a valid variational lower bound, denote our objective as $J ( \mathbf { x } ; \phi , \theta )$ from now on. + +We optimize the above objective using the score function estimator (Glynn, 1990), also referred to as REINFORCE (Williams, 1992) applied to the first term, with a moving average baseline.Finally, while the KL term (in the reward) encourages the model to generate high-likelihood plausible sequences, we also penalize the model for producing syntactically invalid programs. + +Although gradients for the KL term in Equation (2) can be computed by decomposing the KL for the sequence into per-timestep monte-carlo gradients, we found that having one term with per-timestep gradients and another that only receives sequence-level rewards (the likelihood term) skewed the model towards minimizing the KL, leading to auto-decoding solutions. Thus, we apply REINFORCE on the entire sequence and compute rewards using the full monte carlo gradient,i.e., we do not optimize the KL term separately, but instead maximize: + +$$ +E _ { \mathbf { z } \sim q _ { \phi } ( \mathbf { z } | \hat { \mathbf { x } } ) } \left[ \log \left[ \frac { p _ { \theta } ( \hat { \mathbf { x } } | \mathbf { z } ) p ( \mathbf { z } ) ^ { \beta } } { q _ { \phi } ( \mathbf { z } | \hat { \mathbf { x } } ) ^ { \beta } } \right] \right] + \alpha \log r _ { \phi } ( \hat { \mathbf { z } } | \hat { \mathbf { x } } ) \cdot \left[ \left[ \mathbf { x } \in \mathcal { S } \right] \right] +$$ + +where $\hat { \mathbf { x } } , \hat { \mathbf { z } }$ are the questions and programs we observe in the dataset. + +For comparison, the semi-supervised approach in Johnson et al. (2017) can be seen as optimizing the last term on supervised training data, while our question coding stage optimizes the elbo in addition to the last term. Our results in Section 5 show that the proposed question coding formulation significantly improves performance relative to Johnson et al. (2017) in terms of explaining “what” the model is doing on test programs. + +Module Training. Having obtained a code for questions in the latent space, we train the parameters of the NMN for question answering. The network modules are trained to perform their corresponding tasks. For example, if the question coding learned to associate the program token $v$ with the question phrase ‘to the left’, then the goal for this second stage is to train the module corresponding to $v$ to actually perform this operation (e.g. transform[left]) when executed on images. + +We do this by optimizing the joint elb ${ \bf \alpha } ) ( { \bf x } , { \bf a } | \bf { i } )$ in (1) with respect to the parameters $\theta _ { \mathbf { z } }$ of the neural modules. We write this objective for each instance as + +$$ +J ( \mathbf { a } \mid \mathbf { x } , \mathbf { i } ; \theta _ { \mathbf { z } } ) = E _ { z \sim q ( \mathbf { z } \mid \mathbf { x } ) } [ \log p _ { \theta _ { z } } ( \mathbf { a } \mid \mathbf { i } , \mathbf { z } ) ] +$$ + +Notably, while we write this objective as a function of all neural modular network parameters $\theta _ { \mathbf { z } }$ , for a given instance, only those modules present in the program $z \sim q ( \mathbf { z } | \mathbf { x } )$ are updated. In practice, we take approx. argmax decoded programs from $q _ { \phi } ( { \bf z } | { \bf x } )$ to simplify training, but perform sampling during joint training in the next stage. To be more specific, we run beam search (Vijayakumar et al., 2018) with a beam size of 1 to do this. Intuitively, we think of this as a bias $\nu s$ . variance tradeoff, doing argmax enables us to trade off bias for high-variance from a posterior which (in the pathological case) could match the prior from the first stage of training. This is also consistent with standard practice for doing inference in sequence models, where beam search is the standard inference tool (and sampling often has issues due to teacher forcing based MLE training, which is how we do semi-supervised learning) (Bengio et al., 2015). + +Joint Training. Having trained the question code and the neural module network parameters, we tune all terms jointly, with the following objective, which adds scaling factors $\alpha$ , $\beta$ and $\gamma$ , and an extra semi-supervised learning term relative to Equation (1): + +$$ +J ( \mathbf { x } , \mathbf { a } | \mathbf { i } ) = E _ { z \sim q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } \big [ \gamma \log p _ { \theta _ { z } } ( \mathbf { a } | \mathbf { z } , \mathbf { i } ) + \log p _ { \theta } ( \mathbf { x } | \mathbf { z } ) \big ] - \beta \mathrm { K L } \big ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) , p ( \mathbf { z } ) \big ) + \alpha \log r _ { \phi } ( \mathbf { z } | \mathbf { x } ) +$$ + +The gamma factor $\gamma$ controls the scaling of the likelihood term for the answers relative to the question (which has more bits of information), similar scaling factors have been for joint models in Vedantam et al. (2018). Similar to Equation (4), we optimize the full monte carlo gradient for the above objective as well. The full training procedure is shown in Algorithm 1 below. + +# 3 RELATED WORK + +We first explain other related work in neuro-symbolic models, the key differences between our formulation and closely related NMN work (Johnson et al., 2017; Hu et al., 2017; 2018), and then connect our work to other themes and directions of research in literature. + +Neural-symbolic models Valiant (2003) identifies neuro-symbolic models, which are able to reason as well as learn efficiently as one of the key problems in computer science. A rich literature of work exists in this area, around two major threads: firstly, there is a line of work in building hybrid models + +# Algorithm 1 V-NMN Training + +1: Given: $p ( \mathbf { z } )$ , Initialize $q _ { \phi } ( { \bf z } | { \bf x } )$ , $p _ { \theta _ { z } } ( \mathbf { a } | \mathbf { z } , \mathbf { i } )$ , $p _ { \theta } ( \mathbf { x } | \mathbf { z } )$ , moving average baseline $B = 0$ , $\beta = 0 , \tau , \alpha =$ 100.0, $\delta = 0 . 9 9$ . +2: Question Coding: Given $\hat { \mathbf { x } } , \hat { \mathbf { z } } \sim p _ { d a t a } ( \mathbf { x } , \mathbf { z } )$ , sample ${ \mathbf z } _ { \mathbf i } \sim q _ { \phi } ( { \mathbf z } | \hat { \mathbf x } )$ [Repeat until convergence] +3: $\nabla _ { \phi } ( R - B ) \log g _ { \phi } ( z _ { i } | x ) + \alpha \nabla _ { \phi } \log r _ { \phi } ( z | x ) ;$ where $R = \log \left[ \frac { p _ { \theta } ( \hat { \mathbf { x } } | \mathbf { z } _ { i } ) p ( \mathbf { z } _ { i } ) ^ { \beta } } { q _ { \phi } ( \mathbf { z } _ { i } | \hat { \mathbf { x } } ) ^ { \beta } } \right]$ +4: $\nabla _ { \boldsymbol { \theta } } \gets \nabla _ { \boldsymbol { \theta } } \log p _ { \boldsymbol { \theta } } \bigl ( \hat { \mathbf { x } } | \mathbf { z } _ { i } \bigr )$ +5: $B \gets B + ( 1 - \delta ) ( R - B ) ; \nabla _ { \theta } \gets \nabla _ { \theta } \log p ( \hat { \mathbf { x } } | \mathbf { z } _ { i } )$ +6: Question Coding: Given xˆ $, \hat { \mathbf { z } } \sim p _ { d a t a } ( \mathbf { x } , \mathbf { z } )$ , sample ${ \mathbf z } _ { \mathbf i } \sim q _ { \phi } ( { \mathbf z } | \hat { \mathbf x } )$ [Repeat until convergence] +7: Module Training: Given $\begin{array} { l l l } { \hat { \mathbf { x } } , \hat { \mathbf { a } } , \hat { \mathbf { i } } } & { \sim } & { p _ { d a t a } ( \mathbf { x } , \mathbf { a } , \mathbf { i } ) } \end{array}$ , $\begin{array} { r l } { \mathbf { z } _ { i } } & { { } = } \end{array}$ argmax $q ( \mathbf { z } | \hat { \mathbf { x } } )$ [Repeat until convergence] +8: $\nabla _ { \theta _ { z } } \gets \nabla _ { \theta _ { z } } \log p ( \hat { \mathbf { a } } | \mathbf { z } _ { i } , \hat { \mathbf { i } } )$ +9: Joint Training: Given $\begin{array} { r } { \hat { \mathbf { x } } , \hat { \mathbf { a } } , \hat { \mathbf { i } } , \hat { \mathbf { z } } \sim p _ { d a t a } ( \mathbf { x } , \mathbf { a } , \mathbf { i } , \mathbf { z } ) , \mathbf { z } _ { i } \sim q ( \mathbf { z } | \hat { \mathbf { x } } ) , \gamma ; } \end{array}$ Initialize $B = 0$ , [Repeat until convergence] +10: $\begin{array} { r l } & { \nabla _ { \phi } \gets ( R - B ) \nabla _ { \phi } \log g _ { \phi } ( z _ { i } | \hat { x } ) + \alpha \nabla _ { \phi } \log r _ { \phi } ( \hat { z } | \hat { x } ) ; \mathbf { w h e r e ~ } \mathbb { R } = \log \left[ \frac { p _ { \phi } ( \hat { x } | z _ { i } ) p ( \hat { x } | z _ { i } , \hat { \mathbf { H } } ^ { \gamma } p ( z _ { i } ) ) ^ { \phi } } { q _ { \phi } ( z _ { i } | x ) ^ { \phi } } \right] } \\ & { B \gets B + ( 1 - \delta ) ( R - B ) ; \nabla _ { \theta } \gets \nabla _ { \theta } \log p ( \hat { \mathbf { x } } | \mathbf { z } _ { i } ) ; \nabla _ { \theta _ { z } } \gets \nabla _ { \theta _ { z } } \log p ( \hat { \mathbf { a } } | \mathbf { z } _ { i } , \hat { \mathbf { i } } ) } \end{array}$ +11: + +which are hybrids of symbolic reasoning and neural networks (Bader & Hitzler, 2005; Holldobler & ¨ Kalinke), the second thread looks at studying if neural networks can indeed do symbolic reasoning and building models and architectures to achieve this for specific cases (Evans et al., 2018a; Rocktaschel ¨ et al., 2015). Our work is distinct in terms of being motivated by interpretability in the low-sample regime rather than getting connectionist/neural networks to explicitly do symbolic manipulations or building hybrid models that do logic. + +More recently, Yin et al. (2018), propose a probabilistic latent variable model of structured programs which explain textual observations for tasks like parsing, i.e., the model from Yin et al. (2018) does not capture “how” to execute the programs that it generates. Our model is more general in the sense that we have a second modality which is the output of what gets executed when the program runs, and we model both jointly in our model, by learning how to execute the program. Work from Yi et al. (2018) also shares the same goal of doing question answering with limited question program supervision while preserving the semantics of human programs. While they to approach this problem by simplifying the $p ( \mathbf { a } | \mathbf { i } , \mathbf { z } )$ mapping (in context of our model) by first converting the image into a table, we approach this goal by modeling a stochastic latent space. Thus the two approaches are orthogonal, as one can use our probabilistic latent space in conjunction with their model as well to better share statistics between question-program pairs. Overall, compared to these works, our work addresses probabilistic neural-symbolic learning in a more general setting, by learning how to execute programs (on raw-pixels), as well as handling parsing of questions in an end to end manner. + +Connection to prior work on NMNs. Our V-NMN model is inspired in its parameterization of program to answer mapping from prior work on visual question answering using neural module networks such as Johnson et al. (2017); Hu et al. (2017; 2018). Distinct from these previous approaches, our explicit goal is to preserve the semantics of how a human specifies a task to be broken down in terms of instructions, while previous work was motivated by building inductive biases for reasoning (which turn out to be useful for our model as well). Given this goal, we construct a latent variable model that in some sense embeds previous work. The works from Hu et al. (2017; 2018) are more distant from our work, as they use continuous attention modules, which do not capture independence assumptions of the program being a sufficient statistic for answering the question (which makes them harder to specify completely for a human, as they would have to hallucinate attention parameters). However, apart from our focus on preserving the lexicon/specification given by a human, both approaches share the same high level goals of providing interpretable question answering with minimal supervision. + +Beyond Gaussian Latents in Amortized Variational Inference. There has been exciting progress in developing lower-variance estimators for discrete generative models (Tucker et al., 2017; Mnih & Gregor, 2014; Titsias & Lazaro-Gredilla, 2015; Gu & Ling, 2015). In particular, Maddison ´ et al. (2017); Jang et al. (2017) propose the use of the concrete distribution on the probability simplex for providing biased but low variance estimators for discrete latent variable models by considering a reparametrization of samples (which asymptotically approach those from a categorical distribution). The contribution in our work is orthogonal to these approaches, how one can apply such parameterizations to neuro-symbolic models. More relevant to our work is Miao & Blunsom (2016) which proposes a discrete sequential latent variable model for sentence summarization, with summaries as the latent variable. In contrast to their model, ours is a joint model where the latent space consists of programs, not summaries, which have to explain questions and answers to images. + +Joint Variational Autoencoder Models. Vedantam et al. (2018) propose a joint model for images and attributes for “visual imagination”, Suzuki et al. (2017) propose a closely related model and show that one can use the model for conditional generation, while Wang et al. (2016a) propose a deep variational canonical correlation analysis model which learns a joint space between two modalities for denoising and representation learning. In contrast to these approaches, our joint model assumes a discrete structured latent variable (instead of a Gaussian), does semi-supervised training, which is possible for us, but not possible when there are no observations from the latent available for supervision, and is targeted towards neural module networks and question answering. + +Program Induction and Neural Architecture Search. In program induction one is typically interested in learning to write programs given specifications of input-output pairs (Reed & de Freitas, 2015; Kalyan et al., 2018), and optionally language (Neelakantan et al., 2015; Guu et al., 2017) or vision (Gaunt et al., 2017). The key difference between program induction and our problem is that program induction assumes the tokens/instructions in the language are grounded to the executor, i.e., it is assumed that once a valid program is generated there is a black-box execution engine that can execute it, while we learn the execution engine from scratch, and learn the grounding from program symbols to parameters of a neural network for execution. Neural architecture search (Zoph & Le, 2016) is another closely related problem where one is given input output examples for a machine learning task, and the goal is to find a “neural program” that does well on the task. In contrast, ours can be seen as the problem of inducing programs conditioned on particular inputs (questions), where the number of samples seen per question is significantly lower than the number of samples one would observe for a machine learning task of interest. + +# 4 EXPERIMENTS + +Dataset. We perform our experiments on the SHAPES dataset (Andreas et al., 2016a). This dataset has been used in previous works (Andreas et al., 2016a; Hu et al., 2017) on visual question answering using neural module networks, and is intended to be a test for compositional generalization. The dataset has 244 unique binary questions (i.e. having yes/no answers) and 15,616 images in total (Andreas et al., 2016a). The dataset also has annotated programs for each of the questions. We use train, val, and test splits of 13,568, 1,024, and 1,024 $( { \bf x } , { \bf z } , \mathbf { i } , { \bf a } )$ triplets respectively. No questions are repeated across these splits. Further, each question is asked for 64 different images, ensuring that a question can be answered correctly across all these instances only if it is processing the image in a reasonable way (as dictated by the underlying program), as opposed to leveraging any image-question correlations or biases that are unlikely to generalize across images. + +Thus, the dataset explicitly tests for compositional generalization and “reasoning” in the task of visual question answering. The longest questions in the dataset are of length 11 and shortest are of length 4, while the longest programs are of length 6 and shortest programs are of length 4. The size of the question vocabulary $\mathcal { V } _ { \mathbf { x } }$ is 14 and the program vocabulary $\nu _ { \mathbf { z } }$ is 12. + +Training. To simulate a data-sparse regime, we restrict the set of question-aligned programs to 5, 10, 15, or $20 \%$ of unique questions – such that even at the highest level of supervision, programs for $80 \%$ of unique question have never been seen during training. Unless specified explicitly, we train our program prior using a set of 1848 unique programs simulated from the syntax (more details of this can be seen in the appendix). In general, it might not be possible to exhaustively enumerate the space of all valid strings to learn the prior. Thus, we also evaluate the effect of training the prior with varying-sized subsets of the 212 programs present in the training set (without using any corresponding questions or answers) to check how robust overall performance is to the choice of the prior (in the appendix). + +We find sequence-level policy-gradient based training in this setting can be highly variable – with some runs failing spectacularly while others succeed – for both our method and to some extent NMN (Johnson et al., 2017). For fairness and to reduce the impact of stochasticity, we perform the following stage-wise training regime for all models. First, the question-coding stage is run for 5 different random seeds, producing multiple program predictors. Programs taken from the model that performs the best on the validation set are used to train 10 runs of the module training stage. Finally, the best of these module networks (based on val) is trained jointly with the program predictor under settings of $\gamma \in \{ 1 , 1 0 , 1 0 0 \}$ and the max (on val) is chosen. We report mean and variance accuracies on test from this training regime over four random sets of supervisory programs. Our evaluation metrics are as follows: a) question-coding uses reconstruction accuracy (of getting the entire question right) and program prediction accuracy (of matching the ground truth programs) on the training set, since the second stage, module training will use inferences from the first stage (on the training set to train modules), b) module training and joint training: visual question answering accuracy, by counting matches to ground truth answers. + +Implementation Details: Following Johnson et al. (2017) and Hu et al. (2017), we treat the program $\mathbf { z } = \{ z _ { 1 } , \cdots , z _ { K } \}$ as a variable length sequence of tokens with an end of sequence token, which corresponds to the prefix traversal of an execution tree. Given this representational choice, we learn the prior $p ( \mathbf { z } )$ using maximum likelihood training on simulated samples from known syntax of programs. An alternative, interesting approach to try would be to use something like Janz et al. (2017), who acquire a generative model more directly from validity annotations. The $\log p ( \mathbf { a } | \mathbf { z } , \mathbf { i } )$ term in Equation (1) is parameterized by an NMN that is dynamically assembled given the program string, while the generative model $p ( \mathbf { x } | \mathbf { z } )$ , and inference $( q ( \mathbf { z } | \mathbf { x } ) )$ terms are both sequence to sequence models. Our sequence to sequence models are based on LSTM (Hochreiter & Schmidhuber, 1997) cells. For the NMN programs, we use syntax similar to that used by Hu et al. (2017), but we replace their question-attention neural modules with “regular” modules. For clear comparison we adapt the objectives from the baseline approach, Johnson et al. (2017), to our representational choices. More details can be found in appendix. + +# 5 RESULTS + +We focus on evaluating approaches by varying the fraction of question-aligned programs (which we denote $\% \mathbf { x } \mathbf { z }$ ) in Table 1. To put the numbers in context, a baseline $\mathrm { L S T M + }$ image model, which does not use module networks, gets an accuracy of $6 3 . 0 \%$ on Test (see Andreas et al. (2016a); Table. 2). This indicates that SHAPES has highly compositional questions which is challenging to model via conventional methods for Visual Question Answering (Agrawal et al., 2015). Overall, we make the following observations: + +- Our V-NMN approach consistently improves performance in data-sparse regimes. While both methods tend to improve with greater program supervision, V-NMN quickly outpaces NMN (Johnson et al., 2017), achieving test accuracies over $20 \mathrm { - } 3 5 \%$ points higher for ${ > } 5 \%$ program supervision. Notably, both methods perform similarly poorly on the test set given only $5 \%$ program supervision, suggesting this may be too few examples to learn compositional reasoning. Our question coding stage greatly improves initial program prediction. Our V-NMN approach to question coding gets to approximately double the question coding (reconstruction) accuracy as the NMN approach (col 1, question coding). This means that it effectively propagates groundings from question-aligned programs during the coding phase. Consequently, we also see improved VQA performance after the module and joint training stages which are based on predicted programs. Question reconstruction is well-correlated with overall accuracy. Despite the program space being trained directly as a latent code for questions, we find question reconstruction accuracy rises with the amount of supervision – confirming that inference in a space of discrete structured latent variables is difficult to do without some supervision to guide the mapping. + +Figure 3 shows sample programs for each model. We see the limited supervision negatively affects NMN program prediction, with the $5 \%$ model resorting to simple Find[X] Answer structures. Interestingly, we find that the mistakes made by the V-NMN model, e.g., green in $5 \%$ supervision (top-right) are also made when reconstructing the question (also substituting green for blue). Further, when the token does get corrected to blue, the question also eventually gets reconstructed (partially correctly $( 1 0 \% )$ and then fully $( 1 5 \% ) )$ , and the program produces the correct answer. This indicates that there is high fidelity between the learnt question space, the answer space and the latent space. + +Effect of the Program Prior $\mathbf { p } ( \mathbf { z } )$ . Next, we explore the impact of regularizing the program posterior to be close to the prior $p ( \mathbf { z } )$ , for different choices of the prior. Firstly, we disable the KL-divergence term by setting $\beta = 0$ in Equation (2) recovering a deterministic version of our model. Compared to our full model at $10 \%$ supervision, the performance on question coding drops to $1 2 . 7 \%$ program + +
%x←zValidation During Training StagesTest Accuracy (VQA Accuracy)
[I] Question Coding (Reconstruction) ）(Program Prediction)[II] Module Training (VQA Accuracy)[] Joint Training (VQA Accuracy)
NMN (Johnson et a., 2017) V-NMN(Ours)5- 54.86±6.759.28±1.91 17.90±3.4861.56±3.59 63.21±3.1263.08±0.78 66.61±1.5860.06±3.88 61.40±4.41
NMN (Johnsonet a.,017) V-NMN(Ours)10= 87.52±6.4524.30±2.39 65.45±11.8860.31±3.37 81.34±8.6166.51±4.10 92.85±4.8961.99±0.96 88.01±6.45
NMN (Johnson et al.,2017) V-NMN(Ours)15= 99.66±0.1947.67±5.02 86.23±3.2969.47±9.87 93.24±3.1762.43±0.49 97.97±1.2061.32±2.36 94.63±2.65
NMN (Johnson et al.,2017) V-NMN(Ours)201 99.88±0.2058.37±3.30 90.95±0.9066.17±7.02 91.12±2.6780.68±18.00 99.13±0.7278.59±19.27 95.63±1.31
+ +Table 1: Results in $\%$ , on the SHAPES dataset with varying amounts of question-program supervision $( \% \mathbf { x } \mathbf { z } )$ , $\beta = 0 . 1$ . + +prediction accuracy (from $6 5 . 4 5 \%$ ), while the performance on question reconstruction improves from $8 7 . 5 2 \%$ to $9 3 . 8 \%$ . Without the KL term, the model focuses solely on reconstruction and fails to learn compositionality in the latent $\mathbf { z }$ space, such that supervised grounding are poorly propagated to unsupervised questions as evidenced by the drop in program prediction accuracy. Note that this ties well to our high-level goal to better leverage the semantics of the lexicon provided by the user for interpretable question answering. + +Beam search vs. sampling for module training. As explained in Section 2, we use beam search instead of sampling to optimize Equation (5). In this section we empirically study this design choice. We generally find that in low-supervision settings, where the question coding stage might not produce correct programs (for questions) as frequently, using sampling form the sequence model $q ( \mathbf { z } | \mathbf { x } )$ instead of beam search leads to a drop in module training performance. For example, with $10 \%$ supervision, the module training accuracy drops from $8 1 . 3 4 \pm 8 . 6 1$ (with beam search) to 76.31 $\pm ~ 8 . 4 1$ (with sampling). With more supervision the choice of beam search or sampling does not matter, which makes intuitive sense, as the this means that the programs that get sampled tend to be the correct ones more often (that is, the confidence of the model increases with the improvement in correctness). + +Effect of optimizing the true ELBO, $\beta = 1$ . Next, we empirically validate the claim that approximate inference learned using the ELBO leads to autodecoding behavior as predicted by Alemi et al. (2018), and that better program representations for our problem can be found for $\beta < 1$ (as shown above). At $15 \%$ program supervision, VQA accuracy drops from $9 6 . 9 0 \%$ $( \pm 0 . 9 6 )$ at $\beta = 0 . 1$ to $7 8 . 4 9 \%$ $( \pm \ : 1 . 3 7 )$ at $\beta = 1$ (for the empirical prior). Question reconstruction drops from $9 7 . 4 3 \ \%$ $( \pm 3 . 4 2 )$ at $\beta = 0 . 1$ , to $4 3 . 4 0 \%$ $\pm 5 . 3 4 )$ at $\beta = 1$ , showing auto-decoding behavior (as predicted by theory). + +Finally, the N2NMN approach (Hu et al., 2017) evaluates their question-attention based module networks in the fully unsupervised setting, getting to $9 6 . 1 9 \%$ on TEST. However, the programs in this case become non-compositional (see Section 3), as the model leaks information from questions to answers via attention, meaning that programs no longer carry the burden of explaining the observed answers. This makes the modules less interpretable; which is one of the core motivations of this neural-symbolic class of module networks in the first place. + +# 6 DISCUSSION AND CONCLUSION + +In this paper we presented a novel, probabilistic neural symbolic model for interpretable visual question answering, that provides explanations of “what” the model is doing on unseen questions given a minimal number of annotated symbolic traces or programs of “what it should do”. We demonstrate that our formulation provides more interpretable explanations than previous work on visual question answering called neural module networks, on dataset of compositional questions about shapes (Andreas et al., 2016a). + +The key to our approach is a model with programs as a stochastic latent variable, which leads to better sharing of statistics across questions, yielding a latent space where program annotations for known questions propagate effectively to unknown / novel questions without annotations. + +In general, the model family we study is quite rich, and also supports other inference queries such as counter factual explanations of what programs could have led to particular answers for a given image. + +![](images/389c1c47de1104d79e166cd0e7119a70107bf4d9f60d944f61b91a21466606c1.jpg) +Figure 3: Qualitative results: Image with the corresponding question, ground truth program, and ground truth answer (top left). Predicted programs and answers of different models (NMN in white, VNMN in gray) and resconstructed question (V-NMN) for variable amount of supervision. We find that V-NMN finds the right answer as well as the program more often than NMN (Johnson et al., 2017). + +We hope our work inspires more work on applying such ideas to other tasks requiring compositional reasoning such as planning, navigation etc. + +# REFERENCES + +Aishwarya Agrawal, Jiasen Lu, Stanislaw Antol, Margaret Mitchell, C Lawrence Zitnick, Dhruv Batra, and Devi Parikh. VQA: Visual question answering. May 2015. + +Alexander A Alemi, Ben Poole, Ian Fischer, Joshua V Dillon, Rif A Saurous, and Kevin Murphy. Fixing a broken ELBO. November 2018. + +J Andreas, M Rohrbach, T Darrell, and D Klein. Neural module networks. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 39–48, June 2016a. + +Jacob Andreas, Dan Klein, and Sergey Levine. Modular multitask reinforcement learning with policy sketches. 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November 2016. + +# APPENDIX + +# .1 IMPLEMENTATION DETAILS + +We provide more details on the modeling choices and hyperparameters used to optimize the models in the main paper. + +Sequence to Sequence Models: All our sequence to sequence models described in the main paper (Section. 2) are based on LSTM cells with a hidden state of 128 units, single layer depth, and have a word embedding of 32 dimensions for both the question as well as the program vocabulary. + +The when sampling from a model, we sample till the maximum sequence length of 15 for questions and 7 for programs. + +Image CNN: The SHAPES images are of $3 0 \times 3 0$ in size and are processed by a two layered convolutional neural network, which in its first layer has a $\mathsf { 1 0 x 1 0 }$ filters applied at a stride of 10, and in the second layer, it does $\beth \times \beth$ convolution applied at a stride of 1. Channel widths for both the layers are 64 dimensional. + +Moving average baseline For a reward $R$ , we use an action independent baseline $b$ to reduce variance, which tracks the moving average of the rewards seen so far during training. Concretely, the form for the $q ( \mathbf { z } | \mathbf { x } )$ network is: + +$$ +\nabla J ( \theta ) = \mathbf { E _ { z \sim q ( z | x ) } } [ ( R - b _ { t } ) \nabla \log q ( \mathbf { z } | \mathbf { x } ) ] +$$ + +where, given $D$ as the decay rate for the baseline, + +$$ +b _ { t } = b _ { t - 1 } + D * \left( R - b _ { t - 1 } \right) +$$ + +is the update on the baseline performed at every step of training. + +Note that in practice, when estimating the gradient for the variational lower bound, we consider the total derivative (which adds a term having a gradient with respect to $R$ above), since in the case of sample-based optimization of the elbo, the reward itself is also a function of the variational parameters, via. both the sample used estimate the reward, as well as the reward directly being a function of $q ( \mathbf { z } | \mathbf { x } )$ . + +Hyperparameters for training: We use the ADAM optimizer (Kingma & Ba, 2014) with a learning rate of $\beth \in - 3$ , a minibatch size of 576, and use a moving average baseline for REINFORCE, with a decay factor of 0.99. Typical values of $\beta$ are set to 0.1, $\alpha$ is set to 100 and $\gamma$ is chosen on a validation set among (1.0, 10.0, 100.0). + +Simulating programs from known syntax: We follow a simple two-stage heuristic procedure for generating a set of samples to train the program prior in our model. Firstly, we build a list of possible future tokens given a current token, and sample a random token from that list, and repeat the procedure for the new token to get a large set of possible valid sequences – we then pass the set of candidate sampled sequences through a second stage of filtering based on constraints from the work of Hu et al. (2017). + +# .2 ADDITIONAL RESULTS + +Empirical vs. syntactic priors While our default choice of the prior $p ( \mathbf { z } )$ is from programs simulated from the known syntax, it might not be possible to exhaustively enumerate all valid program strings in general. Here, we consider the special case where we train the prior on a set of unaligned, ground truth programs from the dataset. Interestingly, we find that the performance of the question coding stage, especially at reconstructing the questions improves significantly when we have the syntactic prior as opposed to the empirical prior (from $3 8 . 3 9 \pm 1 1 . 9 2 \%$ to $5 4 . 8 6 \pm 6 . 7 5 \%$ for $5 \%$ program supervision). In terms of program prediction accuracy, we also observe marginal improvements in the cases where we have $5 \%$ and $10 \%$ supervision respectively, from $5 6 . 2 3 \pm 2 . 8 1 \%$ to $6 5 . 4 5 \pm$ $1 1 . 8 8 \%$ . When more supervision is available, regularizing with respect to the broader, syntactic prior hurts performance marginally, which makes sense, as one can just treat program supervision as a supervised learning problem in this setting. \ No newline at end of file diff --git a/parse/train/ryxhB3CcK7/ryxhB3CcK7_content_list.json b/parse/train/ryxhB3CcK7/ryxhB3CcK7_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..517bdc336658cdd273d3586ff094646c5684f156 --- /dev/null +++ b/parse/train/ryxhB3CcK7/ryxhB3CcK7_content_list.json @@ -0,0 +1,1853 @@ +[ + { + "type": "text", + "text": "PROBABILISTIC NEURAL-SYMBOLIC MODELS FOR INTERPRETABLE VISUAL QUESTION ANSWERING ", + "text_level": 1, + "bbox": [ + 176, + 98, + 779, + 147 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 174, + 398, + 202 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 238, + 544, + 253 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "We propose a new class of probabilistic neural-symbolic models for visual question answering (VQA) that provide interpretable explanations of their decision making in the form of programs, given a small annotated set of human programs. The key idea of our approach is to learn a rich latent space which effectively propagates program annotations from known questions to novel questions. We do this by formalizing prior work on VQA, called module networks (Andreas, 2016) as discrete, structured, latent variable models on the joint distribution over questions and answers given images, and devise a procedure to train the model effectively. Our results on a dataset of compositional questions about SHAPES (Andreas, 2016) show that our model generates more interpretable programs and obtains better accuracy on VQA in the low-data regime than prior work. ", + "bbox": [ + 233, + 268, + 766, + 421 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 446, + 336, + 462 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Exciting progress has been made in recent years in deep representation learning (Hochreiter & Schmidhuber, 1997; LeCun et al., 1998), which has led to advances in artificial intelligence tasks such as image recognition (Krizhevsky et al., 2012; He et al., 2015), machine translation (Sutskever et al., 2014), visual question answering (Agrawal et al., 2015), visual dialog (Das et al., 2016), and reinforcement learning (Mnih et al., 2015), etc. While deep neural networks achieve such impressive performances, many aspects of human cognition such as compositional generalization and reasoning are harder to model with state-of-the art deep learning approaches (Battaglia et al., 2018; Barrett et al., 2018). ", + "bbox": [ + 174, + 477, + 825, + 588 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Symbolic approaches (Newell & Simon, 1976) on the other hand provide strong compositional and reasoning capabilities which are challenging to model with neural networks (Lake et al., 2017). Consequently, a rich line of work in neuro-symbolic processing sought to build AI systems with both strong learning and reasoning capabilities (Valiant, 2003; Evans et al., 2018b; Bader & Hitzler, 2005). ", + "bbox": [ + 174, + 595, + 825, + 665 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "As we tackle higher-level tasks which involve reasoning, such as visual question answering (Agrawal et al., 2015), planning (Andreas et al., 2016b; Gregor et al., 2016) etc., it is natural to desire the ability to provide instructions to models to guide them. Symbolic instructions, by their very nature are easier to specify and more interpretable than specifying the parameters of a neural network. For such high-level tasks, a sensible approach is to specify “what” to do using symbols and learn how to do the task using modern representation learning techniques. ", + "bbox": [ + 174, + 672, + 825, + 756 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "For the example shown in Figure 1, given a question “Is a square to the left of a green shape?”, one can ask the model to reason about the answer by first applying a find[green] operator, then find[left], then And the result together with a find[square] operator in order to predict the answer, in other words specifying “what” are the computations we desire to be executed, in the form of a “program”. (Johnson et al., 2017; Hu et al., 2017; Andreas et al., 2016a). The network can then learn “how” to execute such a program from data using deep representation learning (Johnson et al., 2017; Hu et al., 2017; Andreas et al., 2016a). ", + "bbox": [ + 174, + 763, + 825, + 861 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "This paper addresses a very natural desiderata for such neuro-symbolic models, in the context of visual question answering – can we retain the intepretability of “what the network did” expressed in terms of the syntax and lexicon that we are interested in, while specifying minimal teaching examples of “what to do” given an input question? ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/2aa1b4dd59dc5a8e445ac2166becf0f18b8fb3f6685e2df672b5e879925a26f6.jpg", + "image_caption": [ + "Figure 1: An illustration of neural module networks for visual question answering. Given a question, the program generator produces a program in prefix notation which is used to construct a neural network from the specified module networks. This network operates on the image and intermediate attention maps to answer the question. " + ], + "image_footnote": [], + "bbox": [ + 235, + 98, + 753, + 205 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We propose to approach this problem with a treatment of programs $\\mathbf { z }$ as a latent variable, embedded into a generative model of questions $\\mathbf { x }$ and answers a given images i as visualized in Figure 2. This model class has certain desirable properties: first, a proper treatment of $\\mathbf { z }$ as a stochastic latent variable is helpful especially in the presence of a limited number of expert human instructions, as we model the uncertainty associated with $\\mathbf { z }$ explicitly, leading to more interepretable latent spaces (Section 5); second, the model makes intuitive independence assumptions with regards to the question answering task – conditioned on the program $\\mathbf { z }$ , questions $\\mathbf { x }$ are independent of the answers a for an image, meaning that the program must be a sufficient statistic for predicting the answer given a question1. Third, we parameterize the $p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } )$ term, which predicts an answer given an image and a program using a popular, related class of question answering models called neural module networks (NMNs) (Hu et al., 2017; Johnson et al., 2017; Andreas et al., 2016a) which previous work has shown leads to better fit between the program (or “what the model is doing”) and the execution (or “how it is doing what it is doing”). Given the close ties of our model to NMNs, we will refer to our model as variational neural module networks (V-NMN) for the rest of the paper. ", + "bbox": [ + 174, + 289, + 825, + 483 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Apart from our modeling contribution of a probabilistic latent variable, neuro-symbolic model, our key technical contribution is to show how to train this model in the context of visual question answering. Optimizing such generative models with structured discrete latent variables is an intractable problem to solve in genereal – for this particular case, we formulate training into stages of grounding questions into programs (question coding), learning how to execute them and then with those initializations train the full variational objective. We demonstrate that this stage-wise optimization allows us to successfully learn a probabilistic neural-symbolic model that can answer compositional questions about shapes in an image (Andreas et al., 2016a). ", + "bbox": [ + 174, + 489, + 825, + 602 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "While the V-NMN model is instantiated for the specific problem of visual question answering, we believe the tools and techniques we develop are more generally applicable to probabilistic, structured, discrete, interpretable latent spaces for interpretable neuro-symbolic models for planning, instruction following, grounding referring expressions etc. ", + "bbox": [ + 174, + 608, + 825, + 664 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "We benchmark our model on the SHAPES (Andreas et al., 2016a) dataset with compositionally novel questions and show that our model is able to provide interpretable explanations of “what it is doing” (measured by program correctness) for unseen questions even in the setting where it is given as few as $\\approx 2 0$ paired examples of programs and corresponding questions from the dataset, while also getting to high question answering accuracy. We find that the proposed approach outperforms a state of the art neural-symbolic model designed for VQA from Johnson et al. (2017), as well as deterministic ablations of our model. ", + "bbox": [ + 174, + 671, + 826, + 768 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 A PROBABILISTIC NEURAL-SYMBOLIC MODEL FOR VQA ", + "text_level": 1, + "bbox": [ + 176, + 789, + 684, + 805 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "For a standard neural-symbolic NMN approach to question answering (Johnson et al., 2017), we are given an input image i and a question ${ \\bf x } = ( x _ { 1 } , \\cdots , x _ { N } )$ as a sequence of words $x _ { i }$ each encoded using a vocabulary $\\mathcal { V } _ { \\mathbf { x } }$ , and must produce answers a by planning and executing a series of modular computations $( { \\bf z } = ( z _ { 1 } , \\cdot \\cdot \\cdot , z _ { N } ) ) - i . e$ . a program of operations $( z _ { i } \\in \\mathcal { V } _ { \\mathbf { z } } )$ ) to be executed in order to answer the question. Note that $\\mathbf { z }$ here is the prefix serialization of the program and programs themselves often have tree-structured topologies. We denote the dataset of $( \\mathbf { i } , \\mathbf { x } , \\mathbf { a } )$ triplets as $\\mathcal { D }$ . Additionally, we are provided programs $\\mathbf { z }$ for some input questions $( \\mathbf { x } \\in S )$ ), which specify the notion of grounding desired by an end user. Our goal is to provide faithful explanations on unseen questions (along with correct answers) consistent with the specified grounding. ", + "bbox": [ + 174, + 820, + 825, + 863 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/22b2bbae87156a9dbb5b0416e0b6f7d0dca7b729f9eb2243043afb4d763daba4.jpg", + "image_caption": [ + "Figure 2: An illustration of the graphical model used for neural module networks (NMNs). " + ], + "image_footnote": [], + "bbox": [ + 266, + 98, + 727, + 215 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 256, + 825, + 340 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We accomplish this by modeling $\\mathbf { z }$ as a latent variable that underlies both question and answer generation given an input image. Intuitively, we expect this will enable us to learn a more meaningful latent representation for questions, as modeling uncertainity should help better handle the potentially underspecified mapping between questions and programs provided by the user. As such, our model factorizes $p ( \\mathbf { x } , \\mathbf { z } , \\bar { \\mathbf { a } } | \\bar { \\mathbf { i } } ) \\stackrel { - } { = } p ( \\mathbf { x } | \\mathbf { z } ) p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } ) p ( \\mathbf { z } )$ as in the graphical model shown in Figure 2, where the prior $p ( \\mathbf { z } )$ is learnt by training a neural sequence model on valid strings simulated from known syntax. Conceptually, this factorization captures the notion that many differently-worded questions are reducible to identical underlying semantics (captured by a program), making the program space, in a sense, a compression of the question space. As mentioned in Section 1 the model also captures the independence of questions from answers for an image given programs, which is also an intuitive desiderata for interpretable neural-symbolic models2. ", + "bbox": [ + 173, + 347, + 825, + 500 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given the proposed model shown in Figure 2, we must address two tasks to put the model to use for VQA – learning model parameters, and performing inference for a specific query. ", + "bbox": [ + 174, + 507, + 821, + 535 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Learning. We will always condition on images i but may not observe underlying programs $\\mathbf { z }$ , so our per-data point training objective is the log-marginal-conditional-likelihood: $\\log p ( \\mathbf { x } , \\mathbf { a } \\mid \\mathbf { i } )$ . Since computing this objective involves an intractable summation over latent programs $\\begin{array} { r } { ( \\log \\sum _ { \\mathbf { z } } p ( \\mathbf { x } _ { j } , \\mathbf { z } , \\mathbf { a } _ { j } \\ | } \\end{array}$ $\\mathbf { i } _ { j }$ )), we use a variational approximation (Jordan et al., 1999). Specifically, let ", + "bbox": [ + 174, + 541, + 825, + 598 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "- $q _ { \\phi } ( \\mathbf { z } \\mid \\mathbf { x } )$ denote an amortized inference neural network parameterized by $\\phi$ ,which instantiates a sequence to sequence LSTM (Hochreiter & Schmidhuber, 1997) model \n- $p _ { \\theta _ { \\mathbf { z } } } ( \\mathbf { a } \\mid \\mathbf { z } , \\mathbf { i } )$ be a NMN parametrized by $\\boldsymbol { \\theta } _ { \\mathbf { z } }$ run on image i with program $\\mathbf { z }$ , \n- $p _ { \\theta } ( \\mathbf { x } \\mid \\mathbf { z } )$ be a recurrent network (sequence to sequence model) parametrized by $\\theta$ that maps programs to questions, and \n- $\\mathcal { D } ^ { ( z ) }$ be a prior over programs learned from data or based on program syntax directly , parameterized as an LSTM (Hochreiter & Schmidhuber, 1997) sequence model ", + "bbox": [ + 179, + 599, + 825, + 695 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Then, for datapoints without program supervision, the objective is: ", + "bbox": [ + 173, + 702, + 611, + 717 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/87a0784c30dd4f9e31fcbb40e45128705b4b439df426ffc2907b7105aad4eaab.jpg", + "text": "$$\n\\mathrm { e l b o } ( { \\mathbf { a } } , { \\mathbf { x } } | { \\mathbf { i } } ) = E _ { z \\sim q _ { \\phi } ( { \\mathbf { z } } | \\mathbf { x } ) } [ \\log p _ { \\theta _ { \\mathbf { z } } } ( { \\mathbf { a } } | { \\mathbf { z } } , { \\mathbf { i } } ) + \\log p _ { \\theta } ( { \\mathbf { x } } | { \\mathbf { z } } ) ] - \\beta { \\mathrm { K L } } ( q _ { \\phi } ( { \\mathbf { z } } | \\mathbf { x } ) , p ( \\mathbf { z } ) )\n$$", + "text_format": "latex", + "bbox": [ + 236, + 723, + 759, + 742 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "It is easy to show that $\\mathrm { e l b o } ( \\mathbf { x } , \\mathbf { a } | \\mathbf { i } ) \\leq \\log p ( \\mathbf { x } , \\mathbf { a } | \\mathbf { i } )$ for $\\beta \\geq 1$ ; as such, maximizing this objective is a form of approximate maximum likelihood training for distribution parameters $\\theta _ { \\mathbf { z } } , \\theta$ , and $\\phi$ . ", + "bbox": [ + 173, + 750, + 826, + 779 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Additionally, for data points with annotations, we add the likelihood of the program given the question $\\log r _ { \\phi } ( \\mathbf { z } | \\mathbf { x } )$ to the objective. Note that $r$ shares parameters $\\phi$ with the inference network $q$ . So our overall objective becomes: ", + "bbox": [ + 174, + 784, + 825, + 827 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/3853467e1bddf9f01da245a4cf51f56647346322a8140628ce692dab61cc8edc.jpg", + "text": "$$\n\\mathrm { e l b o } ( \\mathbf { a } , \\mathbf { x } | \\mathbf { i } ) + \\alpha \\log r ( \\mathbf { z } | \\mathbf { x } ) \\cdot \\left[ [ \\mathbf { x } \\in \\mathcal { S } ] \\right]\n$$", + "text_format": "latex", + "bbox": [ + 372, + 829, + 622, + 847 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where the indicator function $[ [ \\mathbf { x } \\in S ] ]$ is 1 if $\\mathbf { x }$ is in the supervised set and 0 otherwise and $\\alpha$ is a positive scaling factor. The goal is to learn the desired association of questions with programs from the second term, and to propagate them to other questions using the first term. ", + "bbox": [ + 173, + 858, + 825, + 875 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Inference. At test time, we can answer questions for images by computing $p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { x } ) \\ =$ $\\begin{array} { r } { \\sum _ { z } p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } ) p ( \\mathbf { z } | \\mathbf { x } ) } \\end{array}$ using an approximation $p _ { q } ( \\mathbf { \\bar { a } } | \\mathbf { i } , \\mathbf { x } ) \\approx E _ { \\mathbf { z } _ { i } \\sim q ( \\mathbf { z } | \\mathbf { x } ) } p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } _ { i } )$ . ", + "bbox": [ + 171, + 138, + 821, + 169 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Challenges. While the elbo term in Equation (1) appears similar to a standard ELBO for joint latent variable models (Vedantam et al., 2018; Wang et al., 2016b; Suzuki et al., 2017), there are a few important departures: ", + "bbox": [ + 174, + 174, + 825, + 215 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "- Firstly, the latent variable $\\mathbf { z }$ is not a Gaussian or categorical random variable (which admit straightforward reparameterizations (Maddison et al., 2017; Jang et al., 2016)) but rather a sequence of discrete tokens, which complicates the optimization problem in Equation (1). Secondly, the conditional log-likelihood for a is parameterized by a NMN that is dynamically assembled based on sample programs drawn from $q ( \\mathbf { z } | \\mathbf { x } )$ . Depending upon the uncertainty of the program posterior, program tokens may not be consistently grounded across examples, making learning the appropriate computation for the corresponding neural modules difficult. Lastly, our observations x are explained by powerful “autoregressive” sequence models. Learning meaningful latent variable models that are informative of observations for such models is difficult (c.f. (Bowman et al., 2016; Chen et al., 2016; Alemi et al., 2018)) as powerful decoder models can explain the data using stochasticity from sampling (referred to as autodecoding in Alemi et al. (2018)). Consequently, it also becomes hard to learn meaningful NMN parameters from the previous step. ", + "bbox": [ + 176, + 218, + 826, + 398 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "In the following sections, we present a training regime and adjustments to the objective in Equation (1) that make optimization more tractable and enable training NMNs with significantly reduced questionaligned program supervision. While instantiated for question answering, it is not difficult to conceive versions of this regime for other problems relevant to probabilistic neural-symbolic models. ", + "bbox": [ + 174, + 405, + 826, + 462 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.1 OPTIMIZING THE ELBO FOR V-NMN ", + "text_level": 1, + "bbox": [ + 176, + 477, + 475, + 492 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Rather than approaching the joint objective directly, we break ELBO training into three stages: ", + "bbox": [ + 176, + 503, + 794, + 518 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "1. Question coding, where we learn a latent ‘code’ for questions in the latent space of programs, 2. Module training, where we learn the ‘execution’ of the programs from the first stage of training, grounding tokens in programs into the operations they perform on images; and 3. Joint training, where we train both stages jointly given the ‘warm-starts’ from the other phases. ", + "bbox": [ + 178, + 520, + 826, + 577 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Question Coding. The goal of this stage is to learn a code for questions in the latent space of programs $- i . e$ . an informative mapping from questions to programs. Given some set of question-aligned programs, we find this coding process effectively propagates these supervised groundings to unsupervised questions, which enable interpretable explanations $/$ groundings of queries to explanations of “what” the model is doing, which is our primary goal. ", + "bbox": [ + 173, + 578, + 826, + 648 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "During question coding, we drop the $p _ { \\theta _ { z } } ( \\mathbf { a } | \\mathbf { z } , \\mathbf { i } )$ term from Eq. 1, yielding the objective ", + "bbox": [ + 179, + 654, + 748, + 670 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/152d55333f95cdabe41fe967d8e131b253df47ca7a31b573398e9b0ca8937a6f.jpg", + "text": "$$\nJ ( \\mathbf { x } ; \\phi , \\theta ) = E _ { z \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } [ \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) ] - \\beta \\mathrm { K L } ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) , p ( \\mathbf { z } ) ) + \\alpha \\log r _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) \\cdot [ [ \\mathbf { x } \\in \\mathcal { S } ]\n$$", + "text_format": "latex", + "bbox": [ + 200, + 671, + 777, + 689 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Recent theory from Alemi et al. (2018) prescribes changes to the ELBO, setting $\\beta < 1$ as a recipe to learn representations which avoid learning degenerate latent representations in the presence of powerful decoders. Essentially, Alemi et al. (2018) identify that upto a constant factor, the negative log-marignal likelihood term $D { = } { - } E _ { z { \\sim } q _ { \\phi } ( { \\bf z } | { \\bf x } ) } \\big [ \\log p _ { \\theta } ( { \\bf x } | { \\bf z } ) \\big ]$ , and the kl diveregence term $R = \\mathrm { K L } ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) , p ( \\mathbf { z } ) )$ bound the mutual information between the data $\\mathbf { x }$ and the latent variable $\\mathbf { z }$ as follows: ", + "bbox": [ + 173, + 690, + 825, + 775 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/3c6a5a713fb072d09386cfae428568641934b2308100c5d0e20effedee16b3b7.jpg", + "text": "$$\nH - D \\leq I ( \\mathbf { x } , \\mathbf { z } ) \\leq R\n$$", + "text_format": "latex", + "bbox": [ + 421, + 775, + 576, + 791 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $H$ is the entropy of the data, which is a constant. One can immediately notice that the standard e $\\mathrm { b o } = - D - R$ . This means that for the same value of the elbo one can get different models which make drastically different usage of the latent variable (based on the achieved values of $D$ and $R$ ). Thus, one way to achieve a desired behavior (of say high mutual information $I ( \\mathbf { x } , \\mathbf { z } )$ ), is to set $\\beta$ to a lower value than the standard elbo (c.f. Eqn. 6 in Alemi et al. (2018).) This aligns with our goal in question coding, and so we use $\\beta < 1$ . As this no longer a valid variational lower bound, denote our objective as $J ( \\mathbf { x } ; \\phi , \\theta )$ from now on. ", + "bbox": [ + 173, + 791, + 825, + 890 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We optimize the above objective using the score function estimator (Glynn, 1990), also referred to as REINFORCE (Williams, 1992) applied to the first term, with a moving average baseline.Finally, while the KL term (in the reward) encourages the model to generate high-likelihood plausible sequences, we also penalize the model for producing syntactically invalid programs. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 169, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Although gradients for the KL term in Equation (2) can be computed by decomposing the KL for the sequence into per-timestep monte-carlo gradients, we found that having one term with per-timestep gradients and another that only receives sequence-level rewards (the likelihood term) skewed the model towards minimizing the KL, leading to auto-decoding solutions. Thus, we apply REINFORCE on the entire sequence and compute rewards using the full monte carlo gradient,i.e., we do not optimize the KL term separately, but instead maximize: ", + "bbox": [ + 173, + 138, + 825, + 223 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/c4a38f2c0d879d16bbeaabd46889bc590795fe032f4f259dfb89fb59fea9d517.jpg", + "text": "$$\nE _ { \\mathbf { z } \\sim q _ { \\phi } ( \\mathbf { z } | \\hat { \\mathbf { x } } ) } \\left[ \\log \\left[ \\frac { p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { z } ) p ( \\mathbf { z } ) ^ { \\beta } } { q _ { \\phi } ( \\mathbf { z } | \\hat { \\mathbf { x } } ) ^ { \\beta } } \\right] \\right] + \\alpha \\log r _ { \\phi } ( \\hat { \\mathbf { z } } | \\hat { \\mathbf { x } } ) \\cdot \\left[ \\left[ \\mathbf { x } \\in \\mathcal { S } \\right] \\right]\n$$", + "text_format": "latex", + "bbox": [ + 300, + 229, + 697, + 265 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\hat { \\mathbf { x } } , \\hat { \\mathbf { z } }$ are the questions and programs we observe in the dataset. ", + "bbox": [ + 174, + 272, + 622, + 286 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For comparison, the semi-supervised approach in Johnson et al. (2017) can be seen as optimizing the last term on supervised training data, while our question coding stage optimizes the elbo in addition to the last term. Our results in Section 5 show that the proposed question coding formulation significantly improves performance relative to Johnson et al. (2017) in terms of explaining “what” the model is doing on test programs. ", + "bbox": [ + 174, + 292, + 825, + 363 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Module Training. Having obtained a code for questions in the latent space, we train the parameters of the NMN for question answering. The network modules are trained to perform their corresponding tasks. For example, if the question coding learned to associate the program token $v$ with the question phrase ‘to the left’, then the goal for this second stage is to train the module corresponding to $v$ to actually perform this operation (e.g. transform[left]) when executed on images. ", + "bbox": [ + 173, + 369, + 825, + 440 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We do this by optimizing the joint elb ${ \\bf \\alpha } ) ( { \\bf x } , { \\bf a } | \\bf { i } )$ in (1) with respect to the parameters $\\theta _ { \\mathbf { z } }$ of the neural modules. We write this objective for each instance as ", + "bbox": [ + 174, + 445, + 823, + 474 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/25a3b10307f473abd2705dbaed2e8549284420c0b00e57e8e497592dabff2a46.jpg", + "text": "$$\nJ ( \\mathbf { a } \\mid \\mathbf { x } , \\mathbf { i } ; \\theta _ { \\mathbf { z } } ) = E _ { z \\sim q ( \\mathbf { z } \\mid \\mathbf { x } ) } [ \\log p _ { \\theta _ { z } } ( \\mathbf { a } \\mid \\mathbf { i } , \\mathbf { z } ) ]\n$$", + "text_format": "latex", + "bbox": [ + 356, + 481, + 642, + 500 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Notably, while we write this objective as a function of all neural modular network parameters $\\theta _ { \\mathbf { z } }$ , for a given instance, only those modules present in the program $z \\sim q ( \\mathbf { z } | \\mathbf { x } )$ are updated. In practice, we take approx. argmax decoded programs from $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ to simplify training, but perform sampling during joint training in the next stage. To be more specific, we run beam search (Vijayakumar et al., 2018) with a beam size of 1 to do this. Intuitively, we think of this as a bias $\\nu s$ . variance tradeoff, doing argmax enables us to trade off bias for high-variance from a posterior which (in the pathological case) could match the prior from the first stage of training. This is also consistent with standard practice for doing inference in sequence models, where beam search is the standard inference tool (and sampling often has issues due to teacher forcing based MLE training, which is how we do semi-supervised learning) (Bengio et al., 2015). ", + "bbox": [ + 173, + 505, + 825, + 645 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Joint Training. Having trained the question code and the neural module network parameters, we tune all terms jointly, with the following objective, which adds scaling factors $\\alpha$ , $\\beta$ and $\\gamma$ , and an extra semi-supervised learning term relative to Equation (1): ", + "bbox": [ + 174, + 651, + 823, + 694 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/fd3d8a30d4332d07074d746f446cadb9c21e7fee9fbcc98319d3b4688a09540f.jpg", + "text": "$$\nJ ( \\mathbf { x } , \\mathbf { a } | \\mathbf { i } ) = E _ { z \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } \\big [ \\gamma \\log p _ { \\theta _ { z } } ( \\mathbf { a } | \\mathbf { z } , \\mathbf { i } ) + \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) \\big ] - \\beta \\mathrm { K L } \\big ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) , p ( \\mathbf { z } ) \\big ) + \\alpha \\log r _ { \\phi } ( \\mathbf { z } | \\mathbf { x } )\n$$", + "text_format": "latex", + "bbox": [ + 181, + 700, + 799, + 719 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The gamma factor $\\gamma$ controls the scaling of the likelihood term for the answers relative to the question (which has more bits of information), similar scaling factors have been for joint models in Vedantam et al. (2018). Similar to Equation (4), we optimize the full monte carlo gradient for the above objective as well. The full training procedure is shown in Algorithm 1 below. ", + "bbox": [ + 173, + 724, + 825, + 781 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3 RELATED WORK ", + "text_level": 1, + "bbox": [ + 176, + 800, + 344, + 816 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "We first explain other related work in neuro-symbolic models, the key differences between our formulation and closely related NMN work (Johnson et al., 2017; Hu et al., 2017; 2018), and then connect our work to other themes and directions of research in literature. ", + "bbox": [ + 174, + 832, + 825, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Neural-symbolic models Valiant (2003) identifies neuro-symbolic models, which are able to reason as well as learn efficiently as one of the key problems in computer science. A rich literature of work exists in this area, around two major threads: firstly, there is a line of work in building hybrid models ", + "bbox": [ + 174, + 881, + 825, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1 V-NMN Training ", + "text_level": 1, + "bbox": [ + 174, + 103, + 367, + 118 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "1: Given: $p ( \\mathbf { z } )$ , Initialize $q _ { \\phi } ( { \\bf z } | { \\bf x } )$ , $p _ { \\theta _ { z } } ( \\mathbf { a } | \\mathbf { z } , \\mathbf { i } )$ , $p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } )$ , moving average baseline $B = 0$ , $\\beta = 0 , \\tau , \\alpha =$ 100.0, $\\delta = 0 . 9 9$ . \n2: Question Coding: Given $\\hat { \\mathbf { x } } , \\hat { \\mathbf { z } } \\sim p _ { d a t a } ( \\mathbf { x } , \\mathbf { z } )$ , sample ${ \\mathbf z } _ { \\mathbf i } \\sim q _ { \\phi } ( { \\mathbf z } | \\hat { \\mathbf x } )$ [Repeat until convergence] \n3: $\\nabla _ { \\phi } ( R - B ) \\log g _ { \\phi } ( z _ { i } | x ) + \\alpha \\nabla _ { \\phi } \\log r _ { \\phi } ( z | x ) ;$ where $R = \\log \\left[ \\frac { p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { z } _ { i } ) p ( \\mathbf { z } _ { i } ) ^ { \\beta } } { q _ { \\phi } ( \\mathbf { z } _ { i } | \\hat { \\mathbf { x } } ) ^ { \\beta } } \\right]$ \n4: $\\nabla _ { \\boldsymbol { \\theta } } \\gets \\nabla _ { \\boldsymbol { \\theta } } \\log p _ { \\boldsymbol { \\theta } } \\bigl ( \\hat { \\mathbf { x } } | \\mathbf { z } _ { i } \\bigr )$ \n5: $B \\gets B + ( 1 - \\delta ) ( R - B ) ; \\nabla _ { \\theta } \\gets \\nabla _ { \\theta } \\log p ( \\hat { \\mathbf { x } } | \\mathbf { z } _ { i } )$ \n6: Question Coding: Given xˆ $, \\hat { \\mathbf { z } } \\sim p _ { d a t a } ( \\mathbf { x } , \\mathbf { z } )$ , sample ${ \\mathbf z } _ { \\mathbf i } \\sim q _ { \\phi } ( { \\mathbf z } | \\hat { \\mathbf x } )$ [Repeat until convergence] \n7: Module Training: Given $\\begin{array} { l l l } { \\hat { \\mathbf { x } } , \\hat { \\mathbf { a } } , \\hat { \\mathbf { i } } } & { \\sim } & { p _ { d a t a } ( \\mathbf { x } , \\mathbf { a } , \\mathbf { i } ) } \\end{array}$ , $\\begin{array} { r l } { \\mathbf { z } _ { i } } & { { } = } \\end{array}$ argmax $q ( \\mathbf { z } | \\hat { \\mathbf { x } } )$ [Repeat until convergence] \n8: $\\nabla _ { \\theta _ { z } } \\gets \\nabla _ { \\theta _ { z } } \\log p ( \\hat { \\mathbf { a } } | \\mathbf { z } _ { i } , \\hat { \\mathbf { i } } )$ \n9: Joint Training: Given $\\begin{array} { r } { \\hat { \\mathbf { x } } , \\hat { \\mathbf { a } } , \\hat { \\mathbf { i } } , \\hat { \\mathbf { z } } \\sim p _ { d a t a } ( \\mathbf { x } , \\mathbf { a } , \\mathbf { i } , \\mathbf { z } ) , \\mathbf { z } _ { i } \\sim q ( \\mathbf { z } | \\hat { \\mathbf { x } } ) , \\gamma ; } \\end{array}$ Initialize $B = 0$ , [Repeat until convergence] \n10: $\\begin{array} { r l } & { \\nabla _ { \\phi } \\gets ( R - B ) \\nabla _ { \\phi } \\log g _ { \\phi } ( z _ { i } | \\hat { x } ) + \\alpha \\nabla _ { \\phi } \\log r _ { \\phi } ( \\hat { z } | \\hat { x } ) ; \\mathbf { w h e r e ~ } \\mathbb { R } = \\log \\left[ \\frac { p _ { \\phi } ( \\hat { x } | z _ { i } ) p ( \\hat { x } | z _ { i } , \\hat { \\mathbf { H } } ^ { \\gamma } p ( z _ { i } ) ) ^ { \\phi } } { q _ { \\phi } ( z _ { i } | x ) ^ { \\phi } } \\right] } \\\\ & { B \\gets B + ( 1 - \\delta ) ( R - B ) ; \\nabla _ { \\theta } \\gets \\nabla _ { \\theta } \\log p ( \\hat { \\mathbf { x } } | \\mathbf { z } _ { i } ) ; \\nabla _ { \\theta _ { z } } \\gets \\nabla _ { \\theta _ { z } } \\log p ( \\hat { \\mathbf { a } } | \\mathbf { z } _ { i } , \\hat { \\mathbf { i } } ) } \\end{array}$ \n11: ", + "bbox": [ + 178, + 119, + 826, + 330 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "which are hybrids of symbolic reasoning and neural networks (Bader & Hitzler, 2005; Holldobler & ¨ Kalinke), the second thread looks at studying if neural networks can indeed do symbolic reasoning and building models and architectures to achieve this for specific cases (Evans et al., 2018a; Rocktaschel ¨ et al., 2015). Our work is distinct in terms of being motivated by interpretability in the low-sample regime rather than getting connectionist/neural networks to explicitly do symbolic manipulations or building hybrid models that do logic. ", + "bbox": [ + 174, + 361, + 825, + 444 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "More recently, Yin et al. (2018), propose a probabilistic latent variable model of structured programs which explain textual observations for tasks like parsing, i.e., the model from Yin et al. (2018) does not capture “how” to execute the programs that it generates. Our model is more general in the sense that we have a second modality which is the output of what gets executed when the program runs, and we model both jointly in our model, by learning how to execute the program. Work from Yi et al. (2018) also shares the same goal of doing question answering with limited question program supervision while preserving the semantics of human programs. While they to approach this problem by simplifying the $p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } )$ mapping (in context of our model) by first converting the image into a table, we approach this goal by modeling a stochastic latent space. Thus the two approaches are orthogonal, as one can use our probabilistic latent space in conjunction with their model as well to better share statistics between question-program pairs. Overall, compared to these works, our work addresses probabilistic neural-symbolic learning in a more general setting, by learning how to execute programs (on raw-pixels), as well as handling parsing of questions in an end to end manner. ", + "bbox": [ + 174, + 450, + 825, + 632 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Connection to prior work on NMNs. Our V-NMN model is inspired in its parameterization of program to answer mapping from prior work on visual question answering using neural module networks such as Johnson et al. (2017); Hu et al. (2017; 2018). Distinct from these previous approaches, our explicit goal is to preserve the semantics of how a human specifies a task to be broken down in terms of instructions, while previous work was motivated by building inductive biases for reasoning (which turn out to be useful for our model as well). Given this goal, we construct a latent variable model that in some sense embeds previous work. The works from Hu et al. (2017; 2018) are more distant from our work, as they use continuous attention modules, which do not capture independence assumptions of the program being a sufficient statistic for answering the question (which makes them harder to specify completely for a human, as they would have to hallucinate attention parameters). However, apart from our focus on preserving the lexicon/specification given by a human, both approaches share the same high level goals of providing interpretable question answering with minimal supervision. ", + "bbox": [ + 173, + 638, + 825, + 819 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Beyond Gaussian Latents in Amortized Variational Inference. There has been exciting progress in developing lower-variance estimators for discrete generative models (Tucker et al., 2017; Mnih & Gregor, 2014; Titsias & Lazaro-Gredilla, 2015; Gu & Ling, 2015). In particular, Maddison ´ et al. (2017); Jang et al. (2017) propose the use of the concrete distribution on the probability simplex for providing biased but low variance estimators for discrete latent variable models by considering a reparametrization of samples (which asymptotically approach those from a categorical distribution). The contribution in our work is orthogonal to these approaches, how one can apply such parameterizations to neuro-symbolic models. More relevant to our work is Miao & Blunsom (2016) which proposes a discrete sequential latent variable model for sentence summarization, with summaries as the latent variable. In contrast to their model, ours is a joint model where the latent space consists of programs, not summaries, which have to explain questions and answers to images. ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 159 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Joint Variational Autoencoder Models. Vedantam et al. (2018) propose a joint model for images and attributes for “visual imagination”, Suzuki et al. (2017) propose a closely related model and show that one can use the model for conditional generation, while Wang et al. (2016a) propose a deep variational canonical correlation analysis model which learns a joint space between two modalities for denoising and representation learning. In contrast to these approaches, our joint model assumes a discrete structured latent variable (instead of a Gaussian), does semi-supervised training, which is possible for us, but not possible when there are no observations from the latent available for supervision, and is targeted towards neural module networks and question answering. ", + "bbox": [ + 173, + 166, + 825, + 277 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Program Induction and Neural Architecture Search. In program induction one is typically interested in learning to write programs given specifications of input-output pairs (Reed & de Freitas, 2015; Kalyan et al., 2018), and optionally language (Neelakantan et al., 2015; Guu et al., 2017) or vision (Gaunt et al., 2017). The key difference between program induction and our problem is that program induction assumes the tokens/instructions in the language are grounded to the executor, i.e., it is assumed that once a valid program is generated there is a black-box execution engine that can execute it, while we learn the execution engine from scratch, and learn the grounding from program symbols to parameters of a neural network for execution. Neural architecture search (Zoph & Le, 2016) is another closely related problem where one is given input output examples for a machine learning task, and the goal is to find a “neural program” that does well on the task. In contrast, ours can be seen as the problem of inducing programs conditioned on particular inputs (questions), where the number of samples seen per question is significantly lower than the number of samples one would observe for a machine learning task of interest. ", + "bbox": [ + 174, + 285, + 825, + 465 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 491, + 326, + 507 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Dataset. We perform our experiments on the SHAPES dataset (Andreas et al., 2016a). This dataset has been used in previous works (Andreas et al., 2016a; Hu et al., 2017) on visual question answering using neural module networks, and is intended to be a test for compositional generalization. The dataset has 244 unique binary questions (i.e. having yes/no answers) and 15,616 images in total (Andreas et al., 2016a). The dataset also has annotated programs for each of the questions. We use train, val, and test splits of 13,568, 1,024, and 1,024 $( { \\bf x } , { \\bf z } , \\mathbf { i } , { \\bf a } )$ triplets respectively. No questions are repeated across these splits. Further, each question is asked for 64 different images, ensuring that a question can be answered correctly across all these instances only if it is processing the image in a reasonable way (as dictated by the underlying program), as opposed to leveraging any image-question correlations or biases that are unlikely to generalize across images. ", + "bbox": [ + 173, + 526, + 825, + 666 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Thus, the dataset explicitly tests for compositional generalization and “reasoning” in the task of visual question answering. The longest questions in the dataset are of length 11 and shortest are of length 4, while the longest programs are of length 6 and shortest programs are of length 4. The size of the question vocabulary $\\mathcal { V } _ { \\mathbf { x } }$ is 14 and the program vocabulary $\\nu _ { \\mathbf { z } }$ is 12. ", + "bbox": [ + 173, + 672, + 825, + 728 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Training. To simulate a data-sparse regime, we restrict the set of question-aligned programs to 5, 10, 15, or $20 \\%$ of unique questions – such that even at the highest level of supervision, programs for $80 \\%$ of unique question have never been seen during training. Unless specified explicitly, we train our program prior using a set of 1848 unique programs simulated from the syntax (more details of this can be seen in the appendix). In general, it might not be possible to exhaustively enumerate the space of all valid strings to learn the prior. Thus, we also evaluate the effect of training the prior with varying-sized subsets of the 212 programs present in the training set (without using any corresponding questions or answers) to check how robust overall performance is to the choice of the prior (in the appendix). ", + "bbox": [ + 174, + 736, + 825, + 861 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We find sequence-level policy-gradient based training in this setting can be highly variable – with some runs failing spectacularly while others succeed – for both our method and to some extent NMN (Johnson et al., 2017). For fairness and to reduce the impact of stochasticity, we perform the following stage-wise training regime for all models. First, the question-coding stage is run for 5 different random seeds, producing multiple program predictors. Programs taken from the model that performs the best on the validation set are used to train 10 runs of the module training stage. Finally, the best of these module networks (based on val) is trained jointly with the program predictor under settings of $\\gamma \\in \\{ 1 , 1 0 , 1 0 0 \\}$ and the max (on val) is chosen. We report mean and variance accuracies on test from this training regime over four random sets of supervisory programs. Our evaluation metrics are as follows: a) question-coding uses reconstruction accuracy (of getting the entire question right) and program prediction accuracy (of matching the ground truth programs) on the training set, since the second stage, module training will use inferences from the first stage (on the training set to train modules), b) module training and joint training: visual question answering accuracy, by counting matches to ground truth answers. ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 825, + 242 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Implementation Details: Following Johnson et al. (2017) and Hu et al. (2017), we treat the program $\\mathbf { z } = \\{ z _ { 1 } , \\cdots , z _ { K } \\}$ as a variable length sequence of tokens with an end of sequence token, which corresponds to the prefix traversal of an execution tree. Given this representational choice, we learn the prior $p ( \\mathbf { z } )$ using maximum likelihood training on simulated samples from known syntax of programs. An alternative, interesting approach to try would be to use something like Janz et al. (2017), who acquire a generative model more directly from validity annotations. The $\\log p ( \\mathbf { a } | \\mathbf { z } , \\mathbf { i } )$ term in Equation (1) is parameterized by an NMN that is dynamically assembled given the program string, while the generative model $p ( \\mathbf { x } | \\mathbf { z } )$ , and inference $( q ( \\mathbf { z } | \\mathbf { x } ) )$ terms are both sequence to sequence models. Our sequence to sequence models are based on LSTM (Hochreiter & Schmidhuber, 1997) cells. For the NMN programs, we use syntax similar to that used by Hu et al. (2017), but we replace their question-attention neural modules with “regular” modules. For clear comparison we adapt the objectives from the baseline approach, Johnson et al. (2017), to our representational choices. More details can be found in appendix. ", + "bbox": [ + 173, + 250, + 825, + 429 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 RESULTS ", + "text_level": 1, + "bbox": [ + 174, + 450, + 281, + 465 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We focus on evaluating approaches by varying the fraction of question-aligned programs (which we denote $\\% \\mathbf { x } \\mathbf { z }$ ) in Table 1. To put the numbers in context, a baseline $\\mathrm { L S T M + }$ image model, which does not use module networks, gets an accuracy of $6 3 . 0 \\%$ on Test (see Andreas et al. (2016a); Table. 2). This indicates that SHAPES has highly compositional questions which is challenging to model via conventional methods for Visual Question Answering (Agrawal et al., 2015). Overall, we make the following observations: ", + "bbox": [ + 173, + 481, + 825, + 565 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "- Our V-NMN approach consistently improves performance in data-sparse regimes. While both methods tend to improve with greater program supervision, V-NMN quickly outpaces NMN (Johnson et al., 2017), achieving test accuracies over $20 \\mathrm { - } 3 5 \\%$ points higher for ${ > } 5 \\%$ program supervision. Notably, both methods perform similarly poorly on the test set given only $5 \\%$ program supervision, suggesting this may be too few examples to learn compositional reasoning. Our question coding stage greatly improves initial program prediction. Our V-NMN approach to question coding gets to approximately double the question coding (reconstruction) accuracy as the NMN approach (col 1, question coding). This means that it effectively propagates groundings from question-aligned programs during the coding phase. Consequently, we also see improved VQA performance after the module and joint training stages which are based on predicted programs. Question reconstruction is well-correlated with overall accuracy. Despite the program space being trained directly as a latent code for questions, we find question reconstruction accuracy rises with the amount of supervision – confirming that inference in a space of discrete structured latent variables is difficult to do without some supervision to guide the mapping. ", + "bbox": [ + 176, + 568, + 826, + 761 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 3 shows sample programs for each model. We see the limited supervision negatively affects NMN program prediction, with the $5 \\%$ model resorting to simple Find[X] Answer structures. Interestingly, we find that the mistakes made by the V-NMN model, e.g., green in $5 \\%$ supervision (top-right) are also made when reconstructing the question (also substituting green for blue). Further, when the token does get corrected to blue, the question also eventually gets reconstructed (partially correctly $( 1 0 \\% )$ and then fully $( 1 5 \\% ) )$ , and the program produces the correct answer. This indicates that there is high fidelity between the learnt question space, the answer space and the latent space. ", + "bbox": [ + 173, + 763, + 825, + 861 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Effect of the Program Prior $\\mathbf { p } ( \\mathbf { z } )$ . Next, we explore the impact of regularizing the program posterior to be close to the prior $p ( \\mathbf { z } )$ , for different choices of the prior. Firstly, we disable the KL-divergence term by setting $\\beta = 0$ in Equation (2) recovering a deterministic version of our model. Compared to our full model at $10 \\%$ supervision, the performance on question coding drops to $1 2 . 7 \\%$ program ", + "bbox": [ + 174, + 867, + 823, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/a9465e677d4588c8e90a97269fb4139a55bc17698327073d8942e2e20d6d6082.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
%x←zValidation During Training StagesTest Accuracy (VQA Accuracy)
[I] Question Coding (Reconstruction) ）(Program Prediction)[II] Module Training (VQA Accuracy)[] Joint Training (VQA Accuracy)
NMN (Johnson et a., 2017) V-NMN(Ours)5- 54.86±6.759.28±1.91 17.90±3.4861.56±3.59 63.21±3.1263.08±0.78 66.61±1.5860.06±3.88 61.40±4.41
NMN (Johnsonet a.,017) V-NMN(Ours)10= 87.52±6.4524.30±2.39 65.45±11.8860.31±3.37 81.34±8.6166.51±4.10 92.85±4.8961.99±0.96 88.01±6.45
NMN (Johnson et al.,2017) V-NMN(Ours)15= 99.66±0.1947.67±5.02 86.23±3.2969.47±9.87 93.24±3.1762.43±0.49 97.97±1.2061.32±2.36 94.63±2.65
NMN (Johnson et al.,2017) V-NMN(Ours)201 99.88±0.2058.37±3.30 90.95±0.9066.17±7.02 91.12±2.6780.68±18.00 99.13±0.7278.59±19.27 95.63±1.31
", + "bbox": [ + 207, + 88, + 784, + 236 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 1: Results in $\\%$ , on the SHAPES dataset with varying amounts of question-program supervision $( \\% \\mathbf { x } \\mathbf { z } )$ , $\\beta = 0 . 1$ . ", + "bbox": [ + 176, + 246, + 823, + 273 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "prediction accuracy (from $6 5 . 4 5 \\%$ ), while the performance on question reconstruction improves from $8 7 . 5 2 \\%$ to $9 3 . 8 \\%$ . Without the KL term, the model focuses solely on reconstruction and fails to learn compositionality in the latent $\\mathbf { z }$ space, such that supervised grounding are poorly propagated to unsupervised questions as evidenced by the drop in program prediction accuracy. Note that this ties well to our high-level goal to better leverage the semantics of the lexicon provided by the user for interpretable question answering. ", + "bbox": [ + 174, + 273, + 825, + 357 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Beam search vs. sampling for module training. As explained in Section 2, we use beam search instead of sampling to optimize Equation (5). In this section we empirically study this design choice. We generally find that in low-supervision settings, where the question coding stage might not produce correct programs (for questions) as frequently, using sampling form the sequence model $q ( \\mathbf { z } | \\mathbf { x } )$ instead of beam search leads to a drop in module training performance. For example, with $10 \\%$ supervision, the module training accuracy drops from $8 1 . 3 4 \\pm 8 . 6 1$ (with beam search) to 76.31 $\\pm ~ 8 . 4 1$ (with sampling). With more supervision the choice of beam search or sampling does not matter, which makes intuitive sense, as the this means that the programs that get sampled tend to be the correct ones more often (that is, the confidence of the model increases with the improvement in correctness). ", + "bbox": [ + 173, + 363, + 825, + 502 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Effect of optimizing the true ELBO, $\\beta = 1$ . Next, we empirically validate the claim that approximate inference learned using the ELBO leads to autodecoding behavior as predicted by Alemi et al. (2018), and that better program representations for our problem can be found for $\\beta < 1$ (as shown above). At $15 \\%$ program supervision, VQA accuracy drops from $9 6 . 9 0 \\%$ $( \\pm 0 . 9 6 )$ at $\\beta = 0 . 1$ to $7 8 . 4 9 \\%$ $( \\pm \\ : 1 . 3 7 )$ at $\\beta = 1$ (for the empirical prior). Question reconstruction drops from $9 7 . 4 3 \\ \\%$ $( \\pm 3 . 4 2 )$ at $\\beta = 0 . 1$ , to $4 3 . 4 0 \\%$ $\\pm 5 . 3 4 )$ at $\\beta = 1$ , showing auto-decoding behavior (as predicted by theory). ", + "bbox": [ + 173, + 510, + 825, + 607 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Finally, the N2NMN approach (Hu et al., 2017) evaluates their question-attention based module networks in the fully unsupervised setting, getting to $9 6 . 1 9 \\%$ on TEST. However, the programs in this case become non-compositional (see Section 3), as the model leaks information from questions to answers via attention, meaning that programs no longer carry the burden of explaining the observed answers. This makes the modules less interpretable; which is one of the core motivations of this neural-symbolic class of module networks in the first place. ", + "bbox": [ + 174, + 613, + 825, + 698 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 DISCUSSION AND CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 722, + 468, + 738 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper we presented a novel, probabilistic neural symbolic model for interpretable visual question answering, that provides explanations of “what” the model is doing on unseen questions given a minimal number of annotated symbolic traces or programs of “what it should do”. We demonstrate that our formulation provides more interpretable explanations than previous work on visual question answering called neural module networks, on dataset of compositional questions about shapes (Andreas et al., 2016a). ", + "bbox": [ + 174, + 756, + 825, + 839 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "The key to our approach is a model with programs as a stochastic latent variable, which leads to better sharing of statistics across questions, yielding a latent space where program annotations for known questions propagate effectively to unknown / novel questions without annotations. ", + "bbox": [ + 174, + 847, + 825, + 888 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In general, the model family we study is quite rich, and also supports other inference queries such as counter factual explanations of what programs could have led to particular answers for a given image. ", + "bbox": [ + 173, + 895, + 823, + 924 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/389c1c47de1104d79e166cd0e7119a70107bf4d9f60d944f61b91a21466606c1.jpg", + "image_caption": [ + "Figure 3: Qualitative results: Image with the corresponding question, ground truth program, and ground truth answer (top left). Predicted programs and answers of different models (NMN in white, VNMN in gray) and resconstructed question (V-NMN) for variable amount of supervision. We find that V-NMN finds the right answer as well as the program more often than NMN (Johnson et al., 2017). " + ], + "image_footnote": [], + "bbox": [ + 235, + 101, + 758, + 261 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We hope our work inspires more work on applying such ideas to other tasks requiring compositional reasoning such as planning, navigation etc. ", + "bbox": [ + 173, + 358, + 823, + 387 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 102, + 287, + 118 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Aishwarya Agrawal, Jiasen Lu, Stanislaw Antol, Margaret Mitchell, C Lawrence Zitnick, Dhruv Batra, and Devi Parikh. VQA: Visual question answering. May 2015. ", + "bbox": [ + 174, + 126, + 823, + 155 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "Alexander A Alemi, Ben Poole, Ian Fischer, Joshua V Dillon, Rif A Saurous, and Kevin Murphy. Fixing a broken ELBO. November 2018. 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", + "bbox": [ + 174, + 159, + 823, + 188 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Sequence to Sequence Models: All our sequence to sequence models described in the main paper (Section. 2) are based on LSTM cells with a hidden state of 128 units, single layer depth, and have a word embedding of 32 dimensions for both the question as well as the program vocabulary. ", + "bbox": [ + 174, + 194, + 825, + 237 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "The when sampling from a model, we sample till the maximum sequence length of 15 for questions and 7 for programs. ", + "bbox": [ + 173, + 243, + 823, + 272 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Image CNN: The SHAPES images are of $3 0 \\times 3 0$ in size and are processed by a two layered convolutional neural network, which in its first layer has a $\\mathsf { 1 0 x 1 0 }$ filters applied at a stride of 10, and in the second layer, it does $\\beth \\times \\beth$ convolution applied at a stride of 1. Channel widths for both the layers are 64 dimensional. ", + "bbox": [ + 173, + 279, + 825, + 335 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Moving average baseline For a reward $R$ , we use an action independent baseline $b$ to reduce variance, which tracks the moving average of the rewards seen so far during training. Concretely, the form for the $q ( \\mathbf { z } | \\mathbf { x } )$ network is: ", + "bbox": [ + 176, + 340, + 825, + 383 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/20d7edd868bfad3730af54be971353f7de3a46a13785916de2a83e3d34fa15a2.jpg", + "text": "$$\n\\nabla J ( \\theta ) = \\mathbf { E _ { z \\sim q ( z | x ) } } [ ( R - b _ { t } ) \\nabla \\log q ( \\mathbf { z } | \\mathbf { x } ) ]\n$$", + "text_format": "latex", + "bbox": [ + 356, + 391, + 642, + 410 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "where, given $D$ as the decay rate for the baseline, ", + "bbox": [ + 176, + 415, + 498, + 429 + ], + "page_idx": 13 + }, + { + "type": "equation", + "img_path": "images/60d95e2088acc934690fee08f4d0d573f1ee4565f3de3df2bf430aa2de40204c.jpg", + "text": "$$\nb _ { t } = b _ { t - 1 } + D * \\left( R - b _ { t - 1 } \\right)\n$$", + "text_format": "latex", + "bbox": [ + 403, + 435, + 594, + 454 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "is the update on the baseline performed at every step of training. ", + "bbox": [ + 173, + 460, + 594, + 474 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Note that in practice, when estimating the gradient for the variational lower bound, we consider the total derivative (which adds a term having a gradient with respect to $R$ above), since in the case of sample-based optimization of the elbo, the reward itself is also a function of the variational parameters, via. both the sample used estimate the reward, as well as the reward directly being a function of $q ( \\mathbf { z } | \\mathbf { x } )$ . ", + "bbox": [ + 174, + 481, + 825, + 553 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Hyperparameters for training: We use the ADAM optimizer (Kingma & Ba, 2014) with a learning rate of $\\beth \\in - 3$ , a minibatch size of 576, and use a moving average baseline for REINFORCE, with a decay factor of 0.99. Typical values of $\\beta$ are set to 0.1, $\\alpha$ is set to 100 and $\\gamma$ is chosen on a validation set among (1.0, 10.0, 100.0). ", + "bbox": [ + 174, + 558, + 825, + 613 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Simulating programs from known syntax: We follow a simple two-stage heuristic procedure for generating a set of samples to train the program prior in our model. Firstly, we build a list of possible future tokens given a current token, and sample a random token from that list, and repeat the procedure for the new token to get a large set of possible valid sequences – we then pass the set of candidate sampled sequences through a second stage of filtering based on constraints from the work of Hu et al. (2017). ", + "bbox": [ + 174, + 621, + 825, + 704 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": ".2 ADDITIONAL RESULTS ", + "text_level": 1, + "bbox": [ + 176, + 722, + 362, + 736 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Empirical vs. syntactic priors While our default choice of the prior $p ( \\mathbf { z } )$ is from programs simulated from the known syntax, it might not be possible to exhaustively enumerate all valid program strings in general. Here, we consider the special case where we train the prior on a set of unaligned, ground truth programs from the dataset. Interestingly, we find that the performance of the question coding stage, especially at reconstructing the questions improves significantly when we have the syntactic prior as opposed to the empirical prior (from $3 8 . 3 9 \\pm 1 1 . 9 2 \\%$ to $5 4 . 8 6 \\pm 6 . 7 5 \\%$ for $5 \\%$ program supervision). In terms of program prediction accuracy, we also observe marginal improvements in the cases where we have $5 \\%$ and $10 \\%$ supervision respectively, from $5 6 . 2 3 \\pm 2 . 8 1 \\%$ to $6 5 . 4 5 \\pm$ $1 1 . 8 8 \\%$ . When more supervision is available, regularizing with respect to the broader, syntactic prior hurts performance marginally, which makes sense, as one can just treat program supervision as a supervised learning problem in this setting. 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For", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "such high-level tasks, a sensible approach is to specify “what” to do using symbols and learn how to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 588, + 351, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 351, + 601 + ], + "score": 1.0, + "content": "do the task using modern representation learning techniques.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 32.5, + "bbox_fs": [ + 105, + 533, + 506, + 601 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 605, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 506, + 618 + ], + "score": 1.0, + "content": "For the example shown in Figure 1, given a question “Is a square to the left of a green shape?”,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 629 + ], + "score": 1.0, + "content": "one can ask the model to reason about the answer by first applying a find[green] operator, then", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "find[left], then And the result together with a find[square] operator in order to predict", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "the answer, in other words specifying “what” are the computations we desire to be executed, in the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 661 + ], + "score": 1.0, + "content": "form of a “program”. 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Third, we parameterize the", + "type": "text" + }, + { + "bbox": [ + 258, + 317, + 293, + 329 + ], + "score": 0.93, + "content": "p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "term, which predicts an answer given an image and a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "program using a popular, related class of question answering models called neural module networks", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 339, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 351 + ], + "score": 1.0, + "content": "(NMNs) (Hu et al., 2017; Johnson et al., 2017; Andreas et al., 2016a) which previous work has shown", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "leads to better fit between the program (or “what the model is doing”) and the execution (or “how it", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "is doing what it is doing”). 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Third, we parameterize the", + "type": "text" + }, + { + "bbox": [ + 258, + 317, + 293, + 329 + ], + "score": 0.93, + "content": "p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 317, + 506, + 330 + ], + "score": 1.0, + "content": "term, which predicts an answer given an image and a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "score": 1.0, + "content": "program using a popular, related class of question answering models called neural module networks", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 339, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 351 + ], + "score": 1.0, + "content": "(NMNs) (Hu et al., 2017; Johnson et al., 2017; Andreas et al., 2016a) which previous work has shown", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 506, + 362 + ], + "score": 1.0, + "content": "leads to better fit between the program (or “what the model is doing”) and the execution (or “how it", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "is doing what it is doing”). Given the close ties of our model to NMNs, we will refer to our model as", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 371, + 389, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 371, + 389, + 385 + ], + "score": 1.0, + "content": "variational neural module networks (V-NMN) for the rest of the paper.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 228, + 506, + 385 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 388, + 505, + 477 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 505, + 402 + ], + "score": 1.0, + "content": "Apart from our modeling contribution of a probabilistic latent variable, neuro-symbolic model, our key", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 398, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 506, + 414 + ], + "score": 1.0, + "content": "technical contribution is to show how to train this model in the context of visual question answering.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 411, + 505, + 423 + ], + "spans": [ + { + "bbox": [ + 106, + 411, + 505, + 423 + ], + "score": 1.0, + "content": "Optimizing such generative models with structured discrete latent variables is an intractable problem", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 105, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "to solve in genereal – for this particular case, we formulate training into stages of grounding questions", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 432, + 504, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 504, + 444 + ], + "score": 1.0, + "content": "into programs (question coding), learning how to execute them and then with those initializations", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 505, + 456 + ], + "score": 1.0, + "content": "train the full variational objective. 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Learning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 117, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 117, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "meaningful latent variable models that are informative of observations for such models is difficult", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 117, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 117, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "(c.f. 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One can immediately notice that the standard", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 113, + 649 + ], + "score": 1.0, + "content": "e", + "type": "text" + }, + { + "bbox": [ + 114, + 640, + 175, + 649 + ], + "score": 0.81, + "content": "\\mathrm { b o } = - D - R", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 637, + 505, + 649 + ], + "score": 1.0, + "content": ". This means that for the same value of the elbo one can get different models which", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 462, + 660 + ], + "score": 1.0, + "content": "make drastically different usage of the latent variable (based on the achieved values of", + "type": "text" + }, + { + "bbox": [ + 462, + 651, + 471, + 659 + ], + "score": 0.89, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 648, + 491, + 660 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 491, + 651, + 500, + 659 + ], + "score": 0.85, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 648, + 506, + 660 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 410, + 671 + ], + "score": 1.0, + "content": "Thus, one way to achieve a desired behavior (of say high mutual information", + "type": "text" + }, + { + "bbox": [ + 411, + 662, + 439, + 672 + ], + "score": 0.93, + "content": "I ( \\mathbf { x } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 660, + 480, + 671 + ], + "score": 1.0, + "content": "), is to set", + "type": "text" + }, + { + "bbox": [ + 480, + 662, + 487, + 672 + ], + "score": 0.87, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 660, + 506, + 671 + ], + "score": 1.0, + "content": "to a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "lower value than the standard elbo (c.f. Eqn. 6 in Alemi et al. (2018).) This aligns with our goal in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 232, + 694 + ], + "score": 1.0, + "content": "question coding, and so we use", + "type": "text" + }, + { + "bbox": [ + 233, + 684, + 258, + 693 + ], + "score": 0.92, + "content": "\\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 681, + 506, + 694 + ], + "score": 1.0, + "content": ". As this no longer a valid variational lower bound, denote our", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 693, + 254, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 156, + 705 + ], + "score": 1.0, + "content": "objective as", + "type": "text" + }, + { + "bbox": [ + 156, + 694, + 197, + 705 + ], + "score": 0.93, + "content": "J ( \\mathbf { x } ; \\phi , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 693, + 254, + 705 + ], + "score": 1.0, + "content": "from now on.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "We optimize the above objective using the score function estimator (Glynn, 1990), also referred to", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "as REINFORCE (Williams, 1992) applied to the first term, with a moving average baseline.Finally,", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51.5 + } + ], + "page_idx": 3, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 106 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 105, + 110, + 503, + 134 + ], + "lines": [ + { + "bbox": [ + 105, + 108, + 505, + 124 + ], + "spans": [ + { + "bbox": [ + 105, + 108, + 452, + 124 + ], + "score": 1.0, + "content": "Inference. At test time, we can answer questions for images by computing", + "type": "text" + }, + { + "bbox": [ + 452, + 110, + 505, + 123 + ], + "score": 0.89, + "content": "p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { x } ) \\ =", + "type": "inline_equation" + } + ], + "index": 2 + }, + { + "bbox": [ + 107, + 120, + 421, + 136 + ], + "spans": [ + { + "bbox": [ + 107, + 121, + 185, + 134 + ], + "score": 0.92, + "content": "\\begin{array} { r } { \\sum _ { z } p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } ) p ( \\mathbf { z } | \\mathbf { x } ) } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 120, + 283, + 136 + ], + "score": 1.0, + "content": "using an approximation", + "type": "text" + }, + { + "bbox": [ + 283, + 121, + 416, + 135 + ], + "score": 0.92, + "content": "p _ { q } ( \\mathbf { \\bar { a } } | \\mathbf { i } , \\mathbf { x } ) \\approx E _ { \\mathbf { z } _ { i } \\sim q ( \\mathbf { z } | \\mathbf { x } ) } p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } _ { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 416, + 120, + 421, + 136 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5, + "bbox_fs": [ + 105, + 108, + 505, + 136 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 138, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "Challenges. While the elbo term in Equation (1) appears similar to a standard ELBO for joint latent", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 506, + 162 + ], + "score": 1.0, + "content": "variable models (Vedantam et al., 2018; Wang et al., 2016b; Suzuki et al., 2017), there are a few", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 161, + 195, + 173 + ], + "spans": [ + { + "bbox": [ + 106, + 161, + 195, + 173 + ], + "score": 1.0, + "content": "important departures:", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 138, + 506, + 173 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 173, + 506, + 316 + ], + "lines": [ + { + "bbox": [ + 110, + 173, + 505, + 186 + ], + "spans": [ + { + "bbox": [ + 110, + 173, + 229, + 186 + ], + "score": 1.0, + "content": "- Firstly, the latent variable", + "type": "text" + }, + { + "bbox": [ + 229, + 176, + 236, + 183 + ], + "score": 0.47, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 173, + 505, + 186 + ], + "score": 1.0, + "content": "is not a Gaussian or categorical random variable (which admit", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 116, + 183, + 505, + 198 + ], + "spans": [ + { + "bbox": [ + 116, + 183, + 505, + 198 + ], + "score": 1.0, + "content": "straightforward reparameterizations (Maddison et al., 2017; Jang et al., 2016)) but rather a sequence", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 117, + 195, + 438, + 208 + ], + "spans": [ + { + "bbox": [ + 117, + 195, + 438, + 208 + ], + "score": 1.0, + "content": "of discrete tokens, which complicates the optimization problem in Equation (1).", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 115, + 206, + 505, + 219 + ], + "spans": [ + { + "bbox": [ + 115, + 206, + 505, + 219 + ], + "score": 1.0, + "content": "Secondly, the conditional log-likelihood for a is parameterized by a NMN that is dynamically", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 117, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 117, + 217, + 318, + 230 + ], + "score": 1.0, + "content": "assembled based on sample programs drawn from", + "type": "text" + }, + { + "bbox": [ + 318, + 217, + 345, + 229 + ], + "score": 0.92, + "content": "q ( \\mathbf { z } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 346, + 217, + 505, + 230 + ], + "score": 1.0, + "content": ". Depending upon the uncertainty of the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 116, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 116, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "program posterior, program tokens may not be consistently grounded across examples, making", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 117, + 239, + 456, + 252 + ], + "spans": [ + { + "bbox": [ + 117, + 239, + 456, + 252 + ], + "score": 1.0, + "content": "learning the appropriate computation for the corresponding neural modules difficult.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 113, + 248, + 506, + 264 + ], + "spans": [ + { + "bbox": [ + 113, + 248, + 506, + 264 + ], + "score": 1.0, + "content": "Lastly, our observations x are explained by powerful “autoregressive” sequence models. Learning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 117, + 261, + 505, + 273 + ], + "spans": [ + { + "bbox": [ + 117, + 261, + 505, + 273 + ], + "score": 1.0, + "content": "meaningful latent variable models that are informative of observations for such models is difficult", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 117, + 271, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 117, + 271, + 506, + 285 + ], + "score": 1.0, + "content": "(c.f. (Bowman et al., 2016; Chen et al., 2016; Alemi et al., 2018)) as powerful decoder models", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 116, + 282, + 505, + 295 + ], + "spans": [ + { + "bbox": [ + 116, + 282, + 505, + 295 + ], + "score": 1.0, + "content": "can explain the data using stochasticity from sampling (referred to as autodecoding in Alemi", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 117, + 294, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 117, + 294, + 505, + 306 + ], + "score": 1.0, + "content": "et al. (2018)). Consequently, it also becomes hard to learn meaningful NMN parameters from the", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 117, + 306, + 175, + 317 + ], + "spans": [ + { + "bbox": [ + 117, + 306, + 175, + 317 + ], + "score": 1.0, + "content": "previous step.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 13, + "bbox_fs": [ + 110, + 173, + 506, + 317 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 321, + 506, + 366 + ], + "lines": [ + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "spans": [ + { + "bbox": [ + 105, + 321, + 506, + 334 + ], + "score": 1.0, + "content": "In the following sections, we present a training regime and adjustments to the objective in Equation (1)", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "spans": [ + { + "bbox": [ + 105, + 333, + 506, + 345 + ], + "score": 1.0, + "content": "that make optimization more tractable and enable training NMNs with significantly reduced question-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 106, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "aligned program supervision. While instantiated for question answering, it is not difficult to conceive", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 354, + 474, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 474, + 367 + ], + "score": 1.0, + "content": "versions of this regime for other problems relevant to probabilistic neural-symbolic models.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 321, + 506, + 367 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 378, + 291, + 390 + ], + "lines": [ + { + "bbox": [ + 105, + 378, + 292, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 292, + 391 + ], + "score": 1.0, + "content": "2.1 OPTIMIZING THE ELBO FOR V-NMN", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 108, + 399, + 486, + 411 + ], + "lines": [ + { + "bbox": [ + 105, + 398, + 486, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 486, + 413 + ], + "score": 1.0, + "content": "Rather than approaching the joint objective directly, we break ELBO training into three stages:", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 398, + 486, + 413 + ] + }, + { + "type": "text", + "bbox": [ + 109, + 412, + 506, + 457 + ], + "lines": [ + { + "bbox": [ + 108, + 411, + 501, + 425 + ], + "spans": [ + { + "bbox": [ + 108, + 411, + 501, + 425 + ], + "score": 1.0, + "content": "1. Question coding, where we learn a latent ‘code’ for questions in the latent space of programs,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 107, + 422, + 507, + 437 + ], + "spans": [ + { + "bbox": [ + 107, + 422, + 507, + 437 + ], + "score": 1.0, + "content": "2. Module training, where we learn the ‘execution’ of the programs from the first stage of training,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 121, + 434, + 438, + 447 + ], + "spans": [ + { + "bbox": [ + 121, + 434, + 438, + 447 + ], + "score": 1.0, + "content": "grounding tokens in programs into the operations they perform on images; and", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 107, + 444, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 107, + 444, + 506, + 458 + ], + "score": 1.0, + "content": "3. Joint training, where we train both stages jointly given the ‘warm-starts’ from the other phases.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 27.5, + "bbox_fs": [ + 107, + 411, + 507, + 458 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 458, + 506, + 514 + ], + "lines": [ + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 106, + 458, + 506, + 471 + ], + "score": 1.0, + "content": "Question Coding. The goal of this stage is to learn a code for questions in the latent space of pro-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 469, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 132, + 482 + ], + "score": 1.0, + "content": "grams", + "type": "text" + }, + { + "bbox": [ + 132, + 470, + 150, + 480 + ], + "score": 0.33, + "content": "- i . e", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 469, + 506, + 482 + ], + "score": 1.0, + "content": ". an informative mapping from questions to programs. Given some set of question-aligned", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 479, + 507, + 493 + ], + "spans": [ + { + "bbox": [ + 105, + 479, + 507, + 493 + ], + "score": 1.0, + "content": "programs, we find this coding process effectively propagates these supervised groundings to unsuper-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 491, + 506, + 504 + ], + "spans": [ + { + "bbox": [ + 105, + 491, + 334, + 504 + ], + "score": 1.0, + "content": "vised questions, which enable interpretable explanations", + "type": "text" + }, + { + "bbox": [ + 335, + 492, + 340, + 501 + ], + "score": 0.26, + "content": "/", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 491, + 506, + 504 + ], + "score": 1.0, + "content": "groundings of queries to explanations of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 501, + 322, + 515 + ], + "spans": [ + { + "bbox": [ + 104, + 501, + 322, + 515 + ], + "score": 1.0, + "content": "“what” the model is doing, which is our primary goal.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 32, + "bbox_fs": [ + 104, + 458, + 507, + 515 + ] + }, + { + "type": "text", + "bbox": [ + 110, + 518, + 458, + 531 + ], + "lines": [ + { + "bbox": [ + 108, + 517, + 457, + 533 + ], + "spans": [ + { + "bbox": [ + 108, + 517, + 255, + 533 + ], + "score": 1.0, + "content": "During question coding, we drop the", + "type": "text" + }, + { + "bbox": [ + 255, + 519, + 298, + 531 + ], + "score": 0.92, + "content": "p _ { \\theta _ { z } } ( \\mathbf { a } | \\mathbf { z } , \\mathbf { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 517, + 457, + 533 + ], + "score": 1.0, + "content": "term from Eq. 1, yielding the objective", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35, + "bbox_fs": [ + 108, + 517, + 457, + 533 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 123, + 532, + 476, + 546 + ], + "lines": [ + { + "bbox": [ + 123, + 532, + 476, + 546 + ], + "spans": [ + { + "bbox": [ + 123, + 532, + 476, + 546 + ], + "score": 0.81, + "content": "J ( \\mathbf { x } ; \\phi , \\theta ) = E _ { z \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } [ \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) ] - \\beta \\mathrm { K L } ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) , p ( \\mathbf { z } ) ) + \\alpha \\log r _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) \\cdot [ [ \\mathbf { x } \\in \\mathcal { S } ]", + "type": "interline_equation", + "image_path": "152d55333f95cdabe41fe967d8e131b253df47ca7a31b573398e9b0ca8937a6f.jpg" + } + ] + } + ], + "index": 36, + "virtual_lines": [ + { + "bbox": [ + 123, + 532, + 476, + 546 + ], + "spans": [], + "index": 36 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 547, + 505, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 546, + 505, + 560 + ], + "spans": [ + { + "bbox": [ + 105, + 546, + 433, + 560 + ], + "score": 1.0, + "content": "Recent theory from Alemi et al. (2018) prescribes changes to the ELBO, setting", + "type": "text" + }, + { + "bbox": [ + 433, + 547, + 459, + 559 + ], + "score": 0.9, + "content": "\\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 459, + 546, + 505, + 560 + ], + "score": 1.0, + "content": "as a recipe", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 505, + 570 + ], + "score": 1.0, + "content": "to learn representations which avoid learning degenerate latent representations in the presence", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "spans": [ + { + "bbox": [ + 106, + 568, + 505, + 581 + ], + "score": 1.0, + "content": "of powerful decoders. Essentially, Alemi et al. (2018) identify that upto a constant factor, the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 578, + 506, + 594 + ], + "spans": [ + { + "bbox": [ + 104, + 578, + 265, + 594 + ], + "score": 1.0, + "content": "negative log-marignal likelihood term", + "type": "text" + }, + { + "bbox": [ + 266, + 581, + 383, + 594 + ], + "score": 0.93, + "content": "D { = } { - } E _ { z { \\sim } q _ { \\phi } ( { \\bf z } | { \\bf x } ) } \\big [ \\log p _ { \\theta } ( { \\bf x } | { \\bf z } ) \\big ]", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 578, + 506, + 594 + ], + "score": 1.0, + "content": ", and the kl diveregence term", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 590, + 504, + 605 + ], + "spans": [ + { + "bbox": [ + 107, + 594, + 204, + 605 + ], + "score": 0.91, + "content": "R = \\mathrm { K L } ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) , p ( \\mathbf { z } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 204, + 590, + 398, + 605 + ], + "score": 1.0, + "content": "bound the mutual information between the data", + "type": "text" + }, + { + "bbox": [ + 398, + 594, + 405, + 602 + ], + "score": 0.71, + "content": "\\mathbf { x }", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 590, + 497, + 605 + ], + "score": 1.0, + "content": "and the latent variable", + "type": "text" + }, + { + "bbox": [ + 498, + 595, + 504, + 602 + ], + "score": 0.49, + "content": "\\mathbf { z }", + "type": "inline_equation" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 603, + 152, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 603, + 152, + 615 + ], + "score": 1.0, + "content": "as follows:", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 546, + 506, + 615 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 258, + 614, + 353, + 627 + ], + "lines": [ + { + "bbox": [ + 258, + 614, + 353, + 627 + ], + "spans": [ + { + "bbox": [ + 258, + 614, + 353, + 627 + ], + "score": 0.94, + "content": "H - D \\leq I ( \\mathbf { x } , \\mathbf { z } ) \\leq R", + "type": "interline_equation", + "image_path": "3c6a5a713fb072d09386cfae428568641934b2308100c5d0e20effedee16b3b7.jpg" + } + ] + } + ], + "index": 43, + "virtual_lines": [ + { + "bbox": [ + 258, + 614, + 353, + 627 + ], + "spans": [], + "index": 43 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 627, + 505, + 705 + ], + "lines": [ + { + "bbox": [ + 105, + 626, + 506, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 132, + 639 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 629, + 142, + 637 + ], + "score": 0.86, + "content": "H", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 626, + 506, + 639 + ], + "score": 1.0, + "content": "is the entropy of the data, which is a constant. One can immediately notice that the standard", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 637, + 505, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 637, + 113, + 649 + ], + "score": 1.0, + "content": "e", + "type": "text" + }, + { + "bbox": [ + 114, + 640, + 175, + 649 + ], + "score": 0.81, + "content": "\\mathrm { b o } = - D - R", + "type": "inline_equation" + }, + { + "bbox": [ + 175, + 637, + 505, + 649 + ], + "score": 1.0, + "content": ". This means that for the same value of the elbo one can get different models which", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 648, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 106, + 648, + 462, + 660 + ], + "score": 1.0, + "content": "make drastically different usage of the latent variable (based on the achieved values of", + "type": "text" + }, + { + "bbox": [ + 462, + 651, + 471, + 659 + ], + "score": 0.89, + "content": "D", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 648, + 491, + 660 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 491, + 651, + 500, + 659 + ], + "score": 0.85, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 500, + 648, + 506, + 660 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 660, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 410, + 671 + ], + "score": 1.0, + "content": "Thus, one way to achieve a desired behavior (of say high mutual information", + "type": "text" + }, + { + "bbox": [ + 411, + 662, + 439, + 672 + ], + "score": 0.93, + "content": "I ( \\mathbf { x } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 660, + 480, + 671 + ], + "score": 1.0, + "content": "), is to set", + "type": "text" + }, + { + "bbox": [ + 480, + 662, + 487, + 672 + ], + "score": 0.87, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 660, + 506, + 671 + ], + "score": 1.0, + "content": "to a", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 683 + ], + "score": 1.0, + "content": "lower value than the standard elbo (c.f. Eqn. 6 in Alemi et al. (2018).) This aligns with our goal in", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 681, + 506, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 232, + 694 + ], + "score": 1.0, + "content": "question coding, and so we use", + "type": "text" + }, + { + "bbox": [ + 233, + 684, + 258, + 693 + ], + "score": 0.92, + "content": "\\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 681, + 506, + 694 + ], + "score": 1.0, + "content": ". As this no longer a valid variational lower bound, denote our", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 693, + 254, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 156, + 705 + ], + "score": 1.0, + "content": "objective as", + "type": "text" + }, + { + "bbox": [ + 156, + 694, + 197, + 705 + ], + "score": 0.93, + "content": "J ( \\mathbf { x } ; \\phi , \\theta )", + "type": "inline_equation" + }, + { + "bbox": [ + 197, + 693, + 254, + 705 + ], + "score": 1.0, + "content": "from now on.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 626, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "We optimize the above objective using the score function estimator (Glynn, 1990), also referred to", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "as REINFORCE (Williams, 1992) applied to the first term, with a moving average baseline.Finally,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "while the KL term (in the reward) encourages the model to generate high-likelihood plausible", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 444, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 444, + 107 + ], + "score": 1.0, + "content": "sequences, we also penalize the model for producing syntactically invalid programs.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 51.5, + "bbox_fs": [ + 105, + 709, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "while the KL term (in the reward) encourages the model to generate high-likelihood plausible", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 444, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 444, + 107 + ], + "score": 1.0, + "content": "sequences, we also penalize the model for producing syntactically invalid programs.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "score": 1.0, + "content": "Although gradients for the KL term in Equation (2) can be computed by decomposing the KL for the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "score": 1.0, + "content": "sequence into per-timestep monte-carlo gradients, we found that having one term with per-timestep", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "gradients and another that only receives sequence-level rewards (the likelihood term) skewed the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "model towards minimizing the KL, leading to auto-decoding solutions. Thus, we apply REINFORCE", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "on the entire sequence and compute rewards using the full monte carlo gradient,i.e., we do not", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 330, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 330, + 177 + ], + "score": 1.0, + "content": "optimize the KL term separately, but instead maximize:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5 + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 182, + 427, + 210 + ], + "lines": [ + { + "bbox": [ + 184, + 182, + 427, + 210 + ], + "spans": [ + { + "bbox": [ + 184, + 182, + 427, + 210 + ], + "score": 0.94, + "content": "E _ { \\mathbf { z } \\sim q _ { \\phi } ( \\mathbf { z } | \\hat { \\mathbf { x } } ) } \\left[ \\log \\left[ \\frac { p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { z } ) p ( \\mathbf { z } ) ^ { \\beta } } { q _ { \\phi } ( \\mathbf { z } | \\hat { \\mathbf { x } } ) ^ { \\beta } } \\right] \\right] + \\alpha \\log r _ { \\phi } ( \\hat { \\mathbf { z } } | \\hat { \\mathbf { x } } ) \\cdot \\left[ \\left[ \\mathbf { x } \\in \\mathcal { S } \\right] \\right]", + "type": "interline_equation", + "image_path": "c4a38f2c0d879d16bbeaabd46889bc590795fe032f4f259dfb89fb59fea9d517.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 184, + 182, + 427, + 196.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 184, + 196.0, + 427, + 210.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 381, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 382, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 133, + 227 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 218, + 149, + 227 + ], + "score": 0.84, + "content": "\\hat { \\mathbf { x } } , \\hat { \\mathbf { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 215, + 382, + 227 + ], + "score": 1.0, + "content": "are the questions and programs we observe in the dataset.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 246 + ], + "score": 1.0, + "content": "For comparison, the semi-supervised approach in Johnson et al. (2017) can be seen as optimizing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "the last term on supervised training data, while our question coding stage optimizes the elbo in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "addition to the last term. Our results in Section 5 show that the proposed question coding formulation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 506, + 277 + ], + "score": 1.0, + "content": "significantly improves performance relative to Johnson et al. (2017) in terms of explaining “what”", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 276, + 254, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 254, + 289 + ], + "score": 1.0, + "content": "the model is doing on test programs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "Module Training. Having obtained a code for questions in the latent space, we train the parameters", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "of the NMN for question answering. The network modules are trained to perform their corresponding", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 316, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 427, + 327 + ], + "score": 1.0, + "content": "tasks. For example, if the question coding learned to associate the program token", + "type": "text" + }, + { + "bbox": [ + 427, + 317, + 434, + 325 + ], + "score": 0.7, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 316, + 505, + 327 + ], + "score": 1.0, + "content": "with the question", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 487, + 339 + ], + "score": 1.0, + "content": "phrase ‘to the left’, then the goal for this second stage is to train the module corresponding to", + "type": "text" + }, + { + "bbox": [ + 487, + 328, + 493, + 336 + ], + "score": 0.75, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 337, + 454, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 454, + 350 + ], + "score": 1.0, + "content": "actually perform this operation (e.g. transform[left]) when executed on images.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 504, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 261, + 366 + ], + "score": 1.0, + "content": "We do this by optimizing the joint elb", + "type": "text" + }, + { + "bbox": [ + 262, + 354, + 293, + 366 + ], + "score": 0.65, + "content": "{ \\bf \\alpha } ) ( { \\bf x } , { \\bf a } | \\bf { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 353, + 440, + 366 + ], + "score": 1.0, + "content": "in (1) with respect to the parameters", + "type": "text" + }, + { + "bbox": [ + 441, + 354, + 451, + 365 + ], + "score": 0.89, + "content": "\\theta _ { \\mathbf { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "of the neural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 364, + 321, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 321, + 378 + ], + "score": 1.0, + "content": "modules. We write this objective for each instance as", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 381, + 393, + 396 + ], + "lines": [ + { + "bbox": [ + 218, + 381, + 393, + 396 + ], + "spans": [ + { + "bbox": [ + 218, + 381, + 393, + 396 + ], + "score": 0.92, + "content": "J ( \\mathbf { a } \\mid \\mathbf { x } , \\mathbf { i } ; \\theta _ { \\mathbf { z } } ) = E _ { z \\sim q ( \\mathbf { z } \\mid \\mathbf { x } ) } [ \\log p _ { \\theta _ { z } } ( \\mathbf { a } \\mid \\mathbf { i } , \\mathbf { z } ) ]", + "type": "interline_equation", + "image_path": "25a3b10307f473abd2705dbaed2e8549284420c0b00e57e8e497592dabff2a46.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 218, + 381, + 393, + 396 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 400, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 400, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 477, + 413 + ], + "score": 1.0, + "content": "Notably, while we write this objective as a function of all neural modular network parameters", + "type": "text" + }, + { + "bbox": [ + 477, + 401, + 487, + 412 + ], + "score": 0.88, + "content": "\\theta _ { \\mathbf { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 400, + 506, + 413 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 412, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 355, + 424 + ], + "score": 1.0, + "content": "a given instance, only those modules present in the program", + "type": "text" + }, + { + "bbox": [ + 355, + 412, + 402, + 424 + ], + "score": 0.93, + "content": "z \\sim q ( \\mathbf { z } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 412, + 506, + 424 + ], + "score": 1.0, + "content": "are updated. In practice,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 301, + 435 + ], + "score": 1.0, + "content": "we take approx. argmax decoded programs from", + "type": "text" + }, + { + "bbox": [ + 302, + 423, + 334, + 435 + ], + "score": 0.93, + "content": "q _ { \\phi } ( { \\bf z } | { \\bf x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "to simplify training, but perform sampling", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "during joint training in the next stage. To be more specific, we run beam search (Vijayakumar et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 417, + 457 + ], + "score": 1.0, + "content": "2018) with a beam size of 1 to do this. Intuitively, we think of this as a bias", + "type": "text" + }, + { + "bbox": [ + 418, + 447, + 427, + 455 + ], + "score": 0.25, + "content": "\\nu s", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 443, + 506, + 457 + ], + "score": 1.0, + "content": ". variance tradeoff,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "doing argmax enables us to trade off bias for high-variance from a posterior which (in the pathological", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 478 + ], + "score": 1.0, + "content": "case) could match the prior from the first stage of training. This is also consistent with standard", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 478, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 489 + ], + "score": 1.0, + "content": "practice for doing inference in sequence models, where beam search is the standard inference tool", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 488, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 500 + ], + "score": 1.0, + "content": "(and sampling often has issues due to teacher forcing based MLE training, which is how we do", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 299, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 299, + 511 + ], + "score": 1.0, + "content": "semi-supervised learning) (Bengio et al., 2015).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5 + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 504, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 530 + ], + "score": 1.0, + "content": "Joint Training. Having trained the question code and the neural module network parameters, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 426, + 540 + ], + "score": 1.0, + "content": "tune all terms jointly, with the following objective, which adds scaling factors", + "type": "text" + }, + { + "bbox": [ + 427, + 529, + 434, + 537 + ], + "score": 0.61, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 528, + 438, + 540 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 438, + 528, + 446, + 539 + ], + "score": 0.77, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 528, + 464, + 540 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 465, + 529, + 471, + 539 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 528, + 505, + 540 + ], + "score": 1.0, + "content": ", and an", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 540, + 349, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 349, + 550 + ], + "score": 1.0, + "content": "extra semi-supervised learning term relative to Equation (1):", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35 + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 555, + 489, + 570 + ], + "lines": [ + { + "bbox": [ + 111, + 555, + 489, + 570 + ], + "spans": [ + { + "bbox": [ + 111, + 555, + 489, + 570 + ], + "score": 0.84, + "content": "J ( \\mathbf { x } , \\mathbf { a } | \\mathbf { i } ) = E _ { z \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } \\big [ \\gamma \\log p _ { \\theta _ { z } } ( \\mathbf { a } | \\mathbf { z } , \\mathbf { i } ) + \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) \\big ] - \\beta \\mathrm { K L } \\big ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) , p ( \\mathbf { z } ) \\big ) + \\alpha \\log r _ { \\phi } ( \\mathbf { z } | \\mathbf { x } )", + "type": "interline_equation", + "image_path": "fd3d8a30d4332d07074d746f446cadb9c21e7fee9fbcc98319d3b4688a09540f.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 111, + 555, + 489, + 570 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 104, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 179, + 587 + ], + "score": 1.0, + "content": "The gamma factor", + "type": "text" + }, + { + "bbox": [ + 180, + 576, + 187, + 586 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 573, + 505, + 587 + ], + "score": 1.0, + "content": "controls the scaling of the likelihood term for the answers relative to the question", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "(which has more bits of information), similar scaling factors have been for joint models in Vedantam", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "et al. (2018). Similar to Equation (4), we optimize the full monte carlo gradient for the above objective", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 606, + 378, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 378, + 619 + ], + "score": 1.0, + "content": "as well. The full training procedure is shown in Algorithm 1 below.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5 + }, + { + "type": "title", + "bbox": [ + 108, + 634, + 211, + 647 + ], + "lines": [ + { + "bbox": [ + 104, + 633, + 213, + 649 + ], + "spans": [ + { + "bbox": [ + 104, + 633, + 213, + 649 + ], + "score": 1.0, + "content": "3 RELATED WORK", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 673 + ], + "score": 1.0, + "content": "We first explain other related work in neuro-symbolic models, the key differences between our", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "formulation and closely related NMN work (Johnson et al., 2017; Hu et al., 2017; 2018), and then", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 682, + 399, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 399, + 694 + ], + "score": 1.0, + "content": "connect our work to other themes and directions of research in literature.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Neural-symbolic models Valiant (2003) identifies neuro-symbolic models, which are able to reason", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "as well as learn efficiently as one of the key problems in computer science. A rich literature of work", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 505, + 732 + ], + "score": 1.0, + "content": "exists in this area, around two major threads: firstly, there is a line of work in building hybrid models", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 104, + 82, + 504, + 105 + ], + "lines": [], + "index": 0.5, + "bbox_fs": [ + 105, + 82, + 505, + 107 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 110, + 505, + 177 + ], + "lines": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "spans": [ + { + "bbox": [ + 106, + 110, + 506, + 123 + ], + "score": 1.0, + "content": "Although gradients for the KL term in Equation (2) can be computed by decomposing the KL for the", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "spans": [ + { + "bbox": [ + 105, + 120, + 505, + 134 + ], + "score": 1.0, + "content": "sequence into per-timestep monte-carlo gradients, we found that having one term with per-timestep", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "gradients and another that only receives sequence-level rewards (the likelihood term) skewed the", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "model towards minimizing the KL, leading to auto-decoding solutions. Thus, we apply REINFORCE", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 154, + 505, + 166 + ], + "score": 1.0, + "content": "on the entire sequence and compute rewards using the full monte carlo gradient,i.e., we do not", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 164, + 330, + 177 + ], + "spans": [ + { + "bbox": [ + 106, + 164, + 330, + 177 + ], + "score": 1.0, + "content": "optimize the KL term separately, but instead maximize:", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 110, + 506, + 177 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 184, + 182, + 427, + 210 + ], + "lines": [ + { + "bbox": [ + 184, + 182, + 427, + 210 + ], + "spans": [ + { + "bbox": [ + 184, + 182, + 427, + 210 + ], + "score": 0.94, + "content": "E _ { \\mathbf { z } \\sim q _ { \\phi } ( \\mathbf { z } | \\hat { \\mathbf { x } } ) } \\left[ \\log \\left[ \\frac { p _ { \\theta } ( \\hat { \\mathbf { x } } | \\mathbf { z } ) p ( \\mathbf { z } ) ^ { \\beta } } { q _ { \\phi } ( \\mathbf { z } | \\hat { \\mathbf { x } } ) ^ { \\beta } } \\right] \\right] + \\alpha \\log r _ { \\phi } ( \\hat { \\mathbf { z } } | \\hat { \\mathbf { x } } ) \\cdot \\left[ \\left[ \\mathbf { x } \\in \\mathcal { S } \\right] \\right]", + "type": "interline_equation", + "image_path": "c4a38f2c0d879d16bbeaabd46889bc590795fe032f4f259dfb89fb59fea9d517.jpg" + } + ] + } + ], + "index": 8.5, + "virtual_lines": [ + { + "bbox": [ + 184, + 182, + 427, + 196.0 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 184, + 196.0, + 427, + 210.0 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 216, + 381, + 227 + ], + "lines": [ + { + "bbox": [ + 106, + 215, + 382, + 227 + ], + "spans": [ + { + "bbox": [ + 106, + 215, + 133, + 227 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 218, + 149, + 227 + ], + "score": 0.84, + "content": "\\hat { \\mathbf { x } } , \\hat { \\mathbf { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 215, + 382, + 227 + ], + "score": 1.0, + "content": "are the questions and programs we observe in the dataset.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10, + "bbox_fs": [ + 106, + 215, + 382, + 227 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 232, + 505, + 288 + ], + "lines": [ + { + "bbox": [ + 105, + 231, + 505, + 246 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 246 + ], + "score": 1.0, + "content": "For comparison, the semi-supervised approach in Johnson et al. (2017) can be seen as optimizing", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 505, + 256 + ], + "score": 1.0, + "content": "the last term on supervised training data, while our question coding stage optimizes the elbo in", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "addition to the last term. Our results in Section 5 show that the proposed question coding formulation", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 265, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 506, + 277 + ], + "score": 1.0, + "content": "significantly improves performance relative to Johnson et al. (2017) in terms of explaining “what”", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 276, + 254, + 289 + ], + "spans": [ + { + "bbox": [ + 106, + 276, + 254, + 289 + ], + "score": 1.0, + "content": "the model is doing on test programs.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 231, + 506, + 289 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 293, + 505, + 349 + ], + "lines": [ + { + "bbox": [ + 106, + 292, + 505, + 306 + ], + "spans": [ + { + "bbox": [ + 106, + 292, + 505, + 306 + ], + "score": 1.0, + "content": "Module Training. Having obtained a code for questions in the latent space, we train the parameters", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 505, + 317 + ], + "score": 1.0, + "content": "of the NMN for question answering. The network modules are trained to perform their corresponding", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 316, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 106, + 316, + 427, + 327 + ], + "score": 1.0, + "content": "tasks. For example, if the question coding learned to associate the program token", + "type": "text" + }, + { + "bbox": [ + 427, + 317, + 434, + 325 + ], + "score": 0.7, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 316, + 505, + 327 + ], + "score": 1.0, + "content": "with the question", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 325, + 506, + 339 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 487, + 339 + ], + "score": 1.0, + "content": "phrase ‘to the left’, then the goal for this second stage is to train the module corresponding to", + "type": "text" + }, + { + "bbox": [ + 487, + 328, + 493, + 336 + ], + "score": 0.75, + "content": "v", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 325, + 506, + 339 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 337, + 454, + 350 + ], + "spans": [ + { + "bbox": [ + 106, + 337, + 454, + 350 + ], + "score": 1.0, + "content": "actually perform this operation (e.g. transform[left]) when executed on images.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 292, + 506, + 350 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 353, + 504, + 376 + ], + "lines": [ + { + "bbox": [ + 106, + 353, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 261, + 366 + ], + "score": 1.0, + "content": "We do this by optimizing the joint elb", + "type": "text" + }, + { + "bbox": [ + 262, + 354, + 293, + 366 + ], + "score": 0.65, + "content": "{ \\bf \\alpha } ) ( { \\bf x } , { \\bf a } | \\bf { i } )", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 353, + 440, + 366 + ], + "score": 1.0, + "content": "in (1) with respect to the parameters", + "type": "text" + }, + { + "bbox": [ + 441, + 354, + 451, + 365 + ], + "score": 0.89, + "content": "\\theta _ { \\mathbf { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 353, + 505, + 366 + ], + "score": 1.0, + "content": "of the neural", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 364, + 321, + 378 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 321, + 378 + ], + "score": 1.0, + "content": "modules. We write this objective for each instance as", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 353, + 505, + 378 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 218, + 381, + 393, + 396 + ], + "lines": [ + { + "bbox": [ + 218, + 381, + 393, + 396 + ], + "spans": [ + { + "bbox": [ + 218, + 381, + 393, + 396 + ], + "score": 0.92, + "content": "J ( \\mathbf { a } \\mid \\mathbf { x } , \\mathbf { i } ; \\theta _ { \\mathbf { z } } ) = E _ { z \\sim q ( \\mathbf { z } \\mid \\mathbf { x } ) } [ \\log p _ { \\theta _ { z } } ( \\mathbf { a } \\mid \\mathbf { i } , \\mathbf { z } ) ]", + "type": "interline_equation", + "image_path": "25a3b10307f473abd2705dbaed2e8549284420c0b00e57e8e497592dabff2a46.jpg" + } + ] + } + ], + "index": 23, + "virtual_lines": [ + { + "bbox": [ + 218, + 381, + 393, + 396 + ], + "spans": [], + "index": 23 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 400, + 505, + 511 + ], + "lines": [ + { + "bbox": [ + 105, + 400, + 506, + 413 + ], + "spans": [ + { + "bbox": [ + 105, + 400, + 477, + 413 + ], + "score": 1.0, + "content": "Notably, while we write this objective as a function of all neural modular network parameters", + "type": "text" + }, + { + "bbox": [ + 477, + 401, + 487, + 412 + ], + "score": 0.88, + "content": "\\theta _ { \\mathbf { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 488, + 400, + 506, + 413 + ], + "score": 1.0, + "content": ", for", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 412, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 355, + 424 + ], + "score": 1.0, + "content": "a given instance, only those modules present in the program", + "type": "text" + }, + { + "bbox": [ + 355, + 412, + 402, + 424 + ], + "score": 0.93, + "content": "z \\sim q ( \\mathbf { z } | \\mathbf { x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 412, + 506, + 424 + ], + "score": 1.0, + "content": "are updated. In practice,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 301, + 435 + ], + "score": 1.0, + "content": "we take approx. argmax decoded programs from", + "type": "text" + }, + { + "bbox": [ + 302, + 423, + 334, + 435 + ], + "score": 0.93, + "content": "q _ { \\phi } ( { \\bf z } | { \\bf x } )", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "to simplify training, but perform sampling", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 446 + ], + "score": 1.0, + "content": "during joint training in the next stage. To be more specific, we run beam search (Vijayakumar et al.,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 443, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 443, + 417, + 457 + ], + "score": 1.0, + "content": "2018) with a beam size of 1 to do this. Intuitively, we think of this as a bias", + "type": "text" + }, + { + "bbox": [ + 418, + 447, + 427, + 455 + ], + "score": 0.25, + "content": "\\nu s", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 443, + 506, + 457 + ], + "score": 1.0, + "content": ". variance tradeoff,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "spans": [ + { + "bbox": [ + 106, + 456, + 505, + 467 + ], + "score": 1.0, + "content": "doing argmax enables us to trade off bias for high-variance from a posterior which (in the pathological", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 466, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 478 + ], + "score": 1.0, + "content": "case) could match the prior from the first stage of training. This is also consistent with standard", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 478, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 478, + 505, + 489 + ], + "score": 1.0, + "content": "practice for doing inference in sequence models, where beam search is the standard inference tool", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 488, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 506, + 500 + ], + "score": 1.0, + "content": "(and sampling often has issues due to teacher forcing based MLE training, which is how we do", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 500, + 299, + 511 + ], + "spans": [ + { + "bbox": [ + 105, + 500, + 299, + 511 + ], + "score": 1.0, + "content": "semi-supervised learning) (Bengio et al., 2015).", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 28.5, + "bbox_fs": [ + 105, + 400, + 506, + 511 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 516, + 504, + 550 + ], + "lines": [ + { + "bbox": [ + 105, + 515, + 506, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 530 + ], + "score": 1.0, + "content": "Joint Training. Having trained the question code and the neural module network parameters, we", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 528, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 426, + 540 + ], + "score": 1.0, + "content": "tune all terms jointly, with the following objective, which adds scaling factors", + "type": "text" + }, + { + "bbox": [ + 427, + 529, + 434, + 537 + ], + "score": 0.61, + "content": "\\alpha", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 528, + 438, + 540 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 438, + 528, + 446, + 539 + ], + "score": 0.77, + "content": "\\beta", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 528, + 464, + 540 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 465, + 529, + 471, + 539 + ], + "score": 0.81, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 472, + 528, + 505, + 540 + ], + "score": 1.0, + "content": ", and an", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 540, + 349, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 540, + 349, + 550 + ], + "score": 1.0, + "content": "extra semi-supervised learning term relative to Equation (1):", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 515, + 506, + 550 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 111, + 555, + 489, + 570 + ], + "lines": [ + { + "bbox": [ + 111, + 555, + 489, + 570 + ], + "spans": [ + { + "bbox": [ + 111, + 555, + 489, + 570 + ], + "score": 0.84, + "content": "J ( \\mathbf { x } , \\mathbf { a } | \\mathbf { i } ) = E _ { z \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } \\big [ \\gamma \\log p _ { \\theta _ { z } } ( \\mathbf { a } | \\mathbf { z } , \\mathbf { i } ) + \\log p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) \\big ] - \\beta \\mathrm { K L } \\big ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) , p ( \\mathbf { z } ) \\big ) + \\alpha \\log r _ { \\phi } ( \\mathbf { z } | \\mathbf { x } )", + "type": "interline_equation", + "image_path": "fd3d8a30d4332d07074d746f446cadb9c21e7fee9fbcc98319d3b4688a09540f.jpg" + } + ] + } + ], + "index": 37, + "virtual_lines": [ + { + "bbox": [ + 111, + 555, + 489, + 570 + ], + "spans": [], + "index": 37 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 574, + 505, + 619 + ], + "lines": [ + { + "bbox": [ + 104, + 573, + 505, + 587 + ], + "spans": [ + { + "bbox": [ + 104, + 573, + 179, + 587 + ], + "score": 1.0, + "content": "The gamma factor", + "type": "text" + }, + { + "bbox": [ + 180, + 576, + 187, + 586 + ], + "score": 0.82, + "content": "\\gamma", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 573, + 505, + 587 + ], + "score": 1.0, + "content": "controls the scaling of the likelihood term for the answers relative to the question", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 505, + 597 + ], + "score": 1.0, + "content": "(which has more bits of information), similar scaling factors have been for joint models in Vedantam", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 106, + 596, + 505, + 608 + ], + "score": 1.0, + "content": "et al. (2018). Similar to Equation (4), we optimize the full monte carlo gradient for the above objective", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 606, + 378, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 378, + 619 + ], + "score": 1.0, + "content": "as well. The full training procedure is shown in Algorithm 1 below.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 573, + 505, + 619 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 634, + 211, + 647 + ], + "lines": [ + { + "bbox": [ + 104, + 633, + 213, + 649 + ], + "spans": [ + { + "bbox": [ + 104, + 633, + 213, + 649 + ], + "score": 1.0, + "content": "3 RELATED WORK", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 107, + 659, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 506, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 506, + 673 + ], + "score": 1.0, + "content": "We first explain other related work in neuro-symbolic models, the key differences between our", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 682 + ], + "score": 1.0, + "content": "formulation and closely related NMN work (Johnson et al., 2017; Hu et al., 2017; 2018), and then", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 682, + 399, + 694 + ], + "spans": [ + { + "bbox": [ + 105, + 682, + 399, + 694 + ], + "score": 1.0, + "content": "connect our work to other themes and directions of research in literature.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 658, + 506, + 694 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 698, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 698, + 505, + 711 + ], + "score": 1.0, + "content": "Neural-symbolic models Valiant (2003) identifies neuro-symbolic models, which are able to reason", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "as well as learn efficiently as one of the key problems in computer science. 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Our work is distinct in terms of being motivated by interpretability in the low-sample", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "regime rather than getting connectionist/neural networks to explicitly do symbolic manipulations or", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 257, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 257, + 354 + ], + "score": 1.0, + "content": "building hybrid models that do logic.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "More recently, Yin et al. 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Our model is more general in the sense", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "that we have a second modality which is the output of what gets executed when the program runs,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "and we model both jointly in our model, by learning how to execute the program. Work from Yi", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "et al. (2018) also shares the same goal of doing question answering with limited question program", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 106, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "supervision while preserving the semantics of human programs. While they to approach this problem", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 185, + 447 + ], + "score": 1.0, + "content": "by simplifying the", + "type": "text" + }, + { + "bbox": [ + 185, + 436, + 219, + 447 + ], + "score": 0.94, + "content": "p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "mapping (in context of our model) by first converting the image into", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "a table, we approach this goal by modeling a stochastic latent space. Thus the two approaches are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "orthogonal, as one can use our probabilistic latent space in conjunction with their model as well to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "better share statistics between question-program pairs. Overall, compared to these works, our work", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "addresses probabilistic neural-symbolic learning in a more general setting, by learning how to execute", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 475, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 475, + 502 + ], + "score": 1.0, + "content": "programs (on raw-pixels), as well as handling parsing of questions in an end to end manner.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "Connection to prior work on NMNs. Our V-NMN model is inspired in its parameterization of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "program to answer mapping from prior work on visual question answering using neural module", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "score": 1.0, + "content": "networks such as Johnson et al. (2017); Hu et al. (2017; 2018). Distinct from these previous", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "approaches, our explicit goal is to preserve the semantics of how a human specifies a task to be broken", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "down in terms of instructions, while previous work was motivated by building inductive biases for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "reasoning (which turn out to be useful for our model as well). 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However, apart from our focus on preserving the lexicon/specification given", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "score": 1.0, + "content": "by a human, both approaches share the same high level goals of providing interpretable question", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 639, + 255, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 255, + 650 + ], + "score": 1.0, + "content": "answering with minimal supervision.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "Beyond Gaussian Latents in Amortized Variational Inference. There has been exciting progress", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "in developing lower-variance estimators for discrete generative models (Tucker et al., 2017; Mnih", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "& Gregor, 2014; Titsias & Lazaro-Gredilla, 2015; Gu & Ling, 2015). In particular, Maddison ´", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "et al. (2017); Jang et al. 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Our work is distinct in terms of being motivated by interpretability in the low-sample", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 506, + 343 + ], + "score": 1.0, + "content": "regime rather than getting connectionist/neural networks to explicitly do symbolic manipulations or", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 340, + 257, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 257, + 354 + ], + "score": 1.0, + "content": "building hybrid models that do logic.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 286, + 506, + 354 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 357, + 505, + 501 + ], + "lines": [ + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 357, + 506, + 370 + ], + "score": 1.0, + "content": "More recently, Yin et al. 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Our model is more general in the sense", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "that we have a second modality which is the output of what gets executed when the program runs,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 401, + 505, + 414 + ], + "score": 1.0, + "content": "and we model both jointly in our model, by learning how to execute the program. Work from Yi", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 505, + 425 + ], + "score": 1.0, + "content": "et al. 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While they to approach this problem", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 434, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 185, + 447 + ], + "score": 1.0, + "content": "by simplifying the", + "type": "text" + }, + { + "bbox": [ + 185, + 436, + 219, + 447 + ], + "score": 0.94, + "content": "p ( \\mathbf { a } | \\mathbf { i } , \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 434, + 505, + 447 + ], + "score": 1.0, + "content": "mapping (in context of our model) by first converting the image into", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 506, + 458 + ], + "score": 1.0, + "content": "a table, we approach this goal by modeling a stochastic latent space. Thus the two approaches are", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 505, + 469 + ], + "score": 1.0, + "content": "orthogonal, as one can use our probabilistic latent space in conjunction with their model as well to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 466, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 106, + 466, + 506, + 480 + ], + "score": 1.0, + "content": "better share statistics between question-program pairs. Overall, compared to these works, our work", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "addresses probabilistic neural-symbolic learning in a more general setting, by learning how to execute", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 489, + 475, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 475, + 502 + ], + "score": 1.0, + "content": "programs (on raw-pixels), as well as handling parsing of questions in an end to end manner.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 27, + "bbox_fs": [ + 105, + 357, + 506, + 502 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 506, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 106, + 507, + 505, + 518 + ], + "score": 1.0, + "content": "Connection to prior work on NMNs. Our V-NMN model is inspired in its parameterization of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 505, + 531 + ], + "score": 1.0, + "content": "program to answer mapping from prior work on visual question answering using neural module", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 542 + ], + "score": 1.0, + "content": "networks such as Johnson et al. (2017); Hu et al. (2017; 2018). Distinct from these previous", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 539, + 505, + 552 + ], + "score": 1.0, + "content": "approaches, our explicit goal is to preserve the semantics of how a human specifies a task to be broken", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 506, + 563 + ], + "score": 1.0, + "content": "down in terms of instructions, while previous work was motivated by building inductive biases for", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "reasoning (which turn out to be useful for our model as well). Given this goal, we construct a latent", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 572, + 506, + 585 + ], + "score": 1.0, + "content": "variable model that in some sense embeds previous work. The works from Hu et al. (2017; 2018)", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 105, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "are more distant from our work, as they use continuous attention modules, which do not capture", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "independence assumptions of the program being a sufficient statistic for answering the question", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 505, + 618 + ], + "score": 1.0, + "content": "(which makes them harder to specify completely for a human, as they would have to hallucinate", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 629 + ], + "score": 1.0, + "content": "attention parameters). However, apart from our focus on preserving the lexicon/specification given", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 505, + 641 + ], + "score": 1.0, + "content": "by a human, both approaches share the same high level goals of providing interpretable question", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 639, + 255, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 255, + 650 + ], + "score": 1.0, + "content": "answering with minimal supervision.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 507, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 505, + 667 + ], + "score": 1.0, + "content": "Beyond Gaussian Latents in Amortized Variational Inference. There has been exciting progress", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "spans": [ + { + "bbox": [ + 105, + 665, + 506, + 679 + ], + "score": 1.0, + "content": "in developing lower-variance estimators for discrete generative models (Tucker et al., 2017; Mnih", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "& Gregor, 2014; Titsias & Lazaro-Gredilla, 2015; Gu & Ling, 2015). In particular, Maddison ´", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 701 + ], + "score": 1.0, + "content": "et al. (2017); Jang et al. (2017) propose the use of the concrete distribution on the probability", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "simplex for providing biased but low variance estimators for discrete latent variable models by", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "considering a reparametrization of samples (which asymptotically approach those from a categorical", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "distribution). The contribution in our work is orthogonal to these approaches, how one can apply", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "such parameterizations to neuro-symbolic models. More relevant to our work is Miao & Blunsom", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "(2016) which proposes a discrete sequential latent variable model for sentence summarization, with", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "summaries as the latent variable. In contrast to their model, ours is a joint model where the latent", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "space consists of programs, not summaries, which have to explain questions and answers to images.", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 50, + "bbox_fs": [ + 105, + 654, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 83, + 505, + 95 + ], + "score": 1.0, + "content": "such parameterizations to neuro-symbolic models. More relevant to our work is Miao & Blunsom", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 505, + 106 + ], + "score": 1.0, + "content": "(2016) which proposes a discrete sequential latent variable model for sentence summarization, with", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 506, + 117 + ], + "score": 1.0, + "content": "summaries as the latent variable. In contrast to their model, ours is a joint model where the latent", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 506, + 128 + ], + "score": 1.0, + "content": "space consists of programs, not summaries, which have to explain questions and answers to images.", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "Joint Variational Autoencoder Models. Vedantam et al. (2018) propose a joint model for images", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "and attributes for “visual imagination”, Suzuki et al. (2017) propose a closely related model and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "show that one can use the model for conditional generation, while Wang et al. (2016a) propose a deep", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "variational canonical correlation analysis model which learns a joint space between two modalities", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "for denoising and representation learning. In contrast to these approaches, our joint model assumes", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 504, + 199 + ], + "score": 1.0, + "content": "a discrete structured latent variable (instead of a Gaussian), does semi-supervised training, which", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "is possible for us, but not possible when there are no observations from the latent available for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 449, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 449, + 223 + ], + "score": 1.0, + "content": "supervision, and is targeted towards neural module networks and question answering.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 506, + 238 + ], + "score": 1.0, + "content": "Program Induction and Neural Architecture Search. In program induction one is typically inter-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 507, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 507, + 250 + ], + "score": 1.0, + "content": "ested in learning to write programs given specifications of input-output pairs (Reed & de Freitas,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "2015; Kalyan et al., 2018), and optionally language (Neelakantan et al., 2015; Guu et al., 2017) or", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "vision (Gaunt et al., 2017). The key difference between program induction and our problem is that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "program induction assumes the tokens/instructions in the language are grounded to the executor, i.e.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "it is assumed that once a valid program is generated there is a black-box execution engine that can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "execute it, while we learn the execution engine from scratch, and learn the grounding from program", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 507, + 317 + ], + "score": 1.0, + "content": "symbols to parameters of a neural network for execution. Neural architecture search (Zoph & Le,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "2016) is another closely related problem where one is given input output examples for a machine", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "learning task, and the goal is to find a “neural program” that does well on the task. In contrast, ours", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "score": 1.0, + "content": "can be seen as the problem of inducing programs conditioned on particular inputs (questions), where", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "the number of samples seen per question is significantly lower than the number of samples one would", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 296, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 296, + 370 + ], + "score": 1.0, + "content": "observe for a machine learning task of interest.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18 + }, + { + "type": "title", + "bbox": [ + 108, + 389, + 200, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 202, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 202, + 404 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "Dataset. We perform our experiments on the SHAPES dataset (Andreas et al., 2016a). This", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "dataset has been used in previous works (Andreas et al., 2016a; Hu et al., 2017) on visual question", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "answering using neural module networks, and is intended to be a test for compositional generalization.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "The dataset has 244 unique binary questions (i.e. having yes/no answers) and 15,616 images in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "total (Andreas et al., 2016a). The dataset also has annotated programs for each of the questions. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 327, + 485 + ], + "score": 1.0, + "content": "use train, val, and test splits of 13,568, 1,024, and 1,024", + "type": "text" + }, + { + "bbox": [ + 327, + 473, + 369, + 484 + ], + "score": 0.79, + "content": "( { \\bf x } , { \\bf z } , \\mathbf { i } , { \\bf a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "triplets respectively. No questions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 506, + 497 + ], + "score": 1.0, + "content": "are repeated across these splits. Further, each question is asked for 64 different images, ensuring that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "a question can be answered correctly across all these instances only if it is processing the image in a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "reasonable way (as dictated by the underlying program), as opposed to leveraging any image-question", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 375, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 375, + 530 + ], + "score": 1.0, + "content": "correlations or biases that are unlikely to generalize across images.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5 + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "Thus, the dataset explicitly tests for compositional generalization and “reasoning” in the task of visual", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "question answering. The longest questions in the dataset are of length 11 and shortest are of length", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "4, while the longest programs are of length 6 and shortest programs are of length 4. The size of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 566, + 376, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 189, + 578 + ], + "score": 1.0, + "content": "question vocabulary", + "type": "text" + }, + { + "bbox": [ + 189, + 567, + 202, + 578 + ], + "score": 0.89, + "content": "\\mathcal { V } _ { \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 566, + 338, + 578 + ], + "score": 1.0, + "content": "is 14 and the program vocabulary", + "type": "text" + }, + { + "bbox": [ + 339, + 567, + 351, + 577 + ], + "score": 0.89, + "content": "\\nu _ { \\mathbf { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 566, + 376, + 578 + ], + "score": 1.0, + "content": "is 12.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "Training. To simulate a data-sparse regime, we restrict the set of question-aligned programs to 5,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 146, + 606 + ], + "score": 1.0, + "content": "10, 15, or", + "type": "text" + }, + { + "bbox": [ + 146, + 594, + 166, + 605 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "of unique questions – such that even at the highest level of supervision, programs for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 127, + 615 + ], + "score": 0.85, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "of unique question have never been seen during training. Unless specified explicitly, we train", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "our program prior using a set of 1848 unique programs simulated from the syntax (more details of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "this can be seen in the appendix). In general, it might not be possible to exhaustively enumerate the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "space of all valid strings to learn the prior. Thus, we also evaluate the effect of training the prior with", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "varying-sized subsets of the 212 programs present in the training set (without using any corresponding", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "questions or answers) to check how robust overall performance is to the choice of the prior (in the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 671, + 152, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 152, + 683 + ], + "score": 1.0, + "content": "appendix).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "We find sequence-level policy-gradient based training in this setting can be highly variable – with", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "some runs failing spectacularly while others succeed – for both our method and to some extent NMN", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "(Johnson et al., 2017). For fairness and to reduce the impact of stochasticity, we perform the following", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "stage-wise training regime for all models. First, the question-coding stage is run for 5 different", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 50.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 762 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 126 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 105, + 83, + 506, + 128 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 132, + 505, + 220 + ], + "lines": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 131, + 505, + 145 + ], + "score": 1.0, + "content": "Joint Variational Autoencoder Models. Vedantam et al. (2018) propose a joint model for images", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 106, + 144, + 505, + 156 + ], + "score": 1.0, + "content": "and attributes for “visual imagination”, Suzuki et al. (2017) propose a closely related model and", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "show that one can use the model for conditional generation, while Wang et al. (2016a) propose a deep", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 106, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "variational canonical correlation analysis model which learns a joint space between two modalities", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "for denoising and representation learning. In contrast to these approaches, our joint model assumes", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 188, + 504, + 199 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 504, + 199 + ], + "score": 1.0, + "content": "a discrete structured latent variable (instead of a Gaussian), does semi-supervised training, which", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 210 + ], + "score": 1.0, + "content": "is possible for us, but not possible when there are no observations from the latent available for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 449, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 449, + 223 + ], + "score": 1.0, + "content": "supervision, and is targeted towards neural module networks and question answering.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 131, + 506, + 223 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 369 + ], + "lines": [ + { + "bbox": [ + 106, + 227, + 506, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 227, + 506, + 238 + ], + "score": 1.0, + "content": "Program Induction and Neural Architecture Search. In program induction one is typically inter-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 237, + 507, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 507, + 250 + ], + "score": 1.0, + "content": "ested in learning to write programs given specifications of input-output pairs (Reed & de Freitas,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 248, + 506, + 261 + ], + "score": 1.0, + "content": "2015; Kalyan et al., 2018), and optionally language (Neelakantan et al., 2015; Guu et al., 2017) or", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 259, + 506, + 272 + ], + "spans": [ + { + "bbox": [ + 106, + 259, + 506, + 272 + ], + "score": 1.0, + "content": "vision (Gaunt et al., 2017). The key difference between program induction and our problem is that", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 506, + 283 + ], + "score": 1.0, + "content": "program induction assumes the tokens/instructions in the language are grounded to the executor, i.e.,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 505, + 294 + ], + "score": 1.0, + "content": "it is assumed that once a valid program is generated there is a black-box execution engine that can", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "spans": [ + { + "bbox": [ + 105, + 291, + 506, + 305 + ], + "score": 1.0, + "content": "execute it, while we learn the execution engine from scratch, and learn the grounding from program", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 302, + 507, + 317 + ], + "spans": [ + { + "bbox": [ + 105, + 302, + 507, + 317 + ], + "score": 1.0, + "content": "symbols to parameters of a neural network for execution. Neural architecture search (Zoph & Le,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 313, + 505, + 327 + ], + "score": 1.0, + "content": "2016) is another closely related problem where one is given input output examples for a machine", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 324, + 505, + 338 + ], + "score": 1.0, + "content": "learning task, and the goal is to find a “neural program” that does well on the task. In contrast, ours", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "spans": [ + { + "bbox": [ + 105, + 335, + 505, + 349 + ], + "score": 1.0, + "content": "can be seen as the problem of inducing programs conditioned on particular inputs (questions), where", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 106, + 346, + 505, + 360 + ], + "score": 1.0, + "content": "the number of samples seen per question is significantly lower than the number of samples one would", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 358, + 296, + 370 + ], + "spans": [ + { + "bbox": [ + 106, + 358, + 296, + 370 + ], + "score": 1.0, + "content": "observe for a machine learning task of interest.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 227, + 507, + 370 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 389, + 200, + 402 + ], + "lines": [ + { + "bbox": [ + 105, + 388, + 202, + 404 + ], + "spans": [ + { + "bbox": [ + 105, + 388, + 202, + 404 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 417, + 505, + 528 + ], + "lines": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "Dataset. We perform our experiments on the SHAPES dataset (Andreas et al., 2016a). This", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 441 + ], + "score": 1.0, + "content": "dataset has been used in previous works (Andreas et al., 2016a; Hu et al., 2017) on visual question", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 506, + 452 + ], + "score": 1.0, + "content": "answering using neural module networks, and is intended to be a test for compositional generalization.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 450, + 505, + 463 + ], + "score": 1.0, + "content": "The dataset has 244 unique binary questions (i.e. having yes/no answers) and 15,616 images in", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 505, + 474 + ], + "score": 1.0, + "content": "total (Andreas et al., 2016a). The dataset also has annotated programs for each of the questions. We", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 473, + 505, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 473, + 327, + 485 + ], + "score": 1.0, + "content": "use train, val, and test splits of 13,568, 1,024, and 1,024", + "type": "text" + }, + { + "bbox": [ + 327, + 473, + 369, + 484 + ], + "score": 0.79, + "content": "( { \\bf x } , { \\bf z } , \\mathbf { i } , { \\bf a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 473, + 505, + 485 + ], + "score": 1.0, + "content": "triplets respectively. No questions", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 482, + 506, + 497 + ], + "spans": [ + { + "bbox": [ + 104, + 482, + 506, + 497 + ], + "score": 1.0, + "content": "are repeated across these splits. Further, each question is asked for 64 different images, ensuring that", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 104, + 494, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 104, + 494, + 506, + 507 + ], + "score": 1.0, + "content": "a question can be answered correctly across all these instances only if it is processing the image in a", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 506, + 520 + ], + "score": 1.0, + "content": "reasonable way (as dictated by the underlying program), as opposed to leveraging any image-question", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 516, + 375, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 516, + 375, + 530 + ], + "score": 1.0, + "content": "correlations or biases that are unlikely to generalize across images.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30.5, + "bbox_fs": [ + 104, + 417, + 506, + 530 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 533, + 505, + 577 + ], + "lines": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 532, + 506, + 546 + ], + "score": 1.0, + "content": "Thus, the dataset explicitly tests for compositional generalization and “reasoning” in the task of visual", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 505, + 557 + ], + "score": 1.0, + "content": "question answering. The longest questions in the dataset are of length 11 and shortest are of length", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 505, + 568 + ], + "score": 1.0, + "content": "4, while the longest programs are of length 6 and shortest programs are of length 4. The size of the", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 566, + 376, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 189, + 578 + ], + "score": 1.0, + "content": "question vocabulary", + "type": "text" + }, + { + "bbox": [ + 189, + 567, + 202, + 578 + ], + "score": 0.89, + "content": "\\mathcal { V } _ { \\mathbf { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 202, + 566, + 338, + 578 + ], + "score": 1.0, + "content": "is 14 and the program vocabulary", + "type": "text" + }, + { + "bbox": [ + 339, + 567, + 351, + 577 + ], + "score": 0.89, + "content": "\\nu _ { \\mathbf { z } }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 566, + 376, + 578 + ], + "score": 1.0, + "content": "is 12.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 37.5, + "bbox_fs": [ + 105, + 532, + 506, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 583, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 506, + 595 + ], + "score": 1.0, + "content": "Training. To simulate a data-sparse regime, we restrict the set of question-aligned programs to 5,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 146, + 606 + ], + "score": 1.0, + "content": "10, 15, or", + "type": "text" + }, + { + "bbox": [ + 146, + 594, + 166, + 605 + ], + "score": 0.87, + "content": "20 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "of unique questions – such that even at the highest level of supervision, programs for", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 127, + 615 + ], + "score": 0.85, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 127, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "of unique question have never been seen during training. Unless specified explicitly, we train", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 628 + ], + "score": 1.0, + "content": "our program prior using a set of 1848 unique programs simulated from the syntax (more details of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 640 + ], + "score": 1.0, + "content": "this can be seen in the appendix). In general, it might not be possible to exhaustively enumerate the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "space of all valid strings to learn the prior. Thus, we also evaluate the effect of training the prior with", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 505, + 662 + ], + "score": 1.0, + "content": "varying-sized subsets of the 212 programs present in the training set (without using any corresponding", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "questions or answers) to check how robust overall performance is to the choice of the prior (in the", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 671, + 152, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 152, + 683 + ], + "score": 1.0, + "content": "appendix).", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 44, + "bbox_fs": [ + 105, + 583, + 506, + 683 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "We find sequence-level policy-gradient based training in this setting can be highly variable – with", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "some runs failing spectacularly while others succeed – for both our method and to some extent NMN", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "(Johnson et al., 2017). For fairness and to reduce the impact of stochasticity, we perform the following", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 506, + 733 + ], + "score": 1.0, + "content": "stage-wise training regime for all models. First, the question-coding stage is run for 5 different", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "random seeds, producing multiple program predictors. Programs taken from the model that performs", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "the best on the validation set are used to train 10 runs of the module training stage. Finally, the best", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "of these module networks (based on val) is trained jointly with the program predictor under settings", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 117, + 128 + ], + "score": 1.0, + "content": "of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 118, + 115, + 185, + 127 + ], + "score": 0.91, + "content": "\\gamma \\in \\{ 1 , 1 0 , 1 0 0 \\}", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 185, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "and the max (on val) is chosen. We report mean and variance accuracies on test", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "from this training regime over four random sets of supervisory programs. Our evaluation metrics", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "are as follows: a) question-coding uses reconstruction accuracy (of getting the entire question right)", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "and program prediction accuracy (of matching the ground truth programs) on the training set, since", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "the second stage, module training will use inferences from the first stage (on the training set to train", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 184 + ], + "score": 1.0, + "content": "modules), b) module training and joint training: visual question answering accuracy, by counting", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 241, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 241, + 193 + ], + "score": 1.0, + "content": "matches to ground truth answers.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 687, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "random seeds, producing multiple program predictors. Programs taken from the model that performs", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "the best on the validation set are used to train 10 runs of the module training stage. Finally, the best", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "of these module networks (based on val) is trained jointly with the program predictor under settings", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 117, + 128 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 115, + 185, + 127 + ], + "score": 0.91, + "content": "\\gamma \\in \\{ 1 , 1 0 , 1 0 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 114, + 506, + 128 + ], + "score": 1.0, + "content": "and the max (on val) is chosen. We report mean and variance accuracies on test", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 506, + 139 + ], + "score": 1.0, + "content": "from this training regime over four random sets of supervisory programs. Our evaluation metrics", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 506, + 150 + ], + "score": 1.0, + "content": "are as follows: a) question-coding uses reconstruction accuracy (of getting the entire question right)", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 149, + 505, + 160 + ], + "score": 1.0, + "content": "and program prediction accuracy (of matching the ground truth programs) on the training set, since", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 505, + 172 + ], + "score": 1.0, + "content": "the second stage, module training will use inferences from the first stage (on the training set to train", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 169, + 505, + 184 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 184 + ], + "score": 1.0, + "content": "modules), b) module training and joint training: visual question answering accuracy, by counting", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 241, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 241, + 193 + ], + "score": 1.0, + "content": "matches to ground truth answers.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "Implementation Details: Following Johnson et al. (2017) and Hu et al. (2017), we treat the program", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 182, + 221 + ], + "score": 0.92, + "content": "\\mathbf { z } = \\{ z _ { 1 } , \\cdots , z _ { K } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "as a variable length sequence of tokens with an end of sequence token, which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "corresponds to the prefix traversal of an execution tree. Given this representational choice, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 167, + 244 + ], + "score": 1.0, + "content": "learn the prior", + "type": "text" + }, + { + "bbox": [ + 167, + 231, + 186, + 243 + ], + "score": 0.92, + "content": "p ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "using maximum likelihood training on simulated samples from known syntax", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 507, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 507, + 256 + ], + "score": 1.0, + "content": "of programs. An alternative, interesting approach to try would be to use something like Janz et al.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 455, + 267 + ], + "score": 1.0, + "content": "(2017), who acquire a generative model more directly from validity annotations. The", + "type": "text" + }, + { + "bbox": [ + 455, + 253, + 505, + 265 + ], + "score": 0.92, + "content": "\\log p ( \\mathbf { a } | \\mathbf { z } , \\mathbf { i } )", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 263, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 506, + 278 + ], + "score": 1.0, + "content": "term in Equation (1) is parameterized by an NMN that is dynamically assembled given the program", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 240, + 288 + ], + "score": 1.0, + "content": "string, while the generative model", + "type": "text" + }, + { + "bbox": [ + 240, + 275, + 268, + 286 + ], + "score": 0.93, + "content": "p ( \\mathbf { x } | \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 275, + 327, + 288 + ], + "score": 1.0, + "content": ", and inference", + "type": "text" + }, + { + "bbox": [ + 327, + 275, + 360, + 286 + ], + "score": 0.89, + "content": "( q ( \\mathbf { z } | \\mathbf { x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "terms are both sequence to sequence", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "models. Our sequence to sequence models are based on LSTM (Hochreiter & Schmidhuber, 1997)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "cells. For the NMN programs, we use syntax similar to that used by Hu et al. (2017), but we replace", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "their question-attention neural modules with “regular” modules. For clear comparison we adapt the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "objectives from the baseline approach, Johnson et al. (2017), to our representational choices. More", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 330, + 240, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 240, + 343 + ], + "score": 1.0, + "content": "details can be found in appendix.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16 + }, + { + "type": "title", + "bbox": [ + 107, + 357, + 172, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 174, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 174, + 372 + ], + "score": 1.0, + "content": "5 RESULTS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "We focus on evaluating approaches by varying the fraction of question-aligned programs (which we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 393, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 135, + 405 + ], + "score": 1.0, + "content": "denote", + "type": "text" + }, + { + "bbox": [ + 135, + 393, + 172, + 403 + ], + "score": 0.89, + "content": "\\% \\mathbf { x } \\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 393, + 385, + 405 + ], + "score": 1.0, + "content": ") in Table 1. To put the numbers in context, a baseline", + "type": "text" + }, + { + "bbox": [ + 385, + 393, + 422, + 403 + ], + "score": 0.32, + "content": "\\mathrm { L S T M + }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 393, + 506, + 405 + ], + "score": 1.0, + "content": "image model, which", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 309, + 416 + ], + "score": 1.0, + "content": "does not use module networks, gets an accuracy of", + "type": "text" + }, + { + "bbox": [ + 309, + 404, + 336, + 414 + ], + "score": 0.87, + "content": "6 3 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "on Test (see Andreas et al. (2016a); Table.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "2). This indicates that SHAPES has highly compositional questions which is challenging to model", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "via conventional methods for Visual Question Answering (Agrawal et al., 2015). Overall, we make", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 437, + 217, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 217, + 449 + ], + "score": 1.0, + "content": "the following observations:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5 + }, + { + "type": "text", + "bbox": [ + 108, + 450, + 506, + 603 + ], + "lines": [ + { + "bbox": [ + 109, + 448, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 109, + 448, + 506, + 463 + ], + "score": 1.0, + "content": "- Our V-NMN approach consistently improves performance in data-sparse regimes. While", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 114, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 114, + 459, + 506, + 474 + ], + "score": 1.0, + "content": "both methods tend to improve with greater program supervision, V-NMN quickly outpaces", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 115, + 471, + 504, + 485 + ], + "spans": [ + { + "bbox": [ + 115, + 471, + 371, + 485 + ], + "score": 1.0, + "content": "NMN (Johnson et al., 2017), achieving test accuracies over", + "type": "text" + }, + { + "bbox": [ + 371, + 472, + 405, + 483 + ], + "score": 0.88, + "content": "20 \\mathrm { - } 3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 471, + 481, + 485 + ], + "score": 1.0, + "content": "points higher for", + "type": "text" + }, + { + "bbox": [ + 482, + 471, + 504, + 482 + ], + "score": 0.88, + "content": "{ > } 5 \\%", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 115, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 115, + 482, + 490, + 496 + ], + "score": 1.0, + "content": "program supervision. Notably, both methods perform similarly poorly on the test set given only", + "type": "text" + }, + { + "bbox": [ + 490, + 483, + 505, + 493 + ], + "score": 0.83, + "content": "5 \\%", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 115, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 115, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "program supervision, suggesting this may be too few examples to learn compositional reasoning.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 113, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 113, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "Our question coding stage greatly improves initial program prediction. Our V-NMN approach", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 115, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 115, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "to question coding gets to approximately double the question coding (reconstruction) accuracy as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 115, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 115, + 525, + 506, + 540 + ], + "score": 1.0, + "content": "the NMN approach (col 1, question coding). This means that it effectively propagates groundings", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 115, + 536, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 115, + 536, + 506, + 551 + ], + "score": 1.0, + "content": "from question-aligned programs during the coding phase. Consequently, we also see improved", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 115, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 115, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "VQA performance after the module and joint training stages which are based on predicted programs.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 114, + 558, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 114, + 558, + 505, + 573 + ], + "score": 1.0, + "content": "Question reconstruction is well-correlated with overall accuracy. Despite the program space", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 116, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 116, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "being trained directly as a latent code for questions, we find question reconstruction accuracy rises", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 116, + 581, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 116, + 581, + 505, + 593 + ], + "score": 1.0, + "content": "with the amount of supervision – confirming that inference in a space of discrete structured latent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 115, + 591, + 414, + 605 + ], + "spans": [ + { + "bbox": [ + 115, + 591, + 414, + 605 + ], + "score": 1.0, + "content": "variables is difficult to do without some supervision to guide the mapping.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 36.5 + }, + { + "type": "text", + "bbox": [ + 106, + 605, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "Figure 3 shows sample programs for each model. We see the limited supervision negatively affects", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 615, + 507, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 250, + 629 + ], + "score": 1.0, + "content": "NMN program prediction, with the", + "type": "text" + }, + { + "bbox": [ + 250, + 616, + 265, + 627 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 615, + 414, + 629 + ], + "score": 1.0, + "content": "model resorting to simple Find[X]", + "type": "text" + }, + { + "bbox": [ + 414, + 618, + 425, + 626 + ], + "score": 0.7, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 615, + 507, + 629 + ], + "score": 1.0, + "content": "Answer structures.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 439, + 639 + ], + "score": 1.0, + "content": "Interestingly, we find that the mistakes made by the V-NMN model, e.g., green in", + "type": "text" + }, + { + "bbox": [ + 439, + 627, + 455, + 637 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "supervision", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 637, + 507, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 507, + 651 + ], + "score": 1.0, + "content": "(top-right) are also made when reconstructing the question (also substituting green for blue). Further,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 649, + 504, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 504, + 661 + ], + "score": 1.0, + "content": "when the token does get corrected to blue, the question also eventually gets reconstructed (partially", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 145, + 673 + ], + "score": 1.0, + "content": "correctly", + "type": "text" + }, + { + "bbox": [ + 145, + 660, + 171, + 671 + ], + "score": 0.87, + "content": "( 1 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 660, + 230, + 673 + ], + "score": 1.0, + "content": "and then fully", + "type": "text" + }, + { + "bbox": [ + 230, + 660, + 258, + 671 + ], + "score": 0.86, + "content": "( 1 5 \\% ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 660, + 505, + 673 + ], + "score": 1.0, + "content": ", and the program produces the correct answer. This indicates", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 671, + 498, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 498, + 684 + ], + "score": 1.0, + "content": "that there is high fidelity between the learnt question space, the answer space and the latent space.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 224, + 700 + ], + "score": 1.0, + "content": "Effect of the Program Prior", + "type": "text" + }, + { + "bbox": [ + 225, + 687, + 244, + 699 + ], + "score": 0.86, + "content": "\\mathbf { p } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ". Next, we explore the impact of regularizing the program posterior", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 198, + 711 + ], + "score": 1.0, + "content": "to be close to the prior", + "type": "text" + }, + { + "bbox": [ + 198, + 699, + 217, + 711 + ], + "score": 0.91, + "content": "p ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 699, + 506, + 711 + ], + "score": 1.0, + "content": ", for different choices of the prior. Firstly, we disable the KL-divergence", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 169, + 723 + ], + "score": 1.0, + "content": "term by setting", + "type": "text" + }, + { + "bbox": [ + 169, + 710, + 195, + 721 + ], + "score": 0.91, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "in Equation (2) recovering a deterministic version of our model. Compared", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 186, + 734 + ], + "score": 1.0, + "content": "to our full model at", + "type": "text" + }, + { + "bbox": [ + 187, + 721, + 206, + 731 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 720, + 440, + 734 + ], + "score": 1.0, + "content": "supervision, the performance on question coding drops to", + "type": "text" + }, + { + "bbox": [ + 441, + 721, + 468, + 731 + ], + "score": 0.85, + "content": "1 2 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "program", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52.5 + } + ], + "page_idx": 7, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 761 + ], + "score": 1.0, + "content": "8", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 505, + 192 + ], + "lines": [], + "index": 4.5, + "bbox_fs": [ + 105, + 82, + 506, + 193 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 106, + 198, + 505, + 340 + ], + "lines": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 197, + 506, + 211 + ], + "score": 1.0, + "content": "Implementation Details: Following Johnson et al. (2017) and Hu et al. (2017), we treat the program", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 209, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 182, + 221 + ], + "score": 0.92, + "content": "\\mathbf { z } = \\{ z _ { 1 } , \\cdots , z _ { K } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 209, + 505, + 222 + ], + "score": 1.0, + "content": "as a variable length sequence of tokens with an end of sequence token, which", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "corresponds to the prefix traversal of an execution tree. Given this representational choice, we", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 231, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 167, + 244 + ], + "score": 1.0, + "content": "learn the prior", + "type": "text" + }, + { + "bbox": [ + 167, + 231, + 186, + 243 + ], + "score": 0.92, + "content": "p ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 231, + 506, + 244 + ], + "score": 1.0, + "content": "using maximum likelihood training on simulated samples from known syntax", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 507, + 256 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 507, + 256 + ], + "score": 1.0, + "content": "of programs. An alternative, interesting approach to try would be to use something like Janz et al.", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 252, + 505, + 267 + ], + "spans": [ + { + "bbox": [ + 105, + 252, + 455, + 267 + ], + "score": 1.0, + "content": "(2017), who acquire a generative model more directly from validity annotations. The", + "type": "text" + }, + { + "bbox": [ + 455, + 253, + 505, + 265 + ], + "score": 0.92, + "content": "\\log p ( \\mathbf { a } | \\mathbf { z } , \\mathbf { i } )", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 263, + 506, + 278 + ], + "spans": [ + { + "bbox": [ + 104, + 263, + 506, + 278 + ], + "score": 1.0, + "content": "term in Equation (1) is parameterized by an NMN that is dynamically assembled given the program", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 240, + 288 + ], + "score": 1.0, + "content": "string, while the generative model", + "type": "text" + }, + { + "bbox": [ + 240, + 275, + 268, + 286 + ], + "score": 0.93, + "content": "p ( \\mathbf { x } | \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 275, + 327, + 288 + ], + "score": 1.0, + "content": ", and inference", + "type": "text" + }, + { + "bbox": [ + 327, + 275, + 360, + 286 + ], + "score": 0.89, + "content": "( q ( \\mathbf { z } | \\mathbf { x } ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 360, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "terms are both sequence to sequence", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "models. Our sequence to sequence models are based on LSTM (Hochreiter & Schmidhuber, 1997)", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "cells. For the NMN programs, we use syntax similar to that used by Hu et al. (2017), but we replace", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "their question-attention neural modules with “regular” modules. For clear comparison we adapt the", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "objectives from the baseline approach, Johnson et al. (2017), to our representational choices. More", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 330, + 240, + 343 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 240, + 343 + ], + "score": 1.0, + "content": "details can be found in appendix.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 16, + "bbox_fs": [ + 104, + 197, + 507, + 343 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 357, + 172, + 369 + ], + "lines": [ + { + "bbox": [ + 105, + 356, + 174, + 372 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 174, + 372 + ], + "score": 1.0, + "content": "5 RESULTS", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 106, + 381, + 505, + 448 + ], + "lines": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "spans": [ + { + "bbox": [ + 105, + 381, + 506, + 394 + ], + "score": 1.0, + "content": "We focus on evaluating approaches by varying the fraction of question-aligned programs (which we", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 393, + 506, + 405 + ], + "spans": [ + { + "bbox": [ + 106, + 393, + 135, + 405 + ], + "score": 1.0, + "content": "denote", + "type": "text" + }, + { + "bbox": [ + 135, + 393, + 172, + 403 + ], + "score": 0.89, + "content": "\\% \\mathbf { x } \\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 173, + 393, + 385, + 405 + ], + "score": 1.0, + "content": ") in Table 1. To put the numbers in context, a baseline", + "type": "text" + }, + { + "bbox": [ + 385, + 393, + 422, + 403 + ], + "score": 0.32, + "content": "\\mathrm { L S T M + }", + "type": "inline_equation" + }, + { + "bbox": [ + 422, + 393, + 506, + 405 + ], + "score": 1.0, + "content": "image model, which", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 403, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 106, + 403, + 309, + 416 + ], + "score": 1.0, + "content": "does not use module networks, gets an accuracy of", + "type": "text" + }, + { + "bbox": [ + 309, + 404, + 336, + 414 + ], + "score": 0.87, + "content": "6 3 . 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 403, + 506, + 416 + ], + "score": 1.0, + "content": "on Test (see Andreas et al. (2016a); Table.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "2). This indicates that SHAPES has highly compositional questions which is challenging to model", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 426, + 505, + 438 + ], + "score": 1.0, + "content": "via conventional methods for Visual Question Answering (Agrawal et al., 2015). Overall, we make", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 437, + 217, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 217, + 449 + ], + "score": 1.0, + "content": "the following observations:", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26.5, + "bbox_fs": [ + 105, + 381, + 506, + 449 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 450, + 506, + 603 + ], + "lines": [ + { + "bbox": [ + 109, + 448, + 506, + 463 + ], + "spans": [ + { + "bbox": [ + 109, + 448, + 506, + 463 + ], + "score": 1.0, + "content": "- Our V-NMN approach consistently improves performance in data-sparse regimes. While", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 114, + 459, + 506, + 474 + ], + "spans": [ + { + "bbox": [ + 114, + 459, + 506, + 474 + ], + "score": 1.0, + "content": "both methods tend to improve with greater program supervision, V-NMN quickly outpaces", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 115, + 471, + 504, + 485 + ], + "spans": [ + { + "bbox": [ + 115, + 471, + 371, + 485 + ], + "score": 1.0, + "content": "NMN (Johnson et al., 2017), achieving test accuracies over", + "type": "text" + }, + { + "bbox": [ + 371, + 472, + 405, + 483 + ], + "score": 0.88, + "content": "20 \\mathrm { - } 3 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 406, + 471, + 481, + 485 + ], + "score": 1.0, + "content": "points higher for", + "type": "text" + }, + { + "bbox": [ + 482, + 471, + 504, + 482 + ], + "score": 0.88, + "content": "{ > } 5 \\%", + "type": "inline_equation" + } + ], + "index": 32 + }, + { + "bbox": [ + 115, + 482, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 115, + 482, + 490, + 496 + ], + "score": 1.0, + "content": "program supervision. Notably, both methods perform similarly poorly on the test set given only", + "type": "text" + }, + { + "bbox": [ + 490, + 483, + 505, + 493 + ], + "score": 0.83, + "content": "5 \\%", + "type": "inline_equation" + } + ], + "index": 33 + }, + { + "bbox": [ + 115, + 493, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 115, + 493, + 506, + 507 + ], + "score": 1.0, + "content": "program supervision, suggesting this may be too few examples to learn compositional reasoning.", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 113, + 505, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 113, + 505, + 505, + 517 + ], + "score": 1.0, + "content": "Our question coding stage greatly improves initial program prediction. Our V-NMN approach", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 115, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 115, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "to question coding gets to approximately double the question coding (reconstruction) accuracy as", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 115, + 525, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 115, + 525, + 506, + 540 + ], + "score": 1.0, + "content": "the NMN approach (col 1, question coding). This means that it effectively propagates groundings", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 115, + 536, + 506, + 551 + ], + "spans": [ + { + "bbox": [ + 115, + 536, + 506, + 551 + ], + "score": 1.0, + "content": "from question-aligned programs during the coding phase. Consequently, we also see improved", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 115, + 548, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 115, + 548, + 506, + 561 + ], + "score": 1.0, + "content": "VQA performance after the module and joint training stages which are based on predicted programs.", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 114, + 558, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 114, + 558, + 505, + 573 + ], + "score": 1.0, + "content": "Question reconstruction is well-correlated with overall accuracy. Despite the program space", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 116, + 570, + 505, + 583 + ], + "spans": [ + { + "bbox": [ + 116, + 570, + 505, + 583 + ], + "score": 1.0, + "content": "being trained directly as a latent code for questions, we find question reconstruction accuracy rises", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 116, + 581, + 505, + 593 + ], + "spans": [ + { + "bbox": [ + 116, + 581, + 505, + 593 + ], + "score": 1.0, + "content": "with the amount of supervision – confirming that inference in a space of discrete structured latent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 115, + 591, + 414, + 605 + ], + "spans": [ + { + "bbox": [ + 115, + 591, + 414, + 605 + ], + "score": 1.0, + "content": "variables is difficult to do without some supervision to guide the mapping.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 36.5, + "bbox_fs": [ + 109, + 448, + 506, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 605, + 505, + 682 + ], + "lines": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 106, + 605, + 505, + 617 + ], + "score": 1.0, + "content": "Figure 3 shows sample programs for each model. We see the limited supervision negatively affects", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 615, + 507, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 250, + 629 + ], + "score": 1.0, + "content": "NMN program prediction, with the", + "type": "text" + }, + { + "bbox": [ + 250, + 616, + 265, + 627 + ], + "score": 0.85, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 615, + 414, + 629 + ], + "score": 1.0, + "content": "model resorting to simple Find[X]", + "type": "text" + }, + { + "bbox": [ + 414, + 618, + 425, + 626 + ], + "score": 0.7, + "content": "", + "type": "inline_equation" + }, + { + "bbox": [ + 425, + 615, + 507, + 629 + ], + "score": 1.0, + "content": "Answer structures.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 439, + 639 + ], + "score": 1.0, + "content": "Interestingly, we find that the mistakes made by the V-NMN model, e.g., green in", + "type": "text" + }, + { + "bbox": [ + 439, + 627, + 455, + 637 + ], + "score": 0.86, + "content": "5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "supervision", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 637, + 507, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 507, + 651 + ], + "score": 1.0, + "content": "(top-right) are also made when reconstructing the question (also substituting green for blue). Further,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 649, + 504, + 661 + ], + "spans": [ + { + "bbox": [ + 106, + 649, + 504, + 661 + ], + "score": 1.0, + "content": "when the token does get corrected to blue, the question also eventually gets reconstructed (partially", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 660, + 505, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 660, + 145, + 673 + ], + "score": 1.0, + "content": "correctly", + "type": "text" + }, + { + "bbox": [ + 145, + 660, + 171, + 671 + ], + "score": 0.87, + "content": "( 1 0 \\% )", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 660, + 230, + 673 + ], + "score": 1.0, + "content": "and then fully", + "type": "text" + }, + { + "bbox": [ + 230, + 660, + 258, + 671 + ], + "score": 0.86, + "content": "( 1 5 \\% ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 258, + 660, + 505, + 673 + ], + "score": 1.0, + "content": ", and the program produces the correct answer. This indicates", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 106, + 671, + 498, + 684 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 498, + 684 + ], + "score": 1.0, + "content": "that there is high fidelity between the learnt question space, the answer space and the latent space.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 605, + 507, + 684 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 224, + 700 + ], + "score": 1.0, + "content": "Effect of the Program Prior", + "type": "text" + }, + { + "bbox": [ + 225, + 687, + 244, + 699 + ], + "score": 0.86, + "content": "\\mathbf { p } ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 687, + 506, + 700 + ], + "score": 1.0, + "content": ". Next, we explore the impact of regularizing the program posterior", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 105, + 699, + 506, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 198, + 711 + ], + "score": 1.0, + "content": "to be close to the prior", + "type": "text" + }, + { + "bbox": [ + 198, + 699, + 217, + 711 + ], + "score": 0.91, + "content": "p ( \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 699, + 506, + 711 + ], + "score": 1.0, + "content": ", for different choices of the prior. Firstly, we disable the KL-divergence", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 169, + 723 + ], + "score": 1.0, + "content": "term by setting", + "type": "text" + }, + { + "bbox": [ + 169, + 710, + 195, + 721 + ], + "score": 0.91, + "content": "\\beta = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "in Equation (2) recovering a deterministic version of our model. Compared", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 186, + 734 + ], + "score": 1.0, + "content": "to our full model at", + "type": "text" + }, + { + "bbox": [ + 187, + 721, + 206, + 731 + ], + "score": 0.86, + "content": "10 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 207, + 720, + 440, + 734 + ], + "score": 1.0, + "content": "supervision, the performance on question coding drops to", + "type": "text" + }, + { + "bbox": [ + 441, + 721, + 468, + 731 + ], + "score": 0.85, + "content": "1 2 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 720, + 506, + 734 + ], + "score": 1.0, + "content": "program", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 52.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 127, + 70, + 480, + 187 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 127, + 70, + 480, + 187 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 70, + 480, + 187 + ], + "spans": [ + { + "bbox": [ + 127, + 70, + 480, + 187 + ], + "score": 0.984, + "html": "
%x←zValidation During Training StagesTest Accuracy (VQA Accuracy)
[I] Question Coding (Reconstruction) ）(Program Prediction)[II] Module Training (VQA Accuracy)[] Joint Training (VQA Accuracy)
NMN (Johnson et a., 2017) V-NMN(Ours)5- 54.86±6.759.28±1.91 17.90±3.4861.56±3.59 63.21±3.1263.08±0.78 66.61±1.5860.06±3.88 61.40±4.41
NMN (Johnsonet a.,017) V-NMN(Ours)10= 87.52±6.4524.30±2.39 65.45±11.8860.31±3.37 81.34±8.6166.51±4.10 92.85±4.8961.99±0.96 88.01±6.45
NMN (Johnson et al.,2017) V-NMN(Ours)15= 99.66±0.1947.67±5.02 86.23±3.2969.47±9.87 93.24±3.1762.43±0.49 97.97±1.2061.32±2.36 94.63±2.65
NMN (Johnson et al.,2017) V-NMN(Ours)201 99.88±0.2058.37±3.30 90.95±0.9066.17±7.02 91.12±2.6780.68±18.00 99.13±0.7278.59±19.27 95.63±1.31
", + "type": "table", + "image_path": "a9465e677d4588c8e90a97269fb4139a55bc17698327073d8942e2e20d6d6082.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 127, + 70, + 480, + 109.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 127, + 109.0, + 480, + 148.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 127, + 148.0, + 480, + 187.0 + ], + "spans": [], + "index": 2 + } + ] + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 108, + 195, + 504, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 505, + 208 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 181, + 208 + ], + "score": 1.0, + "content": "Table 1: Results in", + "type": "text" + }, + { + "bbox": [ + 181, + 195, + 190, + 205 + ], + "score": 0.83, + "content": "\\%", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 194, + 505, + 208 + ], + "score": 1.0, + "content": ", on the SHAPES dataset with varying amounts of question-program supervision", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 108, + 205, + 191, + 219 + ], + "spans": [ + { + "bbox": [ + 108, + 206, + 148, + 217 + ], + "score": 0.85, + "content": "( \\% \\mathbf { x } \\mathbf { z } )", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 205, + 153, + 219 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 153, + 206, + 186, + 217 + ], + "score": 0.9, + "content": "\\beta = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 186, + 205, + 191, + 219 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 107, + 217, + 505, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 217, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 210, + 229 + ], + "score": 1.0, + "content": "prediction accuracy (from", + "type": "text" + }, + { + "bbox": [ + 210, + 217, + 242, + 227 + ], + "score": 0.8, + "content": "6 5 . 4 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 242, + 217, + 505, + 229 + ], + "score": 1.0, + "content": "), while the performance on question reconstruction improves from", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 227, + 505, + 240 + ], + "spans": [ + { + "bbox": [ + 106, + 228, + 139, + 238 + ], + "score": 0.89, + "content": "8 7 . 5 2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 227, + 151, + 240 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 152, + 228, + 179, + 238 + ], + "score": 0.87, + "content": "9 3 . 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 179, + 227, + 505, + 240 + ], + "score": 1.0, + "content": ". Without the KL term, the model focuses solely on reconstruction and fails to", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 238, + 505, + 252 + ], + "spans": [ + { + "bbox": [ + 105, + 238, + 246, + 252 + ], + "score": 1.0, + "content": "learn compositionality in the latent", + "type": "text" + }, + { + "bbox": [ + 246, + 241, + 253, + 249 + ], + "score": 0.44, + "content": "\\mathbf { z }", + "type": "inline_equation" + }, + { + "bbox": [ + 253, + 238, + 505, + 252 + ], + "score": 1.0, + "content": "space, such that supervised grounding are poorly propagated to", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 262 + ], + "score": 1.0, + "content": "unsupervised questions as evidenced by the drop in program prediction accuracy. Note that this ties", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "well to our high-level goal to better leverage the semantics of the lexicon provided by the user for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 271, + 241, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 241, + 285 + ], + "score": 1.0, + "content": "interpretable question answering.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "text", + "bbox": [ + 106, + 288, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "Beam search vs. sampling for module training. As explained in Section 2, we use beam search", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 299, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 506, + 311 + ], + "score": 1.0, + "content": "instead of sampling to optimize Equation (5). In this section we empirically study this design choice.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 310, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 323 + ], + "score": 1.0, + "content": "We generally find that in low-supervision settings, where the question coding stage might not produce", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 478, + 334 + ], + "score": 1.0, + "content": "correct programs (for questions) as frequently, using sampling form the sequence model", + "type": "text" + }, + { + "bbox": [ + 478, + 323, + 505, + 333 + ], + "score": 0.92, + "content": "q ( \\mathbf { z } | \\mathbf { x } )", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 331, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 485, + 344 + ], + "score": 1.0, + "content": "instead of beam search leads to a drop in module training performance. For example, with", + "type": "text" + }, + { + "bbox": [ + 486, + 334, + 504, + 343 + ], + "score": 0.83, + "content": "10 \\%", + "type": "inline_equation" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 343, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 343, + 330, + 354 + ], + "score": 1.0, + "content": "supervision, the module training accuracy drops from", + "type": "text" + }, + { + "bbox": [ + 330, + 343, + 386, + 354 + ], + "score": 0.87, + "content": "8 1 . 3 4 \\pm 8 . 6 1", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 343, + 505, + 354 + ], + "score": 1.0, + "content": "(with beam search) to 76.31", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 354, + 505, + 366 + ], + "spans": [ + { + "bbox": [ + 106, + 354, + 136, + 366 + ], + "score": 0.79, + "content": "\\pm ~ 8 . 4 1", + "type": "inline_equation" + }, + { + "bbox": [ + 137, + 354, + 505, + 366 + ], + "score": 1.0, + "content": "(with sampling). With more supervision the choice of beam search or sampling does not", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "matter, which makes intuitive sense, as the this means that the programs that get sampled tend to be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "the correct ones more often (that is, the confidence of the model increases with the improvement in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 388, + 159, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 159, + 398 + ], + "score": 1.0, + "content": "correctness).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 507, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 261, + 418 + ], + "score": 1.0, + "content": "Effect of optimizing the true ELBO,", + "type": "text" + }, + { + "bbox": [ + 262, + 404, + 288, + 415 + ], + "score": 0.9, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 403, + 507, + 418 + ], + "score": 1.0, + "content": ". Next, we empirically validate the claim that approxi-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "mate inference learned using the ELBO leads to autodecoding behavior as predicted by Alemi et al.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 435, + 439 + ], + "score": 1.0, + "content": "(2018), and that better program representations for our problem can be found for", + "type": "text" + }, + { + "bbox": [ + 435, + 426, + 461, + 437 + ], + "score": 0.9, + "content": "\\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "(as shown", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 437, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 154, + 449 + ], + "score": 1.0, + "content": "above). At", + "type": "text" + }, + { + "bbox": [ + 155, + 437, + 174, + 447 + ], + "score": 0.86, + "content": "15 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 437, + 377, + 449 + ], + "score": 1.0, + "content": "program supervision, VQA accuracy drops from", + "type": "text" + }, + { + "bbox": [ + 378, + 437, + 411, + 448 + ], + "score": 0.83, + "content": "9 6 . 9 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 437, + 445, + 448 + ], + "score": 0.73, + "content": "( \\pm 0 . 9 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 437, + 457, + 449 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 458, + 437, + 493, + 448 + ], + "score": 0.9, + "content": "\\beta = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 493, + 437, + 506, + 449 + ], + "score": 1.0, + "content": "to", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 106, + 448, + 139, + 459 + ], + "score": 0.82, + "content": "7 8 . 4 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 448, + 177, + 459 + ], + "score": 0.65, + "content": "( \\pm \\ : 1 . 3 7 )", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 447, + 188, + 460 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 189, + 448, + 215, + 459 + ], + "score": 0.9, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 447, + 469, + 460 + ], + "score": 1.0, + "content": "(for the empirical prior). 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However, the programs in this", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "spans": [ + { + "bbox": [ + 106, + 509, + 505, + 520 + ], + "score": 1.0, + "content": "case become non-compositional (see Section 3), as the model leaks information from questions to", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 505, + 532 + ], + "score": 1.0, + "content": "answers via attention, meaning that programs no longer carry the burden of explaining the observed", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "spans": [ + { + "bbox": [ + 106, + 531, + 505, + 542 + ], + "score": 1.0, + "content": "answers. This makes the modules less interpretable; which is one of the core motivations of this", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 541, + 347, + 554 + ], + "spans": [ + { + "bbox": [ + 105, + 541, + 347, + 554 + ], + "score": 1.0, + "content": "neural-symbolic class of module networks in the first place.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 30.5 + }, + { + "type": "title", + "bbox": [ + 108, + 572, + 287, + 585 + ], + "lines": [ + { + "bbox": [ + 105, + 570, + 290, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 570, + 290, + 588 + ], + "score": 1.0, + "content": "6 DISCUSSION AND CONCLUSION", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 665 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "In this paper we presented a novel, probabilistic neural symbolic model for interpretable visual", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 609, + 505, + 623 + ], + "score": 1.0, + "content": "question answering, that provides explanations of “what” the model is doing on unseen questions", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 506, + 633 + ], + "score": 1.0, + "content": "given a minimal number of annotated symbolic traces or programs of “what it should do”. We", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "demonstrate that our formulation provides more interpretable explanations than previous work on", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "score": 1.0, + "content": "visual question answering called neural module networks, on dataset of compositional questions", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 654, + 256, + 666 + ], + "spans": [ + { + "bbox": [ + 106, + 654, + 256, + 666 + ], + "score": 1.0, + "content": "about shapes (Andreas et al., 2016a).", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 671, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 671, + 505, + 683 + ], + "score": 1.0, + "content": "The key to our approach is a model with programs as a stochastic latent variable, which leads to better", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 681, + 505, + 695 + ], + "score": 1.0, + "content": "sharing of statistics across questions, yielding a latent space where program annotations for known", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 693, + 436, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 436, + 705 + ], + "score": 1.0, + "content": "questions propagate effectively to unknown / novel questions without annotations.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 42 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In general, the model family we study is quite rich, and also supports other inference queries such as", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 505, + 734 + ], + "score": 1.0, + "content": "counter factual explanations of what programs could have led to particular answers for a given image.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 44.5 + } + ], + "page_idx": 8, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2019", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 759 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 309, + 762 + ], + "score": 1.0, + "content": "9", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 127, + 70, + 480, + 187 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 127, + 70, + 480, + 187 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 127, + 70, + 480, + 187 + ], + "spans": [ + { + "bbox": [ + 127, + 70, + 480, + 187 + ], + "score": 0.984, + "html": "
%x←zValidation During Training StagesTest Accuracy (VQA Accuracy)
[I] Question Coding (Reconstruction) ）(Program Prediction)[II] Module Training (VQA Accuracy)[] Joint Training (VQA Accuracy)
NMN (Johnson et a., 2017) V-NMN(Ours)5- 54.86±6.759.28±1.91 17.90±3.4861.56±3.59 63.21±3.1263.08±0.78 66.61±1.5860.06±3.88 61.40±4.41
NMN (Johnsonet a.,017) V-NMN(Ours)10= 87.52±6.4524.30±2.39 65.45±11.8860.31±3.37 81.34±8.6166.51±4.10 92.85±4.8961.99±0.96 88.01±6.45
NMN (Johnson et al.,2017) V-NMN(Ours)15= 99.66±0.1947.67±5.02 86.23±3.2969.47±9.87 93.24±3.1762.43±0.49 97.97±1.2061.32±2.36 94.63±2.65
NMN (Johnson et al.,2017) V-NMN(Ours)201 99.88±0.2058.37±3.30 90.95±0.9066.17±7.02 91.12±2.6780.68±18.00 99.13±0.7278.59±19.27 95.63±1.31
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Note that this ties", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 261, + 506, + 273 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 506, + 273 + ], + "score": 1.0, + "content": "well to our high-level goal to better leverage the semantics of the lexicon provided by the user for", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 271, + 241, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 241, + 285 + ], + "score": 1.0, + "content": "interpretable question answering.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5, + "bbox_fs": [ + 105, + 217, + 506, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 288, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 106, + 287, + 505, + 300 + ], + "score": 1.0, + "content": "Beam search vs. sampling for module training. As explained in Section 2, we use beam search", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 299, + 506, + 311 + ], + "spans": [ + { + "bbox": [ + 106, + 299, + 506, + 311 + ], + "score": 1.0, + "content": "instead of sampling to optimize Equation (5). In this section we empirically study this design choice.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 310, + 505, + 323 + ], + "spans": [ + { + "bbox": [ + 106, + 310, + 505, + 323 + ], + "score": 1.0, + "content": "We generally find that in low-supervision settings, where the question coding stage might not produce", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 321, + 505, + 334 + ], + "spans": [ + { + "bbox": [ + 106, + 321, + 478, + 334 + ], + "score": 1.0, + "content": "correct programs (for questions) as frequently, using sampling form the sequence model", + "type": "text" + }, + { + "bbox": [ + 478, + 323, + 505, + 333 + ], + "score": 0.92, + "content": "q ( \\mathbf { z } | \\mathbf { x } )", + "type": "inline_equation" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 331, + 504, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 331, + 485, + 344 + ], + "score": 1.0, + "content": "instead of beam search leads to a drop in module training performance. 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With more supervision the choice of beam search or sampling does not", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "spans": [ + { + "bbox": [ + 106, + 365, + 505, + 378 + ], + "score": 1.0, + "content": "matter, which makes intuitive sense, as the this means that the programs that get sampled tend to be", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 506, + 388 + ], + "score": 1.0, + "content": "the correct ones more often (that is, the confidence of the model increases with the improvement in", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 388, + 159, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 388, + 159, + 398 + ], + "score": 1.0, + "content": "correctness).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 287, + 506, + 398 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 404, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 403, + 507, + 418 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 261, + 418 + ], + "score": 1.0, + "content": "Effect of optimizing the true ELBO,", + "type": "text" + }, + { + "bbox": [ + 262, + 404, + 288, + 415 + ], + "score": 0.9, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 403, + 507, + 418 + ], + "score": 1.0, + "content": ". Next, we empirically validate the claim that approxi-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 506, + 428 + ], + "score": 1.0, + "content": "mate inference learned using the ELBO leads to autodecoding behavior as predicted by Alemi et al.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 425, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 435, + 439 + ], + "score": 1.0, + "content": "(2018), and that better program representations for our problem can be found for", + "type": "text" + }, + { + "bbox": [ + 435, + 426, + 461, + 437 + ], + "score": 0.9, + "content": "\\beta < 1", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 425, + 506, + 439 + ], + "score": 1.0, + "content": "(as shown", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 437, + 506, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 154, + 449 + ], + "score": 1.0, + "content": "above). 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Question reconstruction drops from", + "type": "text" + }, + { + "bbox": [ + 469, + 448, + 505, + 459 + ], + "score": 0.84, + "content": "9 7 . 4 3 \\ \\%", + "type": "inline_equation" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 458, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 459, + 140, + 470 + ], + "score": 0.8, + "content": "( \\pm 3 . 4 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 458, + 150, + 472 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 150, + 459, + 183, + 470 + ], + "score": 0.9, + "content": "\\beta = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 184, + 458, + 198, + 472 + ], + "score": 1.0, + "content": ", to", + "type": "text" + }, + { + "bbox": [ + 198, + 459, + 232, + 470 + ], + "score": 0.76, + "content": "4 3 . 4 0 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 459, + 266, + 470 + ], + "score": 0.58, + "content": "\\pm 5 . 3 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 458, + 277, + 472 + ], + "score": 1.0, + "content": "at", + "type": "text" + }, + { + "bbox": [ + 278, + 459, + 303, + 470 + ], + "score": 0.91, + "content": "\\beta = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 458, + 505, + 472 + ], + "score": 1.0, + "content": ", showing auto-decoding behavior (as predicted by", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 468, + 142, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 468, + 142, + 484 + ], + "score": 1.0, + "content": "theory).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 403, + 507, + 484 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 486, + 505, + 553 + ], + "lines": [ + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 498 + ], + "score": 1.0, + "content": "Finally, the N2NMN approach (Hu et al., 2017) evaluates their question-attention based module", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 497, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 497, + 313, + 510 + ], + "score": 1.0, + "content": "networks in the fully unsupervised setting, getting to", + "type": "text" + }, + { + "bbox": [ + 313, + 498, + 345, + 508 + ], + "score": 0.88, + "content": "9 6 . 1 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 345, + 497, + 505, + 510 + ], + "score": 1.0, + "content": "on TEST. 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When more supervision is available, regularizing with respect to the broader, syntactic prior", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 691, + 506, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 691, + 506, + 704 + ], + "score": 1.0, + "content": "hurts performance marginally, which makes sense, as one can just treat program supervision as a", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 702, + 281, + 716 + ], + "spans": [ + { + "bbox": [ + 106, + 702, + 281, + 716 + ], + "score": 1.0, + "content": "supervised learning problem in this setting.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 41, + "bbox_fs": [ + 105, + 592, + 506, + 716 + ] + } + ] + } + ], + "_backend": "pipeline", + "_version_name": "2.2.2" +} \ No newline at end of file diff --git a/parse/train/ryxhB3CcK7/ryxhB3CcK7_model.json b/parse/train/ryxhB3CcK7/ryxhB3CcK7_model.json new file mode 100644 index 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%x←zValidation During Training StagesTest Accuracy (VQA Accuracy)
[I] Question Coding (Reconstruction) ）(Program Prediction)[II] Module Training (VQA Accuracy)[] Joint Training (VQA Accuracy)
NMN (Johnson et a., 2017) V-NMN(Ours)5- 54.86±6.759.28±1.91 17.90±3.4861.56±3.59 63.21±3.1263.08±0.78 66.61±1.5860.06±3.88 61.40±4.41
NMN (Johnsonet a.,017) V-NMN(Ours)10= 87.52±6.4524.30±2.39 65.45±11.8860.31±3.37 81.34±8.6166.51±4.10 92.85±4.8961.99±0.96 88.01±6.45
NMN (Johnson et al.,2017) V-NMN(Ours)15= 99.66±0.1947.67±5.02 86.23±3.2969.47±9.87 93.24±3.1762.43±0.49 97.97±1.2061.32±2.36 94.63±2.65
NMN (Johnson et al.,2017) V-NMN(Ours)201 99.88±0.2058.37±3.30 90.95±0.9066.17±7.02 91.12±2.6780.68±18.00 99.13±0.7278.59±19.27 95.63±1.31
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