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+# DATA AUGMENTATION INSTEAD OFEXPLICIT REGULARIZATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Modern deep artificial neural networks have achieved impressive results through models with very large capacity—compared to the number of training examples— that control overfitting with the help of different forms of regularization. Regularization can be implicit, as is the case of stochastic gradient descent or parameter sharing in convolutional layers, or explicit. Most common explicit regularization techniques, such as dropout and weight decay, reduce the effective capacity of the model and typically require the use of deeper and wider architectures to compensate for the reduced capacity. Although these techniques have been proven successful in terms of results, they seem to waste capacity. In contrast, data augmentation techniques reduce the generalization error by increasing the number of training examples and without reducing the effective capacity. In this paper we systematically analyze the effect of data augmentation on some popular architectures and conclude that data augmentation alone—without any other explicit regularization techniques—can achieve the same performance or higher as regularized models, especially when training with fewer examples.
+
+# 1 INTRODUCTION
+
+Regularization plays a central role in machine learning. Loosely defined, regularization is any modification applied to a learning algorithm that helps prevent overfitting and improve generalization. Whereas in simple machine learning algorithms the sources of regularization can be easily identified as explicit terms in the objective function, in modern deep neural networks the sources of regularization are multiple and some of them are not explicit, but implicit.
+
+Although the terms explicit and implicit regularization have been used recently in the literature (Neyshabur et al., 2014; Zhang et al., 2017), their distinction is rather subjective. We propose the following definitions:
+
+• Explicit regularization techniques are those specifically and solely designed to constrain the effective capacity of a given model in order to reduce overfitting. Furthermore, explicit regularizers are not a structural or essential part of the network architecture, the data or the learning algorithm and can typically be added or removed easily. Implicit regularization is the reduction of the generalization error or overfitting provided by characteristics of the network architecture, the training data or the learning algorithm, which are not specifically designed to constrain the effective capacity of the given model.
+
+Examples of explicit regularizers are weight decay (Hanson & Pratt, 1989), which penalizes large parameters; dropout (Srivastava et al., 2014), which randomly removes a fraction of the neural connections during training; or stochastic depth (Huang et al., 2016), which drops whole layers instead. Implicit regularization effects are provided by the popular stochastic gradient descent (SGD) algorithm, which tends to converge to solutions with small norm (Zhang et al., 2017); convolutional layers, which impose parameter sharing based on prior knowledge about the data; batch normalization (Ioffe & Szegedy, 2015), whose main goal is reducing the the internal covariate shift, but also implicitly regularizes the model due to the noise in the batch estimates for mean and variance.
+
+Driven by the efficient use and development of GPUs, much research efforts have been devoted to finding ways of training deeper and wider networks of larger capacity (Simonyan & Zisserman,
+
+2014; He et al., 2016; Zagoruyko & Komodakis, 2016), Ironically, their effective capacity is eventually reduced in practice by the use of weight decay and dropout, among other explicit regularizers. It is known, for instance, that the gain in generalization provided by dropout comes at the cost of using larger models and training for longer (Goodfellow et al., 2016). Hence, it seems that with such an approach deep networks are wasting capacity (Dauphin & Bengio, 2013). As a matter of fact, unlike traditional machine learning models, deep neural networks seem not to need explicit regularizers to generalize well, as recently suggested by Zhang et al. (2017).
+
+One popular technique that also improves generalization is data augmentation. Importantly, it differs from explicit regularizers mainly in that it does not reduce the effective capacity of the model. Data augmentation is a very old practice in machine learning (Simard et al., 1992) and it has been identified as a critical component of many models (Ciresan et al., 2010; Krizhevsky et al., 2012; LeCun et al., 2015). However, although some authors have reported the impact of data augmentation on the performance of their models and, in some cases, a comparison of different amount of augmentation (Graham, 2014) the literature lacks, to our knowledge, a systematic analysis of the impact of data augmentation on deep neural networks compared to the most popular regularization techniques.
+
+# 1.1 OUR CONTRIBUTIONS
+
+In this paper, we systematically analyze the role of data augmentation in deep neural networks for object recognition, compare it to some popular explicit regularization techniques, discuss its relationship with model capacity and test its potential to enhance learning from less training data and adapt to different architectures.
+
+# 1.1.1 DATA AUGMENTATION AND EXPLICIT REGULARIZATION
+
+Zhang et al. (2017) recently raised the thought-provoking idea that explicit regularization may improve generalization performance, but is neither necessary nor by itself sufficient for controlling generalization error. The authors came to this conclusion from the observation that turning off the explicit regularizers of a model does not prevent the model from generalizing—although the performance does become degraded. This contrasts with traditional machine learning involving convex optimization, where regularization is necessary to avoid overfitting and generalize.
+
+However, Zhang et al. (2017) consider data augmentation an explicit form of regularization comparable to weight decay and dropout. We argue instead that data augmentation deserves a different classification due to some fundamental properties: Notably, data augmentation does not reduce the effective capacity of the model. Explicit regularizers are often used to counteract overfitting, but as a side effect the architecture needs to be larger and the training longer (Krizhevsky et al., 2012; Goodfellow et al., 2016). In contrast, data augmentation increases the number of training examples— although not in an independently distributed way—and the robustness against input variability. This has the welcome side-effect of implicitly regularizing the model and improving generalization.
+
+Here, we build upon some of the ideas and procedures from Zhang et al. (2017) and perform some experiments to assess the role of data augmentation in deep neural networks and in particular in contrast to explicit regularizers (weight decay and dropout). In our experiments, we consider two levels of augmentation, light and heavier, as well as no augmentation at all. Then, we test them on two popular successful network architectures: the relatively shallow all convolutional network net (Springenberg et al., 2014) and the deeper wide residual network (Zagoruyko & Komodakis, 2016), trained on CIFAR-10 and CIFAR-100 (Krizhevsky & Hinton, 2009), with and and without explicit regularization. Our central conclusion can be summarized as:
+
+In a deep convolutional neural network trained with sufficient level of data augmentation, optimized by SGD, explicit regularizers (weight decay and dropout) might not provide any additional generalization improvement.
+
+# 1.1.2 DATA AUGMENTATION AND TRAINING WITH FEWER EXAMPLES
+
+Augmented data might be regarded as artificial and very similar to the source examples, therefore with limited contribution for making a network learn more useful representations. However, it has proven to be very useful in extreme cases such as one-shot learning, where only one or few training examples are available (Vinyals et al., 2016).
+
+In order to provide a better insight of the usefulness of data augmentation, we train the networks with only $80 \%$ , $50 \%$ , $10 \%$ and $1 \%$ of the available training data and test the effect of data augmentation, again in contrast to explicit regularizers. The summary of our findings in this regard can be summarized as:
+
+When a deep neural network is trained with a subset of the training data, heavier data augmentation achieves a smaller gap with respect to the baseline model, especially if no explicit regularization is used. Thus, data augmentation seems to serve as true data to a great extent.
+
+# 1.1.3 DATA AUGMENTATION AND ADAPTABILITY
+
+One of the disadvantages of explicit regularization is that the parameters highly depend on the network architecture, the amount of training data and other factors. Therefore, if the architecture or other factors change, one has to tune the regularization hyperparameters to achieve comparable results. In order to analyze how data augmentation adapts to different architectures, we test several augmentation schemes on shallower and deeper versions of the network, with and without explicit regularization. Our finding is the following:
+
+Data augmentation easily adapts to different depths without tuning its parameters. If no explicit regularization is used, we observe that a shallower network achieves slightly worse results and a deeper architecture achieves better results.
+
+# 1.2 RELATED WORK
+
+Regularization is a central research topic in machine learning as it is a key component for ensuring good generalization (Girosi et al., 1995; Muller, 2012). In the case of deep learning, where networks ¨ tend to have several orders of magnitude more parameters than training examples, statistical learning theory (Vapnik & Chervonenkis, 1971) indicates that regularization becomes even more crucial. Accordingly, a myriad of tools and techniques have been proposed as regularizers: early stopping (Plaut et al., 1986), weight decay (Hanson & Pratt, 1989) and other $L ^ { p }$ penalties, dropout (Srivastava et al., 2014) and stochastic depth (Huang et al., 2016), to name a few examples. Besides, other successful techniques have been studied for their regularization effect, despite not being explicitly intended as such. That is the case of unsupervised pre-training (Erhan et al., 2010), multi-task learning (Caruana, 1998), convolutional layers (LeCun et al., 1990), batch normalization (Ioffe & Szegedy, 2015) or adversarial training (Szegedy et al., 2013).
+
+Data augmentation is another almost ubiquitous technique in deep learning, especially for computer vision tasks, which can be regarded as an implicit regularizer because it improves regularization. It was already used in the late 80’s and early 90’s for handwritten digit recognition (Simard et al., 1992) and it has been identified as a very important element of many modern successful models, like AlexNet (Krizhevsky et al., 2012), All-CNN (Springenberg et al., 2014) or ResNet (He et al., 2016), for instance. In some cases, data augmentation has been applied heavily with successful results (Wu et al., 2015). In domains other than computer vision, data augmentation has also been proven effective, for example in speech recognition (Jaitly & Hinton, 2013), music source separation (Uhlich et al., 2017) or text categorization (Lu et al., 2006).
+
+Bengio et al. (2011) focused on the importance of data augmentation for recognizing handwritten digits (MNIST) through greedy layer-wise unsupervised pre-training (Bengio et al., 2007). The main conclusion of that work was that deeper architectures benefit more from data augmentation than shallow networks. Zhang et al. (2017) included data augmentation in their analysis of the role of regularization in the generalization of deep networks, although it was considered an explicit regularizer similar to weight decay and dropout. A few works have reported the performance of their models when trained with different types of data augmentation levels, as is the case of Graham (2014). Recently, the deep learning community seems to have become more aware of the importance of data augmentation and new techniques, such as cutout (DeVries & Taylor, 2017a) or augmentation in the feature space (DeVries & Taylor, 2017b), have been proposed. Very interestingly, models that automatically learn useful data transformations have also been published recently (Hauberg et al., 2016; Lemley et al., 2017; Ratner et al., 2017).
+
+# 2 EXPERIMENTS AND RESULTS
+
+This section describes the experimental setup for systematically analyzing the role of data augmentation in modern deep neural networks and presents the most relevant and interesting results.
+
+# 2.1 SETUP
+
+All the experiments are performed on the neural networks API Keras (Chollet et al., 2015) on top of TensorFlow (Abadi et al., 2015) and on a single GPU NVIDIA GeForce GTX 1080 Ti.
+
+# 2.1.1 NETWORK ACRCHITECTURES
+
+We perform our experiments on two popular architectures that have achieved successful results in object recognition tasks: the all convolutional network, All-CNN (Springenberg et al., 2014) and the wide residual network, WRN (Zagoruyko & Komodakis, 2016). We choose these networks not only because of their effectiveness, but also because they have simple architectures, which is convenient for drawing clearer conclusions. All-CNN has a relatively small number of layers and parameters, whereas WRN is rather deep and has many more parameters.
+
+All convolutional net. All-CNN consists of only convolutional layers with ReLU activations (Glorot et al., 2011), it is relatively shallow (12 layers) and has about $1 . 3 { \bf M }$ parameters. The architecture can be described as follows:
+
+where $K C D ( S )$ is a $D \times D$ convolutional layer with $K$ channels and stride $S$ , followed by batch normalization and a ReLU non-linearity. $N . C l .$ is the number of classes and Gl.Avg. refers to global average pooling. The network is identical to the All-CNN-C architecture in the original paper, except for the introduction of the batch normalization layers. We set the same training parameters as in the original paper in the cases they are reported. Specifically, in all experiments the All-CNN networks are trained using stochastic gradient descent with batch size of 128, during 350 epochs, with fixed momentum 0.9 and learning rate of 0.01 multiplied by 0.1 at epochs 200, 250 and 300. The kernel parameters are initialized according to the Xavier uniform initialization (Glorot & Bengio, 2010).
+
+Wide Residual Network. WRN is a modification of ResNet (He et al., 2016) that achieves better performance with fewer layers, but more units per layer. Although in the original paper several combinations of depth and width are tested, here we choose for our experiments the WRN-28-10 version (28 layers and about $3 6 . 5 \mathrm { ~ M ~ }$ parameters), which is reported to achieve the best results on CIFAR. It has the following architecture:
+
+# 16C3(1)–4×160R–4×320R–4×640R–BN–ReLU–Avg.(8)–FC–Softmax
+
+where $K \mathbf { R }$ is a residual block with residual function BN–ReLU–KC3(1)–BN–ReLU–KC3(1). BN is batch normalization, Avg.(8) is spatial average pooling of size 8 and FC is a fully connected layer. The stride of the first convolution within the residual blocks is 1 except in the first block of the series of 4, where it is 2 to subsample the feature maps. As before, we try to replicate the training parameters of the original paper: we use SGD with batch size of 128, during 200 epochs, with fixed Nesterov momentum 0.9 and learning rate of 0.1 multiplied by 0.2 at epochs 60, 120 and 160. The kernel parameters are initialized according to the He normal initialization (He et al., 2015).
+
+# 2.1.2 DATA
+
+We perform the experiments on the two highly benchmarked data sets CIFAR-10 and CIFAR-100 (Krizhevsky & Hinton, 2009), which are labeled according to 10 and 100 object classes respectively. Both data sets consist of $6 0 , 0 0 0 \ 3 2 \mathrm { ~ x ~ } \ 3 2$ color images split into 50,000 for training and 10,000 for testing. In all our experiments, the input images are fed into the network with pixel values normalized to the range $[ 0 , 1 ]$ and with floating precision of 32 bits. So as to analyze the role of data augmentation, we test the network architectures presented above with two different augmentation schemes as well as with no data augmentation at all:
+
+![](images/cd3cbfd948a1943779f97b4a51f142e747f15f667a9bf6c5d60e6f0854cea858.jpg)  
+Figure 1: Random images from CIFAR10 transformed according to the augmentation schemes used in our experiments, choosing extreme values from the augmentation parameters. Note that these images are very unlikely to be used during training.
+
+Table 1: Description and range of possible values of the parameters used for the heavier augmentation. $B ( p )$ denotes a Bernouilli distribution and $\textstyle { \mathcal { U } } ( a , b )$ a uniform distribution.   
+
+<table><tr><td>Parameter</td><td>Description</td><td>Range</td></tr><tr><td></td><td>Horizontal flip</td><td>1-2B(0.5)</td></tr><tr><td>fh t</td><td>Horizontal translation</td><td>u(-0.1,0.1)</td></tr><tr><td>ty</td><td>Vertical translation</td><td>u(-0.1,0.1)</td></tr><tr><td></td><td>Horizontal scale</td><td>u(0.85,1.15)</td></tr><tr><td>2x</td><td>Vertical scale</td><td>u(0.85,1.15)</td></tr><tr><td>福</td><td>Rotation angle</td><td></td></tr><tr><td>?</td><td>Shear angle</td><td>（22.5,22.5）</td></tr><tr><td>？</td><td>Contrast</td><td>u(-0.15,0.15)</td></tr><tr><td>8</td><td></td><td>u(0.5,1.5)</td></tr><tr><td></td><td>Brightness</td><td>U(-0.25,0.25)</td></tr></table>
+
+Light augmentation. This scheme is adopted from the literature, for example (Goodfellow et al., 2013; Springenberg et al., 2014), and performs only horizontal flips and horizontal and vertical translations of $10 \%$ of the image size.
+
+Heavier augmentation. This scheme performs a larger range of affine transformations, as well as contrast and brightness adjustment:
+
+• Affine transformatio $\operatorname { n s : } \left[ { y ^ { \prime } } \right] = { \left[ \begin{array} { l l l } { f _ { h } z _ { x } \cos ( \theta ) } & { - z _ { y } \sin ( \theta + \phi ) } & { t _ { x } } \\ { z _ { x } \sin ( \theta ) } & { \ : z _ { y } \cos ( \theta + \phi ) } & { t _ { y } } \\ { 0 } & { 0 } & { 1 } \end{array} \right] } { \left[ \begin{array} { l } { x } \\ { y } \\ { 1 } \end{array} \right] }$   
+• Contrast adjustment: $x ^ { \prime } = \gamma ( x - { \overline { { x } } } ) + { \overline { { x } } }$   
+• Brightness adjustment: $x ^ { \prime } = x + \delta$
+
+The description and range of values of the parameters are specified in Table 1 and some examples of transformed images with extreme values of the parameters are provided in Figure 1. The choice of the parameters is arbitrary and the only criterion was that the objects are still recognizable, by visually inspecting a few images. We deliberately avoid designing a particularly successful scheme.
+
+# 2.2 A SUBSTITUTE FOR EXPLICIT REGULARIZATION
+
+![](images/af0d73144298b60f98d1c6d008990d676327a81a2be64210dafc6cd7ccc0e2d8.jpg)  
+Figure 2: Test accuracy of the networks All-CNN and WRN on CIFAR-10 and CIFAR-100, trained without any explicit regularization (upper groups of bars) and with both dropout and weight decay (lower groups), as in the original papers. The different bars represent different models (original, deeper and shallower) and different percentage of training images (100, 50 and $10 \%$ ). The different shades within each bar show the result of training with each data augmentation scheme (none, light and heavier). In most cases, the models trained without regularization achieve the same performance as the explicitly regularized models, or even significantly higher accuracy, as is the case of the shallower and deeper models and when training with fewer examples.
+
+In order to analyze the role of data augmentation and test the hypothesis that it might serve as a substitute for explicit regularization techniques, we first try to replicate the results of All-CNN and WRN provided in the original papers, achieved with both weight decay and dropout. Then, we train the models without weight decay and finally without neither weight decay nor dropout. We test all these different models with the three data augmentation schemes: light, heavier and no augmentation. Additionally, we test the effect of removing the batch normalization (see Appendix A).
+
+As reported by previous works (Krizhevsky et al., 2012; Simonyan & Zisserman, 2014), when the models are trained with data augmentation, at test time slightly better results are obtained by augmenting the test set as well. Therefore, the test accuracy reported here comes from averaging the softmax posteriors over 10 random light augmentations.
+
+The main results of the different experiments are shown in Figure 2 (blue bars) and the full report of all experiments can be found in Table 2 of the Appendix A. As expected, both explicit regularization —weight decay and dropout—and data augmentation are successful in reducing the generalization error. However, some relevant observations can be made. Most notably, it seems that data augmentation alone is able to regularize the model as much as in combination with weight decay and dropout and in some cases it clearly achieves better performance, as in the case of All-CNN. Another observation is that without explicit regularization, heavier augmentation always provides better results than light augmentation, whereas with regularization this effect is counterintuitively not consistent.
+
+# 2.3 FEWER AVAILABLE TRAINING EXAMPLES
+
+We extend the analysis of the data augmentation role by training the same networks with fewer training examples. Similarly, we also analyze the combination of data augmentation and explicit regularization in this case. All models are trained with the same random subset of data and tested in the same test set as the previous experiments in order to enable fairer comparisons. Figure 2 (green and red bars) shows the main results with $50 \%$ and $10 \%$ of the available data and the full report, as well as additional experiments with $80 \%$ and $1 \%$ of the data, is given in Table 3 of the Appendix A.
+
+As expected, the performance decays with the number of available training examples. However, as the level of data augmentation increases, the difference with respect to the baseline performance (by training with all examples) significantly decreases. This indicates that data augmentation serves, to a great extent, as true data. Therefore, this confirms the effectiveness of this technique when not many training examples are available.
+
+Furthermore, in all the experiments with a reduced set of the available data, the observations presented above become even clearer. It seems that if explicit regularization is removed, data augmentation alone better resists the lack of data. This can probably be explained by the fact that explicit regularization reduces the effective capacity, preventing the model from taking advantage of the augmented data.
+
+# 2.4 SHALLOWER AND DEEPER ARCHITECTURES
+
+Finally, we perform the same experiments on shallower and deeper versions of All-CNN, so as to analyze how data augmentation and regularization are handled by architectures of different depth. We test a shallower network with 9 layers instead of 12 and $3 7 4 \mathrm { K }$ parameters instead of $1 . 3 { \bf M }$ :
+
+and a deeper network with 15 layers and $2 . 4 \mathbf { M }$ parameters:
+
+2×96C3(1)–96C3(2)–2×192C3(1)–192C3(2)–2×192C3(1)–192C3(2)–192C3(1)–192C1(1) –N.Cl.C1(1)–Gl.Avg.–Softmax
+
+The results in Figure 2 (purple and brown bars) together with the detailed report of results in Table 4 of the Appendix A show that if the explicit regularization is removed and data augmentation applied, the shallower network achieves slightly worse results and the deeper network slightly better results than the original network. This behavior can be explained by the reduced or increased depth and number of parameters. However, with the explicit regularization active, the results dramatically decrease in both cases. The most probable explanation is that the regularization parameters are not adjusted to the architecture, whereas in the original models the parameters where finely tuned by the authors to obtain state of the art results. This highlights another important advantage of data augmentation: the adjustment of its parameters depends mostly on the training data, rather than on the particular architecture, which offers much more flexibility compared to using explicit regularization. In Appendix B we provide ananalysis of the norm of the weight matrices that helps shed some more light on how the different levels of regularization and data augmentation affect the learned models.
+
+# 3 DISCUSSION AND CONCLUSION
+
+In this work, we have presented a systematic analysis of the role of data augmentation in deep neural networks for object recognition, focusing on the comparison with popular techniques of explicit regularization. We have built upon the work by Zhang et al. (2017), where the authors concluded that explicit regularization is not necessary, although it improves generalization performance. Here, we have shown that it is not only unnecessary, but also that the generalization gain provided by explicit regularization can be achieved by data augmentation alone.
+
+The importance of these results lies in the fact that explicit regularization is the standard tool to enable the generalization of most machine learning methods. However, according to Zhang et al. (2017), explicit regularization plays a different role in deep learning, not explained by statistical learning theory (Vapnik & Chervonenkis, 1971). We argue instead that the theory still holds in deep learning, but one has to properly consider the crucial role of implicit regularization. Explicit regularization is no longer necessary because its contribution is already provided by the many elements that implicitly regularize the models: SGD, convolutional layers or data augmentation, among others.
+
+Whereas explicit regularizers, such as weight decay and dropout, succeed in mitigating overfitting by blindly reducing the effective capacity of a model, implicit regularization operates more effectively at capturing important characteristics of the data (Neyshabur et al., 2014). For instance, convolutional layers successfully reduce the capacity of a model by imposing a parameter sharing strategy that incorporates some essential prior domain knowledge, as well as data augmentation by transforming the training examples in a meaningful and plausible way.
+
+In this regard it is worth highlighting some of the advantages of data augmentation: Not only does it not reduce the effective capacity of the model, but it increases the number of training examples, which, according to statistical learning theories, reduces the generalization error. Furthermore, if the transformations are such that they reflect plausible variations of the real objects, it increases the robustness of the model and it can be regarded as a data-dependent prior, similarly to unsupervised pre-training (Erhan et al., 2010). Besides, unlike explicit regularization techniques, data augmentation does not increase the computational complexity because it can be performed in parallel to the gradient updates on the CPU, making it a computationally free operation. Finally, in Section 2.4 we have shown how data augmentation transparently adapts to architectures of different depth, whereas explicitly regularized models need manual adjustment of the regularization parameters.
+
+Deep neural networks can especially benefit from data augmentation because they do not rely on precomputed features and because the large number of parameters allows them to shatter the augmented training set. Actually, if data augmentation is included for training, we might have to reconsider whether deep learning operates in an overparameterization regime, since the model capacity should take into account the amount of training data, which is exponentially increased by augmentation.
+
+Some argue that despite these advantages, data augmentation is a highly limited approach because it depends on some prior expert knowledge and it cannot be applied to all domains. However, we argue instead that expert knowledge should not be disregarded but exploited. A single data augmentation scheme can be designed for a broad family of data, e.g. natural images, and effectively applied to a broad set of tasks, e.g. object recognition, segmentation, localization, etc. Besides, some recent works show that it is possible to learn the data augmentation strategies (Lemley et al., 2017; Ratner et al., 2017) and future research will probably yield even better results in different domains.
+
+Finally, it is important to note that, due to computational limitations, we have performed a systematic analysis only on CIFAR-10 and CIFAR-100, which consist of very small images. These data sets do not allow performing more agressive data augmentation since the low resolution images can easily show distortions that hinder the recognition of the object. However, some previous works (Graham, 2014; Springenberg et al., 2014) have shown impressive results by performing heavier data augmentation on higher resolution versions on CIFAR-10. We plan to extend this analysis to higher resolution data sets such as ImageNet and one could expect even more benefits from data augmentation compared to explicit regularization techniques.
+
+# ACKNOWLEDGMENTS
+
+This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 641805.
+
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+
+# A DETAILED AND EXTENDED EXPERIMENTAL RESULTS
+
+This appendix details the results of the main experiments shown in Figure 2 and provides the results of many other experiments. For example, the results of the models trained with dropout, but without weight decay and the results of training with $80 \%$ and $1 \%$ of the data are not shown in Figure 2 in order not to clutter the visualization.
+
+Additionally, for most experiments we train a version of the network without batch normalization. These results are provided within brackets in the tables. Note that the original All-CNN results published by Springenberg et al. (2014) did not include batch normalization. In the case of WRN, we remove all batch normalization layers except the top-most one, before the spatial average pooling, since otherwise many models would not converge.
+
+Table 2: Test accuracy of the networks All-CNN and WRN on CIFAR-10 and CIFAR-100, comparing the performance with and without explicit regularizers and the different augmentation schemes. Results within brackets show the performance of the models without batch normalization   
+
+<table><tr><td>Network</td><td>WD</td><td>Dropout</td><td>Aug.scheme</td><td>Test CIFAR-10</td><td>Test CIFAR-100</td></tr><tr><td rowspan="9">All-CNN</td><td>yes</td><td>yes</td><td>no</td><td>90.04 (88.35)</td><td>66.50 (60.54)</td></tr><tr><td>yes</td><td>yes</td><td>light</td><td>93.26 (91.97)</td><td>70.85 (65.57)</td></tr><tr><td>yes</td><td>yes</td><td>heavier</td><td>93.08 (92.44)</td><td>70.59 (68.62)</td></tr><tr><td>no</td><td>yes</td><td>no</td><td>77.99 (87.59)</td><td>52.39 (60.96)</td></tr><tr><td>no</td><td>yes</td><td>light</td><td>77.20 (92.01)</td><td>69.71 (68.01)</td></tr><tr><td>no</td><td>yes</td><td>heavier</td><td>88.29 (92.18)</td><td>70.56 (68.40)</td></tr><tr><td>no</td><td>no</td><td>no</td><td>84.53 (71.98)</td><td>57.99 (39.03)</td></tr><tr><td>no</td><td>no</td><td>light</td><td>93.26 (90.10)</td><td>69.26 (63.00)</td></tr><tr><td>no</td><td>no</td><td>heavier</td><td>93.55 (91.48)</td><td>71.25 (71.46)</td></tr><tr><td rowspan="8">WRN</td><td>yes</td><td>yes</td><td>no</td><td>91.44 (89.30)</td><td>71.67 (67.42)</td></tr><tr><td>yes</td><td>yes</td><td>light</td><td>95.01 (93.48)</td><td>77.58 (74.23)</td></tr><tr><td>yes</td><td>yes</td><td>heavier</td><td>95.60 (94.38)</td><td>76.96 (74.79)</td></tr><tr><td>no</td><td>yes</td><td>no</td><td>91.47 (89.38)</td><td>71.31 (66.85)</td></tr><tr><td>no</td><td>yes</td><td>light</td><td>94.76 (93.52)</td><td>77.42 (74.62)</td></tr><tr><td>no</td><td>yes</td><td>heavier</td><td>95.58 (94.52)</td><td>77.47 (73.96)</td></tr><tr><td>no</td><td>no</td><td>no</td><td>89.56 (85.45)</td><td>68.16 (59.90)</td></tr><tr><td>no</td><td>no</td><td>light</td><td>94.71 (93.69)</td><td>77.08 (75.27)</td></tr><tr><td></td><td>no</td><td>no</td><td>heavier</td><td>95.47 (94.95)</td><td>77.30 (75.69)</td></tr></table>
+
+An important observation from Table 2 is that the interaction of weight decay and dropout is not always consistent, since in some cases better results are obtained with both explicit regularizers active and in other cases, only dropout achieves better generalization. However, the effect of data augmentation seems to be clearer: just some light augmentation achieves much better results than training only with the original data set and performing heavier augmentation almost always further improves the test accuracy, without the need for explicit regularization.
+
+Not surprisingly, batch normalization also contributes to improve the generalization of All-CNN and it seems to combine well with data augmentation. On the contrary, when combined with explicit regularization the results are interestingly not consistent in the case of All-CNN: it seems to improve the generalization of the model trained with both weight decay and dropout, but it drastically reduces the performance with only dropout, in the case of CIFAR-10 and CIFAR-100 without augmentation. A probable explanation is, again, that the regularization hyperparameteres would need to be readjusted with a change of the architecture.
+
+Furthermore, it seems that the gap between the performance of the models trained with and without batch normalization is smaller when they are trained without explicit regularization and when they include heavier data augmentation. This can be observed in both Table 2 and Table 3, which contains the results of the models trained with fewer examples. It is important to note as well the benefits of batch normalization for obtaining better results when training with fewer examples. However, it is surprising that there is only a small drop in the performance of WRN— $9 5 . 4 7 \ \%$ to $9 4 . 9 5 ~ \%$ without regularization— from removing the batch normalization layers of the residual blocks, given
+
+that they were identified as key components for training deep residual networks (He et al., 2016;   
+Zagoruyko & Komodakis, 2016).
+
+Table 3: Test accuracy when training with only $80 \%$ , $50 \%$ , $10 \%$ and $1 \%$ of the available training examples. Results within brackets show the performance of the models without batch normalization   
+
+<table><tr><td>Network</td><td>Pct. Data</td><td>Explicit Reg.</td><td>Aug. scheme</td><td>Test CIFAR-10 89.41 (86.61)</td><td>Test CIFAR-100</td></tr><tr><td rowspan="18"></td><td rowspan="4">80 %</td><td>yes yes</td><td>no light</td><td>92.20 (91.25)</td><td>63.93 (52.51) 67.63 (63.24)</td></tr><tr><td>yes</td><td>heavier</td><td>92.83 (91.42)</td><td>68.01 (65.89)</td></tr><tr><td>no</td><td>no</td><td>83.04 (75.00)</td><td>55.78 (35.95)</td></tr><tr><td>no</td><td>light</td><td>92.25 (88.75)</td><td>69.05 (56.81)</td></tr><tr><td rowspan="6"></td><td>no</td><td>heavier</td><td>92.80 (90.55)</td><td>69.40 (63.57)</td></tr><tr><td>yes</td><td>no</td><td>85.88 (82.33)</td><td>58.24 (44.94)</td></tr><tr><td>yes</td><td>light</td><td>90.30 (87.37)</td><td>61.03 (54.68)</td></tr><tr><td></td><td>heavier</td><td>90.09 (88.94)</td><td>63.25 (57.91)</td></tr><tr><td>yes no</td><td>no</td><td>78.61 (69.46)</td><td>48.62 (31.81)</td></tr><tr><td>no</td><td>light</td><td>90.21 (84.38)</td><td>62.83 (47.84)</td></tr><tr><td rowspan="6">All-CNN 10 %</td><td></td><td>heavier</td><td>90.76 (87.44)</td><td>64.41 (55.27)</td></tr><tr><td>no yes</td><td>no</td><td></td><td></td></tr><tr><td></td><td>light</td><td>67.19 (61.61)</td><td>33.77 (19.79) 38.51 (22.79)</td></tr><tr><td>yes</td><td>heavier</td><td>76.03 (69.18)</td><td>38.34 (26.29)</td></tr><tr><td>yes no</td><td></td><td>78.69 (64.14) 60.97 (41.07)</td><td>26.05 (17.55)</td></tr><tr><td>no</td><td>no light</td><td>78.29 (67.65)</td><td>37.84 (24.34)</td></tr><tr><td rowspan="6">1%</td><td>no</td><td>heavier</td><td>79.87 (70.64)</td><td>39.85 (26.31)</td></tr><tr><td>yes</td><td>no</td><td></td><td>9.16 (3.60)</td></tr><tr><td></td><td>light</td><td>27.53 (29.90) 37.18 (26.85)</td><td>9.64 (3.65)</td></tr><tr><td>yes yes</td><td>heavier</td><td>42.73 (26.87)</td><td>9.14 (2.52)</td></tr><tr><td>no</td><td>no</td><td>38.89 (35.68)</td><td>9.50 (5.51)</td></tr><tr><td>no</td><td>light</td><td>44.35 (29.29)</td><td>9.87 (5.36)</td></tr><tr><td rowspan="10">80% 50%</td><td></td><td>heavier</td><td>47.60 (33.72)</td><td>11.45 (3.57)</td></tr><tr><td>yes</td><td>no</td><td>90.27</td><td>70.41</td></tr><tr><td>yes</td><td>light</td><td>94.07</td><td>75.66</td></tr><tr><td>yes</td><td>heavier</td><td>94.57</td><td>75.51</td></tr><tr><td>no</td><td>no</td><td>88.98</td><td>66.10</td></tr><tr><td></td><td>no light</td><td>93.97</td><td>75.07</td></tr><tr><td></td><td>no heavier</td><td>94.84</td><td>75.38</td></tr><tr><td></td><td>yes no</td><td>86.96</td><td>63.60</td></tr><tr><td>yes</td><td>light</td><td>92.65</td><td>70.83</td></tr><tr><td>yes no</td><td>heavier</td><td>92.86</td><td>70.33</td></tr><tr><td rowspan="6">WRN</td><td></td><td>no</td><td>85.56</td><td>60.64</td></tr><tr><td>no</td><td></td><td></td><td>69.97</td></tr><tr><td>no</td><td>light</td><td>91.87</td><td>70.72</td></tr><tr><td></td><td>heavier</td><td>92.77</td><td></td></tr><tr><td>yes</td><td>no</td><td>70.73</td><td>34.11</td></tr><tr><td>yes</td><td>light</td><td>76.00</td><td>36.65</td></tr><tr><td rowspan="6">10 %</td><td>yes</td><td>heavier</td><td>78.10</td><td>38.93</td></tr><tr><td>no</td><td>no</td><td>60.39</td><td>23.65</td></tr><tr><td>no</td><td>light</td><td>79.19</td><td>39.24</td></tr><tr><td>no</td><td>heavier</td><td>80.29</td><td>41.44</td></tr><tr><td>yes</td><td></td><td>33.45</td><td>7.47</td></tr><tr><td></td><td>no</td><td>34.13</td><td>7.50</td></tr><tr><td rowspan="5">1%</td><td>yes</td><td>light</td><td></td><td></td></tr><tr><td>yes</td><td>heavier</td><td>41.02</td><td>8.37</td></tr><tr><td>no</td><td>no</td><td>38.63</td><td>9.47</td></tr><tr><td>no</td><td>light</td><td>43.84</td><td>9.91</td></tr><tr><td>no</td><td>heavier</td><td>47.14</td><td>11.03</td></tr></table>
+
+The results in Table 3 clearly support the conclusion presented in Section 2.3: data augmentation alone resists better the lack of training data compared to explicit regularizers. Already with $80 \%$ and
+
+Table 4: Test accuracy of the shallower and deeper versions of the All-CNN network on CIFAR-10 and CIFAR-100. Results in parentheses show the difference with respect to the original model.   
+
+<table><tr><td>Network</td><td>Explicit Reg.</td><td>Aug. scheme</td><td>Test CIFAR-10</td><td>Test CIFAR-100</td></tr><tr><td rowspan="5">All-CNN shallower</td><td>yes</td><td>no</td><td>76.45 (-13.59)</td><td>51.31 (-9.23)</td></tr><tr><td>yes</td><td>light</td><td>82.02 (-11.24)</td><td>56.81 (-8.76)</td></tr><tr><td>yes</td><td>heavier</td><td>86.66 (-6.42)</td><td>58.64 (-9.98)</td></tr><tr><td>no</td><td>no</td><td>85.22 (+0.69)</td><td>58.95 (+0.96)</td></tr><tr><td>no</td><td>light</td><td>90.02 (-3.24)</td><td>65.51 (-3.75)</td></tr><tr><td rowspan="6">All-CNN deeper</td><td>no</td><td>heavier</td><td>90.34 (-3.21)</td><td>65.87 (-5.38)</td></tr><tr><td>yes</td><td>no</td><td>86.26 (-3.78)</td><td>49.06 (-11.48)</td></tr><tr><td>yes</td><td>light</td><td>85.04 (-8.22)</td><td>52.03 (-13.54)</td></tr><tr><td>yes</td><td>heavier</td><td>88.46 (-4.62)</td><td>51.78 (-16.84)</td></tr><tr><td>no</td><td>no</td><td>83.30 (-1.23)</td><td>54.22 (-3.77)</td></tr><tr><td>no no</td><td>light heavier</td><td>93.46 (+0.20) 94.19 (+0.64)</td><td>72.16 (+2.90) 73.30 (+2.35)</td></tr></table>
+
+$50 \%$ of the data better results are obtained in some cases, but the differences become much bigger when training with only $10 \%$ and $1 \%$ of the available data. It seems that explicit regularization prevents the model from both fitting the data and generalizing well, whereas data augmentation provides useful transformed examples. Interestingly, with only $1 \%$ of the data, even without data augmentation the models without explicit regularization perform better.
+
+The same effect can be observed in Table 4, where both the shallower and deeper versions of AllCNN perform much worse when trained with explicit regularization, even when trained without data augmentation. This is another piece of evidence that explicit regularization needs to be used very carefully, it requires a proper tuning of the hyperparameters and is not always benefitial.
+
+# B NORM OF THE WEIGHT MATRIX
+
+Table 5: Frobenius norm of the weight matrices learned by the networks All-CNN and WRN on CIFAR-10 and CIFAR-100, trained with and without explicit regularizers and the different augmentation schemes. Norms within brackets correspond to the models without batch normalization   
+
+<table><tr><td>Network</td><td>WD</td><td>Dropout</td><td>Aug. scheme</td><td>Norm CIFAR-10</td><td>Norm( CIFAR-100</td></tr><tr><td rowspan="9">All-CNN</td><td>yes</td><td>yes</td><td>no</td><td>48.7 (64.9)</td><td>76.5 (97.9)</td></tr><tr><td>yes</td><td>yes</td><td>light</td><td>52.7 (63.2)</td><td>77.6 (86.8)</td></tr><tr><td>yes</td><td>yes</td><td>heavier</td><td>57.6 (62.8)</td><td>78.1 (83.1)</td></tr><tr><td>no</td><td>yes</td><td>no</td><td>52.4 (70.5)</td><td>79.7 (103.3)</td></tr><tr><td>no</td><td>yes</td><td>light</td><td>57.0 (67.9)</td><td>83.6 (93.0)</td></tr><tr><td>no</td><td>yes</td><td>heavier</td><td>62.8 (67.5)</td><td>84.0 (88.0)</td></tr><tr><td>no</td><td>no</td><td>no</td><td>37.3 (63.7)</td><td>47.6 (102.7)</td></tr><tr><td>no</td><td>no</td><td>light</td><td>47.0 (69.5)</td><td>80.0 (108.9)</td></tr><tr><td>no</td><td>no</td><td>heavier</td><td>62.0 (71.7)</td><td>91.7 (91.7)</td></tr><tr><td rowspan="8">WRN</td><td>yes</td><td>yes</td><td>no</td><td>101.4 (122.6)</td><td>134.8 (126.5)</td></tr><tr><td>yes</td><td>yes</td><td>light</td><td>106.1 (123.9)</td><td>140.8 (129.3)</td></tr><tr><td>yes</td><td>yes</td><td>heavier</td><td>119.3 (125.3)</td><td>164.2 (132.5)</td></tr><tr><td>no</td><td>yes</td><td>no</td><td>153.3 (122.5)</td><td>185.1 (126.5)</td></tr><tr><td>no</td><td>yes</td><td>light</td><td>160.6 (123.9)</td><td>199.0 (129.4)</td></tr><tr><td>no</td><td>yes</td><td>heavier</td><td>175.1 (125.2)</td><td>225.4 (132.5)</td></tr><tr><td>no</td><td>no</td><td>no</td><td>139.0 (120.4)</td><td>157.9 (122.0)</td></tr><tr><td>no</td><td>no</td><td>light</td><td>153.6 (123.2)</td><td>187.0 (127.2)</td></tr><tr><td></td><td>no</td><td>no</td><td>heavier</td><td>170.4 (125.4)</td><td>217.6 (132.9)</td></tr></table>
+
+One of the simplest way of getting a rough idea of the complexity of the learned models is computing the norm of the weight matrix. Table 5 shows the Frobenius norm of the weight matrices of the models trained with different levels of explicit regularization and data augmentation. The clearest conclusion is that heavier data augmentation seems to yield solutions with larger norm. This is always true except in some All-CNN models trained without batch normalization. Another observation is that, as expected, weight decay constrains the norm of the learned function. Besides, the models trained without batch normalization exhibit smaller differences between different levels of regularization and augmentation and, in the case of All-CNN, less consistency.
+
+Table 6: Frobenius norm of the weight matrices learned by the shallower and deeper versions of the All-CNN network on CIFAR-10 and CIFAR-100.   
+
+<table><tr><td>Network</td><td>Explicit Reg.</td><td>Aug.scheme</td><td>Norm CIFAR-10</td><td>Norm CIFAR-100</td></tr><tr><td rowspan="6">All-CNN shallower</td><td>yes</td><td>no</td><td>47.9</td><td>68.9</td></tr><tr><td>yes</td><td>light</td><td>49.7</td><td>67.1</td></tr><tr><td>yes</td><td>heavier</td><td>51.9</td><td>66.2</td></tr><tr><td>no</td><td>no</td><td>34.8</td><td>64.7</td></tr><tr><td>no</td><td>light</td><td>45.6</td><td>68.8</td></tr><tr><td>no</td><td>heavier</td><td>53.1</td><td>68.3</td></tr><tr><td rowspan="6">All-CNN deeper</td><td>yes</td><td>no</td><td>62.3</td><td>92.1</td></tr><tr><td>yes</td><td>light</td><td>66.5</td><td>95.7</td></tr><tr><td>yes</td><td>heavier</td><td>71.5</td><td>96.9</td></tr><tr><td>no</td><td>no</td><td>45.4</td><td>53.4</td></tr><tr><td>no</td><td>light</td><td>57.3</td><td>77.3</td></tr><tr><td>no</td><td>heavier</td><td>70.7</td><td>97.5</td></tr></table>
+
+One of the relevant results presented in this paper is the poor performance of the regularized models on the shallower and deeper versions of All-CNN, compared to the models without explicit regularization (see Table 4). One hypothesis is that the amount of regularization is not properly adjusted through the hyperparameters. This could be reflected in the norm of the learned weights, shown in Table 6. However, the norm alone does not seem to fully explain the large performance differences between the different models. Finding the exact reasons why the regularized models not able to generalize well might require a much thourough analysis and we leave it as future work.

parse/train/ByJWeR1AW/ByJWeR1AW_content_list.json

@@ -0,0 +1,1568 @@
+[
+    {
+        "type": "text",
+        "text": "DATA AUGMENTATION INSTEAD OFEXPLICIT REGULARIZATION",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
+            593,
+            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": "Modern deep artificial neural networks have achieved impressive results through models with very large capacity—compared to the number of training examples— that control overfitting with the help of different forms of regularization. Regularization can be implicit, as is the case of stochastic gradient descent or parameter sharing in convolutional layers, or explicit. Most common explicit regularization techniques, such as dropout and weight decay, reduce the effective capacity of the model and typically require the use of deeper and wider architectures to compensate for the reduced capacity. Although these techniques have been proven successful in terms of results, they seem to waste capacity. In contrast, data augmentation techniques reduce the generalization error by increasing the number of training examples and without reducing the effective capacity. In this paper we systematically analyze the effect of data augmentation on some popular architectures and conclude that data augmentation alone—without any other explicit regularization techniques—can achieve the same performance or higher as regularized models, especially when training with fewer examples. ",
+        "bbox": [
+            233,
+            266,
+            764,
+            473
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            500,
+            336,
+            515
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Regularization plays a central role in machine learning. Loosely defined, regularization is any modification applied to a learning algorithm that helps prevent overfitting and improve generalization. Whereas in simple machine learning algorithms the sources of regularization can be easily identified as explicit terms in the objective function, in modern deep neural networks the sources of regularization are multiple and some of them are not explicit, but implicit. ",
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+        "type": "text",
+        "text": "Although the terms explicit and implicit regularization have been used recently in the literature (Neyshabur et al., 2014; Zhang et al., 2017), their distinction is rather subjective. We propose the following definitions: ",
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+    {
+        "type": "text",
+        "text": "• Explicit regularization techniques are those specifically and solely designed to constrain the effective capacity of a given model in order to reduce overfitting. Furthermore, explicit regularizers are not a structural or essential part of the network architecture, the data or the learning algorithm and can typically be added or removed easily. Implicit regularization is the reduction of the generalization error or overfitting provided by characteristics of the network architecture, the training data or the learning algorithm, which are not specifically designed to constrain the effective capacity of the given model. ",
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+    {
+        "type": "text",
+        "text": "Examples of explicit regularizers are weight decay (Hanson & Pratt, 1989), which penalizes large parameters; dropout (Srivastava et al., 2014), which randomly removes a fraction of the neural connections during training; or stochastic depth (Huang et al., 2016), which drops whole layers instead. Implicit regularization effects are provided by the popular stochastic gradient descent (SGD) algorithm, which tends to converge to solutions with small norm (Zhang et al., 2017); convolutional layers, which impose parameter sharing based on prior knowledge about the data; batch normalization (Ioffe & Szegedy, 2015), whose main goal is reducing the the internal covariate shift, but also implicitly regularizes the model due to the noise in the batch estimates for mean and variance. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Driven by the efficient use and development of GPUs, much research efforts have been devoted to finding ways of training deeper and wider networks of larger capacity (Simonyan & Zisserman, ",
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+    {
+        "type": "text",
+        "text": "2014; He et al., 2016; Zagoruyko & Komodakis, 2016), Ironically, their effective capacity is eventually reduced in practice by the use of weight decay and dropout, among other explicit regularizers. It is known, for instance, that the gain in generalization provided by dropout comes at the cost of using larger models and training for longer (Goodfellow et al., 2016). Hence, it seems that with such an approach deep networks are wasting capacity (Dauphin & Bengio, 2013). As a matter of fact, unlike traditional machine learning models, deep neural networks seem not to need explicit regularizers to generalize well, as recently suggested by Zhang et al. (2017). ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "One popular technique that also improves generalization is data augmentation. Importantly, it differs from explicit regularizers mainly in that it does not reduce the effective capacity of the model. Data augmentation is a very old practice in machine learning (Simard et al., 1992) and it has been identified as a critical component of many models (Ciresan et al., 2010; Krizhevsky et al., 2012; LeCun et al., 2015). However, although some authors have reported the impact of data augmentation on the performance of their models and, in some cases, a comparison of different amount of augmentation (Graham, 2014) the literature lacks, to our knowledge, a systematic analysis of the impact of data augmentation on deep neural networks compared to the most popular regularization techniques. ",
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+    {
+        "type": "text",
+        "text": "1.1 OUR CONTRIBUTIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we systematically analyze the role of data augmentation in deep neural networks for object recognition, compare it to some popular explicit regularization techniques, discuss its relationship with model capacity and test its potential to enhance learning from less training data and adapt to different architectures. ",
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+    {
+        "type": "text",
+        "text": "1.1.1 DATA AUGMENTATION AND EXPLICIT REGULARIZATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Zhang et al. (2017) recently raised the thought-provoking idea that explicit regularization may improve generalization performance, but is neither necessary nor by itself sufficient for controlling generalization error. The authors came to this conclusion from the observation that turning off the explicit regularizers of a model does not prevent the model from generalizing—although the performance does become degraded. This contrasts with traditional machine learning involving convex optimization, where regularization is necessary to avoid overfitting and generalize. ",
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+    {
+        "type": "text",
+        "text": "However, Zhang et al. (2017) consider data augmentation an explicit form of regularization comparable to weight decay and dropout. We argue instead that data augmentation deserves a different classification due to some fundamental properties: Notably, data augmentation does not reduce the effective capacity of the model. Explicit regularizers are often used to counteract overfitting, but as a side effect the architecture needs to be larger and the training longer (Krizhevsky et al., 2012; Goodfellow et al., 2016). In contrast, data augmentation increases the number of training examples— although not in an independently distributed way—and the robustness against input variability. This has the welcome side-effect of implicitly regularizing the model and improving generalization. ",
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+    {
+        "type": "text",
+        "text": "Here, we build upon some of the ideas and procedures from Zhang et al. (2017) and perform some experiments to assess the role of data augmentation in deep neural networks and in particular in contrast to explicit regularizers (weight decay and dropout). In our experiments, we consider two levels of augmentation, light and heavier, as well as no augmentation at all. Then, we test them on two popular successful network architectures: the relatively shallow all convolutional network net (Springenberg et al., 2014) and the deeper wide residual network (Zagoruyko & Komodakis, 2016), trained on CIFAR-10 and CIFAR-100 (Krizhevsky & Hinton, 2009), with and and without explicit regularization. Our central conclusion can be summarized as: ",
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+    {
+        "type": "text",
+        "text": "In a deep convolutional neural network trained with sufficient level of data augmentation, optimized by SGD, explicit regularizers (weight decay and dropout) might not provide any additional generalization improvement. ",
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+    {
+        "type": "text",
+        "text": "1.1.2 DATA AUGMENTATION AND TRAINING WITH FEWER EXAMPLES",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Augmented data might be regarded as artificial and very similar to the source examples, therefore with limited contribution for making a network learn more useful representations. However, it has proven to be very useful in extreme cases such as one-shot learning, where only one or few training examples are available (Vinyals et al., 2016). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In order to provide a better insight of the usefulness of data augmentation, we train the networks with only $80 \\%$ , $50 \\%$ , $10 \\%$ and $1 \\%$ of the available training data and test the effect of data augmentation, again in contrast to explicit regularizers. The summary of our findings in this regard can be summarized as: ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "When a deep neural network is trained with a subset of the training data, heavier data augmentation achieves a smaller gap with respect to the baseline model, especially if no explicit regularization is used. Thus, data augmentation seems to serve as true data to a great extent. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "1.1.3 DATA AUGMENTATION AND ADAPTABILITY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One of the disadvantages of explicit regularization is that the parameters highly depend on the network architecture, the amount of training data and other factors. Therefore, if the architecture or other factors change, one has to tune the regularization hyperparameters to achieve comparable results. In order to analyze how data augmentation adapts to different architectures, we test several augmentation schemes on shallower and deeper versions of the network, with and without explicit regularization. Our finding is the following: ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Data augmentation easily adapts to different depths without tuning its parameters. If no explicit regularization is used, we observe that a shallower network achieves slightly worse results and a deeper architecture achieves better results. ",
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+    {
+        "type": "text",
+        "text": "1.2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Regularization is a central research topic in machine learning as it is a key component for ensuring good generalization (Girosi et al., 1995; Muller, 2012). In the case of deep learning, where networks ¨ tend to have several orders of magnitude more parameters than training examples, statistical learning theory (Vapnik & Chervonenkis, 1971) indicates that regularization becomes even more crucial. Accordingly, a myriad of tools and techniques have been proposed as regularizers: early stopping (Plaut et al., 1986), weight decay (Hanson & Pratt, 1989) and other $L ^ { p }$ penalties, dropout (Srivastava et al., 2014) and stochastic depth (Huang et al., 2016), to name a few examples. Besides, other successful techniques have been studied for their regularization effect, despite not being explicitly intended as such. That is the case of unsupervised pre-training (Erhan et al., 2010), multi-task learning (Caruana, 1998), convolutional layers (LeCun et al., 1990), batch normalization (Ioffe & Szegedy, 2015) or adversarial training (Szegedy et al., 2013). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Data augmentation is another almost ubiquitous technique in deep learning, especially for computer vision tasks, which can be regarded as an implicit regularizer because it improves regularization. It was already used in the late 80’s and early 90’s for handwritten digit recognition (Simard et al., 1992) and it has been identified as a very important element of many modern successful models, like AlexNet (Krizhevsky et al., 2012), All-CNN (Springenberg et al., 2014) or ResNet (He et al., 2016), for instance. In some cases, data augmentation has been applied heavily with successful results (Wu et al., 2015). In domains other than computer vision, data augmentation has also been proven effective, for example in speech recognition (Jaitly & Hinton, 2013), music source separation (Uhlich et al., 2017) or text categorization (Lu et al., 2006). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Bengio et al. (2011) focused on the importance of data augmentation for recognizing handwritten digits (MNIST) through greedy layer-wise unsupervised pre-training (Bengio et al., 2007). The main conclusion of that work was that deeper architectures benefit more from data augmentation than shallow networks. Zhang et al. (2017) included data augmentation in their analysis of the role of regularization in the generalization of deep networks, although it was considered an explicit regularizer similar to weight decay and dropout. A few works have reported the performance of their models when trained with different types of data augmentation levels, as is the case of Graham (2014). Recently, the deep learning community seems to have become more aware of the importance of data augmentation and new techniques, such as cutout (DeVries & Taylor, 2017a) or augmentation in the feature space (DeVries & Taylor, 2017b), have been proposed. Very interestingly, models that automatically learn useful data transformations have also been published recently (Hauberg et al., 2016; Lemley et al., 2017; Ratner et al., 2017). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "2 EXPERIMENTS AND RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "This section describes the experimental setup for systematically analyzing the role of data augmentation in modern deep neural networks and presents the most relevant and interesting results. ",
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+    {
+        "type": "text",
+        "text": "2.1 SETUP ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "All the experiments are performed on the neural networks API Keras (Chollet et al., 2015) on top of TensorFlow (Abadi et al., 2015) and on a single GPU NVIDIA GeForce GTX 1080 Ti. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "2.1.1 NETWORK ACRCHITECTURES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We perform our experiments on two popular architectures that have achieved successful results in object recognition tasks: the all convolutional network, All-CNN (Springenberg et al., 2014) and the wide residual network, WRN (Zagoruyko & Komodakis, 2016). We choose these networks not only because of their effectiveness, but also because they have simple architectures, which is convenient for drawing clearer conclusions. All-CNN has a relatively small number of layers and parameters, whereas WRN is rather deep and has many more parameters. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "All convolutional net. All-CNN consists of only convolutional layers with ReLU activations (Glorot et al., 2011), it is relatively shallow (12 layers) and has about $1 . 3 { \\bf M }$ parameters. The architecture can be described as follows: ",
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+    },
+    {
+        "type": "text",
+        "text": "where $K C D ( S )$ is a $D \\times D$ convolutional layer with $K$ channels and stride $S$ , followed by batch normalization and a ReLU non-linearity. $N . C l .$ is the number of classes and Gl.Avg. refers to global average pooling. The network is identical to the All-CNN-C architecture in the original paper, except for the introduction of the batch normalization layers. We set the same training parameters as in the original paper in the cases they are reported. Specifically, in all experiments the All-CNN networks are trained using stochastic gradient descent with batch size of 128, during 350 epochs, with fixed momentum 0.9 and learning rate of 0.01 multiplied by 0.1 at epochs 200, 250 and 300. The kernel parameters are initialized according to the Xavier uniform initialization (Glorot & Bengio, 2010). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Wide Residual Network. WRN is a modification of ResNet (He et al., 2016) that achieves better performance with fewer layers, but more units per layer. Although in the original paper several combinations of depth and width are tested, here we choose for our experiments the WRN-28-10 version (28 layers and about $3 6 . 5 \\mathrm { ~ M ~ }$ parameters), which is reported to achieve the best results on CIFAR. It has the following architecture: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "16C3(1)–4×160R–4×320R–4×640R–BN–ReLU–Avg.(8)–FC–Softmax ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $K \\mathbf { R }$ is a residual block with residual function BN–ReLU–KC3(1)–BN–ReLU–KC3(1). BN is batch normalization, Avg.(8) is spatial average pooling of size 8 and FC is a fully connected layer. The stride of the first convolution within the residual blocks is 1 except in the first block of the series of 4, where it is 2 to subsample the feature maps. As before, we try to replicate the training parameters of the original paper: we use SGD with batch size of 128, during 200 epochs, with fixed Nesterov momentum 0.9 and learning rate of 0.1 multiplied by 0.2 at epochs 60, 120 and 160. The kernel parameters are initialized according to the He normal initialization (He et al., 2015). ",
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+    {
+        "type": "text",
+        "text": "2.1.2 DATA ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We perform the experiments on the two highly benchmarked data sets CIFAR-10 and CIFAR-100 (Krizhevsky & Hinton, 2009), which are labeled according to 10 and 100 object classes respectively. Both data sets consist of $6 0 , 0 0 0 \\ 3 2 \\mathrm { ~ x ~ } \\ 3 2$ color images split into 50,000 for training and 10,000 for testing. In all our experiments, the input images are fed into the network with pixel values normalized to the range $[ 0 , 1 ]$ and with floating precision of 32 bits. So as to analyze the role of data augmentation, we test the network architectures presented above with two different augmentation schemes as well as with no data augmentation at all: ",
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+    {
+        "type": "image",
+        "img_path": "images/cd3cbfd948a1943779f97b4a51f142e747f15f667a9bf6c5d60e6f0854cea858.jpg",
+        "image_caption": [
+            "Figure 1: Random images from CIFAR10 transformed according to the augmentation schemes used in our experiments, choosing extreme values from the augmentation parameters. Note that these images are very unlikely to be used during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/bc5385656e454271431b6051ac209d02bcb19677b9163decd84ca314f71327bc.jpg",
+        "table_caption": [
+            "Table 1: Description and range of possible values of the parameters used for the heavier augmentation. $B ( p )$ denotes a Bernouilli distribution and $\\textstyle { \\mathcal { U } } ( a , b )$ a uniform distribution. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Parameter</td><td>Description</td><td>Range</td></tr><tr><td></td><td>Horizontal flip</td><td>1-2B(0.5)</td></tr><tr><td>fh t</td><td>Horizontal translation</td><td>u(-0.1,0.1)</td></tr><tr><td>ty</td><td>Vertical translation</td><td>u(-0.1,0.1)</td></tr><tr><td></td><td>Horizontal scale</td><td>u(0.85,1.15)</td></tr><tr><td>2x</td><td>Vertical scale</td><td>u(0.85,1.15)</td></tr><tr><td>福</td><td>Rotation angle</td><td></td></tr><tr><td>?</td><td>Shear angle</td><td>（22.5,22.5）</td></tr><tr><td>？</td><td>Contrast</td><td>u(-0.15,0.15)</td></tr><tr><td>8</td><td></td><td>u(0.5,1.5)</td></tr><tr><td></td><td>Brightness</td><td>U(-0.25,0.25)</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Light augmentation. This scheme is adopted from the literature, for example (Goodfellow et al., 2013; Springenberg et al., 2014), and performs only horizontal flips and horizontal and vertical translations of $10 \\%$ of the image size. ",
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+    {
+        "type": "text",
+        "text": "Heavier augmentation. This scheme performs a larger range of affine transformations, as well as contrast and brightness adjustment: ",
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+    {
+        "type": "text",
+        "text": "• Affine transformatio $\\operatorname { n s : } \\left[ { y ^ { \\prime } } \\right] = { \\left[ \\begin{array} { l l l } { f _ { h } z _ { x } \\cos ( \\theta ) } & { - z _ { y } \\sin ( \\theta + \\phi ) } & { t _ { x } } \\\\ { z _ { x } \\sin ( \\theta ) } & { \\ : z _ { y } \\cos ( \\theta + \\phi ) } & { t _ { y } } \\\\ { 0 } & { 0 } & { 1 } \\end{array} \\right] } { \\left[ \\begin{array} { l } { x } \\\\ { y } \\\\ { 1 } \\end{array} \\right] }$   \n• Contrast adjustment: $x ^ { \\prime } = \\gamma ( x - { \\overline { { x } } } ) + { \\overline { { x } } }$   \n• Brightness adjustment: $x ^ { \\prime } = x + \\delta$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The description and range of values of the parameters are specified in Table 1 and some examples of transformed images with extreme values of the parameters are provided in Figure 1. The choice of the parameters is arbitrary and the only criterion was that the objects are still recognizable, by visually inspecting a few images. We deliberately avoid designing a particularly successful scheme. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.2 A SUBSTITUTE FOR EXPLICIT REGULARIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/af0d73144298b60f98d1c6d008990d676327a81a2be64210dafc6cd7ccc0e2d8.jpg",
+        "image_caption": [
+            "Figure 2: Test accuracy of the networks All-CNN and WRN on CIFAR-10 and CIFAR-100, trained without any explicit regularization (upper groups of bars) and with both dropout and weight decay (lower groups), as in the original papers. The different bars represent different models (original, deeper and shallower) and different percentage of training images (100, 50 and $10 \\%$ ). The different shades within each bar show the result of training with each data augmentation scheme (none, light and heavier). In most cases, the models trained without regularization achieve the same performance as the explicitly regularized models, or even significantly higher accuracy, as is the case of the shallower and deeper models and when training with fewer examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In order to analyze the role of data augmentation and test the hypothesis that it might serve as a substitute for explicit regularization techniques, we first try to replicate the results of All-CNN and WRN provided in the original papers, achieved with both weight decay and dropout. Then, we train the models without weight decay and finally without neither weight decay nor dropout. We test all these different models with the three data augmentation schemes: light, heavier and no augmentation. Additionally, we test the effect of removing the batch normalization (see Appendix A). ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "As reported by previous works (Krizhevsky et al., 2012; Simonyan & Zisserman, 2014), when the models are trained with data augmentation, at test time slightly better results are obtained by augmenting the test set as well. Therefore, the test accuracy reported here comes from averaging the softmax posteriors over 10 random light augmentations. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The main results of the different experiments are shown in Figure 2 (blue bars) and the full report of all experiments can be found in Table 2 of the Appendix A. As expected, both explicit regularization —weight decay and dropout—and data augmentation are successful in reducing the generalization error. However, some relevant observations can be made. Most notably, it seems that data augmentation alone is able to regularize the model as much as in combination with weight decay and dropout and in some cases it clearly achieves better performance, as in the case of All-CNN. Another observation is that without explicit regularization, heavier augmentation always provides better results than light augmentation, whereas with regularization this effect is counterintuitively not consistent. ",
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+    {
+        "type": "text",
+        "text": "2.3 FEWER AVAILABLE TRAINING EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We extend the analysis of the data augmentation role by training the same networks with fewer training examples. Similarly, we also analyze the combination of data augmentation and explicit regularization in this case. All models are trained with the same random subset of data and tested in the same test set as the previous experiments in order to enable fairer comparisons. Figure 2 (green and red bars) shows the main results with $50 \\%$ and $10 \\%$ of the available data and the full report, as well as additional experiments with $80 \\%$ and $1 \\%$ of the data, is given in Table 3 of the Appendix A. ",
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+    {
+        "type": "text",
+        "text": "As expected, the performance decays with the number of available training examples. However, as the level of data augmentation increases, the difference with respect to the baseline performance (by training with all examples) significantly decreases. This indicates that data augmentation serves, to a great extent, as true data. Therefore, this confirms the effectiveness of this technique when not many training examples are available. ",
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+    {
+        "type": "text",
+        "text": "Furthermore, in all the experiments with a reduced set of the available data, the observations presented above become even clearer. It seems that if explicit regularization is removed, data augmentation alone better resists the lack of data. This can probably be explained by the fact that explicit regularization reduces the effective capacity, preventing the model from taking advantage of the augmented data. ",
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+    {
+        "type": "text",
+        "text": "2.4 SHALLOWER AND DEEPER ARCHITECTURES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Finally, we perform the same experiments on shallower and deeper versions of All-CNN, so as to analyze how data augmentation and regularization are handled by architectures of different depth. We test a shallower network with 9 layers instead of 12 and $3 7 4 \\mathrm { K }$ parameters instead of $1 . 3 { \\bf M }$ : ",
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+    },
+    {
+        "type": "text",
+        "text": "and a deeper network with 15 layers and $2 . 4 \\mathbf { M }$ parameters: ",
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+    {
+        "type": "text",
+        "text": "2×96C3(1)–96C3(2)–2×192C3(1)–192C3(2)–2×192C3(1)–192C3(2)–192C3(1)–192C1(1) –N.Cl.C1(1)–Gl.Avg.–Softmax ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results in Figure 2 (purple and brown bars) together with the detailed report of results in Table 4 of the Appendix A show that if the explicit regularization is removed and data augmentation applied, the shallower network achieves slightly worse results and the deeper network slightly better results than the original network. This behavior can be explained by the reduced or increased depth and number of parameters. However, with the explicit regularization active, the results dramatically decrease in both cases. The most probable explanation is that the regularization parameters are not adjusted to the architecture, whereas in the original models the parameters where finely tuned by the authors to obtain state of the art results. This highlights another important advantage of data augmentation: the adjustment of its parameters depends mostly on the training data, rather than on the particular architecture, which offers much more flexibility compared to using explicit regularization. In Appendix B we provide ananalysis of the norm of the weight matrices that helps shed some more light on how the different levels of regularization and data augmentation affect the learned models. ",
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+    {
+        "type": "text",
+        "text": "3 DISCUSSION AND CONCLUSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this work, we have presented a systematic analysis of the role of data augmentation in deep neural networks for object recognition, focusing on the comparison with popular techniques of explicit regularization. We have built upon the work by Zhang et al. (2017), where the authors concluded that explicit regularization is not necessary, although it improves generalization performance. Here, we have shown that it is not only unnecessary, but also that the generalization gain provided by explicit regularization can be achieved by data augmentation alone. ",
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+        "type": "text",
+        "text": "The importance of these results lies in the fact that explicit regularization is the standard tool to enable the generalization of most machine learning methods. However, according to Zhang et al. (2017), explicit regularization plays a different role in deep learning, not explained by statistical learning theory (Vapnik & Chervonenkis, 1971). We argue instead that the theory still holds in deep learning, but one has to properly consider the crucial role of implicit regularization. Explicit regularization is no longer necessary because its contribution is already provided by the many elements that implicitly regularize the models: SGD, convolutional layers or data augmentation, among others. ",
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+    {
+        "type": "text",
+        "text": "Whereas explicit regularizers, such as weight decay and dropout, succeed in mitigating overfitting by blindly reducing the effective capacity of a model, implicit regularization operates more effectively at capturing important characteristics of the data (Neyshabur et al., 2014). For instance, convolutional layers successfully reduce the capacity of a model by imposing a parameter sharing strategy that incorporates some essential prior domain knowledge, as well as data augmentation by transforming the training examples in a meaningful and plausible way. ",
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+    {
+        "type": "text",
+        "text": "In this regard it is worth highlighting some of the advantages of data augmentation: Not only does it not reduce the effective capacity of the model, but it increases the number of training examples, which, according to statistical learning theories, reduces the generalization error. Furthermore, if the transformations are such that they reflect plausible variations of the real objects, it increases the robustness of the model and it can be regarded as a data-dependent prior, similarly to unsupervised pre-training (Erhan et al., 2010). Besides, unlike explicit regularization techniques, data augmentation does not increase the computational complexity because it can be performed in parallel to the gradient updates on the CPU, making it a computationally free operation. Finally, in Section 2.4 we have shown how data augmentation transparently adapts to architectures of different depth, whereas explicitly regularized models need manual adjustment of the regularization parameters. ",
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+    {
+        "type": "text",
+        "text": "Deep neural networks can especially benefit from data augmentation because they do not rely on precomputed features and because the large number of parameters allows them to shatter the augmented training set. Actually, if data augmentation is included for training, we might have to reconsider whether deep learning operates in an overparameterization regime, since the model capacity should take into account the amount of training data, which is exponentially increased by augmentation. ",
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+    {
+        "type": "text",
+        "text": "Some argue that despite these advantages, data augmentation is a highly limited approach because it depends on some prior expert knowledge and it cannot be applied to all domains. However, we argue instead that expert knowledge should not be disregarded but exploited. A single data augmentation scheme can be designed for a broad family of data, e.g. natural images, and effectively applied to a broad set of tasks, e.g. object recognition, segmentation, localization, etc. Besides, some recent works show that it is possible to learn the data augmentation strategies (Lemley et al., 2017; Ratner et al., 2017) and future research will probably yield even better results in different domains. ",
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+    {
+        "type": "text",
+        "text": "Finally, it is important to note that, due to computational limitations, we have performed a systematic analysis only on CIFAR-10 and CIFAR-100, which consist of very small images. These data sets do not allow performing more agressive data augmentation since the low resolution images can easily show distortions that hinder the recognition of the object. However, some previous works (Graham, 2014; Springenberg et al., 2014) have shown impressive results by performing heavier data augmentation on higher resolution versions on CIFAR-10. We plan to extend this analysis to higher resolution data sets such as ImageNet and one could expect even more benefits from data augmentation compared to explicit regularization techniques. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+        "type": "text",
+        "text": "This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 641805. ",
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+        "text": "Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. In International Conference on Learning Representations, ICLR, arXiv:1611.03530, 2017. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A DETAILED AND EXTENDED EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            647,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This appendix details the results of the main experiments shown in Figure 2 and provides the results of many other experiments. For example, the results of the models trained with dropout, but without weight decay and the results of training with $80 \\%$ and $1 \\%$ of the data are not shown in Figure 2 in order not to clutter the visualization. ",
+        "bbox": [
+            174,
+            135,
+            825,
+            190
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Additionally, for most experiments we train a version of the network without batch normalization. These results are provided within brackets in the tables. Note that the original All-CNN results published by Springenberg et al. (2014) did not include batch normalization. In the case of WRN, we remove all batch normalization layers except the top-most one, before the spatial average pooling, since otherwise many models would not converge. ",
+        "bbox": [
+            174,
+            198,
+            825,
+            267
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/3da9894d7892c0d4de345093684990b56ef235f8a62a568379ebee5bc2b80ee8.jpg",
+        "table_caption": [
+            "Table 2: Test accuracy of the networks All-CNN and WRN on CIFAR-10 and CIFAR-100, comparing the performance with and without explicit regularizers and the different augmentation schemes. Results within brackets show the performance of the models without batch normalization "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>WD</td><td>Dropout</td><td>Aug.scheme</td><td>Test CIFAR-10</td><td>Test CIFAR-100</td></tr><tr><td rowspan=\"9\">All-CNN</td><td>yes</td><td>yes</td><td>no</td><td>90.04 (88.35)</td><td>66.50 (60.54)</td></tr><tr><td>yes</td><td>yes</td><td>light</td><td>93.26 (91.97)</td><td>70.85 (65.57)</td></tr><tr><td>yes</td><td>yes</td><td>heavier</td><td>93.08 (92.44)</td><td>70.59 (68.62)</td></tr><tr><td>no</td><td>yes</td><td>no</td><td>77.99 (87.59)</td><td>52.39 (60.96)</td></tr><tr><td>no</td><td>yes</td><td>light</td><td>77.20 (92.01)</td><td>69.71 (68.01)</td></tr><tr><td>no</td><td>yes</td><td>heavier</td><td>88.29 (92.18)</td><td>70.56 (68.40)</td></tr><tr><td>no</td><td>no</td><td>no</td><td>84.53 (71.98)</td><td>57.99 (39.03)</td></tr><tr><td>no</td><td>no</td><td>light</td><td>93.26 (90.10)</td><td>69.26 (63.00)</td></tr><tr><td>no</td><td>no</td><td>heavier</td><td>93.55 (91.48)</td><td>71.25 (71.46)</td></tr><tr><td rowspan=\"8\">WRN</td><td>yes</td><td>yes</td><td>no</td><td>91.44 (89.30)</td><td>71.67 (67.42)</td></tr><tr><td>yes</td><td>yes</td><td>light</td><td>95.01 (93.48)</td><td>77.58 (74.23)</td></tr><tr><td>yes</td><td>yes</td><td>heavier</td><td>95.60 (94.38)</td><td>76.96 (74.79)</td></tr><tr><td>no</td><td>yes</td><td>no</td><td>91.47 (89.38)</td><td>71.31 (66.85)</td></tr><tr><td>no</td><td>yes</td><td>light</td><td>94.76 (93.52)</td><td>77.42 (74.62)</td></tr><tr><td>no</td><td>yes</td><td>heavier</td><td>95.58 (94.52)</td><td>77.47 (73.96)</td></tr><tr><td>no</td><td>no</td><td>no</td><td>89.56 (85.45)</td><td>68.16 (59.90)</td></tr><tr><td>no</td><td>no</td><td>light</td><td>94.71 (93.69)</td><td>77.08 (75.27)</td></tr><tr><td></td><td>no</td><td>no</td><td>heavier</td><td>95.47 (94.95)</td><td>77.30 (75.69)</td></tr></table>",
+        "bbox": [
+            220,
+            340,
+            779,
+            609
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "An important observation from Table 2 is that the interaction of weight decay and dropout is not always consistent, since in some cases better results are obtained with both explicit regularizers active and in other cases, only dropout achieves better generalization. However, the effect of data augmentation seems to be clearer: just some light augmentation achieves much better results than training only with the original data set and performing heavier augmentation almost always further improves the test accuracy, without the need for explicit regularization. ",
+        "bbox": [
+            174,
+            631,
+            825,
+            715
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Not surprisingly, batch normalization also contributes to improve the generalization of All-CNN and it seems to combine well with data augmentation. On the contrary, when combined with explicit regularization the results are interestingly not consistent in the case of All-CNN: it seems to improve the generalization of the model trained with both weight decay and dropout, but it drastically reduces the performance with only dropout, in the case of CIFAR-10 and CIFAR-100 without augmentation. A probable explanation is, again, that the regularization hyperparameteres would need to be readjusted with a change of the architecture. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            819
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, it seems that the gap between the performance of the models trained with and without batch normalization is smaller when they are trained without explicit regularization and when they include heavier data augmentation. This can be observed in both Table 2 and Table 3, which contains the results of the models trained with fewer examples. It is important to note as well the benefits of batch normalization for obtaining better results when training with fewer examples. However, it is surprising that there is only a small drop in the performance of WRN— $9 5 . 4 7 \\ \\%$ to $9 4 . 9 5 ~ \\%$ without regularization— from removing the batch normalization layers of the residual blocks, given ",
+        "bbox": [
+            174,
+            827,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "that they were identified as key components for training deep residual networks (He et al., 2016;   \nZagoruyko & Komodakis, 2016). ",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/c153e8d3b62d2d6a1eac1239ca29184aea5acc9f71142d99fa6721335d273126.jpg",
+        "table_caption": [
+            "Table 3: Test accuracy when training with only $80 \\%$ , $50 \\%$ , $10 \\%$ and $1 \\%$ of the available training examples. Results within brackets show the performance of the models without batch normalization "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>Pct. Data</td><td>Explicit Reg.</td><td>Aug. scheme</td><td>Test CIFAR-10 89.41 (86.61)</td><td>Test CIFAR-100</td></tr><tr><td rowspan=\"18\"></td><td rowspan=\"4\">80 %</td><td>yes yes</td><td>no light</td><td>92.20 (91.25)</td><td>63.93 (52.51) 67.63 (63.24)</td></tr><tr><td>yes</td><td>heavier</td><td>92.83 (91.42)</td><td>68.01 (65.89)</td></tr><tr><td>no</td><td>no</td><td>83.04 (75.00)</td><td>55.78 (35.95)</td></tr><tr><td>no</td><td>light</td><td>92.25 (88.75)</td><td>69.05 (56.81)</td></tr><tr><td rowspan=\"6\"></td><td>no</td><td>heavier</td><td>92.80 (90.55)</td><td>69.40 (63.57)</td></tr><tr><td>yes</td><td>no</td><td>85.88 (82.33)</td><td>58.24 (44.94)</td></tr><tr><td>yes</td><td>light</td><td>90.30 (87.37)</td><td>61.03 (54.68)</td></tr><tr><td></td><td>heavier</td><td>90.09 (88.94)</td><td>63.25 (57.91)</td></tr><tr><td>yes no</td><td>no</td><td>78.61 (69.46)</td><td>48.62 (31.81)</td></tr><tr><td>no</td><td>light</td><td>90.21 (84.38)</td><td>62.83 (47.84)</td></tr><tr><td rowspan=\"6\">All-CNN 10 %</td><td></td><td>heavier</td><td>90.76 (87.44)</td><td>64.41 (55.27)</td></tr><tr><td>no yes</td><td>no</td><td></td><td></td></tr><tr><td></td><td>light</td><td>67.19 (61.61)</td><td>33.77 (19.79) 38.51 (22.79)</td></tr><tr><td>yes</td><td>heavier</td><td>76.03 (69.18)</td><td>38.34 (26.29)</td></tr><tr><td>yes no</td><td></td><td>78.69 (64.14) 60.97 (41.07)</td><td>26.05 (17.55)</td></tr><tr><td>no</td><td>no light</td><td>78.29 (67.65)</td><td>37.84 (24.34)</td></tr><tr><td rowspan=\"6\">1%</td><td>no</td><td>heavier</td><td>79.87 (70.64)</td><td>39.85 (26.31)</td></tr><tr><td>yes</td><td>no</td><td></td><td>9.16 (3.60)</td></tr><tr><td></td><td>light</td><td>27.53 (29.90) 37.18 (26.85)</td><td>9.64 (3.65)</td></tr><tr><td>yes yes</td><td>heavier</td><td>42.73 (26.87)</td><td>9.14 (2.52)</td></tr><tr><td>no</td><td>no</td><td>38.89 (35.68)</td><td>9.50 (5.51)</td></tr><tr><td>no</td><td>light</td><td>44.35 (29.29)</td><td>9.87 (5.36)</td></tr><tr><td rowspan=\"10\">80% 50%</td><td></td><td>heavier</td><td>47.60 (33.72)</td><td>11.45 (3.57)</td></tr><tr><td>yes</td><td>no</td><td>90.27</td><td>70.41</td></tr><tr><td>yes</td><td>light</td><td>94.07</td><td>75.66</td></tr><tr><td>yes</td><td>heavier</td><td>94.57</td><td>75.51</td></tr><tr><td>no</td><td>no</td><td>88.98</td><td>66.10</td></tr><tr><td></td><td>no light</td><td>93.97</td><td>75.07</td></tr><tr><td></td><td>no heavier</td><td>94.84</td><td>75.38</td></tr><tr><td></td><td>yes no</td><td>86.96</td><td>63.60</td></tr><tr><td>yes</td><td>light</td><td>92.65</td><td>70.83</td></tr><tr><td>yes no</td><td>heavier</td><td>92.86</td><td>70.33</td></tr><tr><td rowspan=\"6\">WRN</td><td></td><td>no</td><td>85.56</td><td>60.64</td></tr><tr><td>no</td><td></td><td></td><td>69.97</td></tr><tr><td>no</td><td>light</td><td>91.87</td><td>70.72</td></tr><tr><td></td><td>heavier</td><td>92.77</td><td></td></tr><tr><td>yes</td><td>no</td><td>70.73</td><td>34.11</td></tr><tr><td>yes</td><td>light</td><td>76.00</td><td>36.65</td></tr><tr><td rowspan=\"6\">10 %</td><td>yes</td><td>heavier</td><td>78.10</td><td>38.93</td></tr><tr><td>no</td><td>no</td><td>60.39</td><td>23.65</td></tr><tr><td>no</td><td>light</td><td>79.19</td><td>39.24</td></tr><tr><td>no</td><td>heavier</td><td>80.29</td><td>41.44</td></tr><tr><td>yes</td><td></td><td>33.45</td><td>7.47</td></tr><tr><td></td><td>no</td><td>34.13</td><td>7.50</td></tr><tr><td rowspan=\"5\">1%</td><td>yes</td><td>light</td><td></td><td></td></tr><tr><td>yes</td><td>heavier</td><td>41.02</td><td>8.37</td></tr><tr><td>no</td><td>no</td><td>38.63</td><td>9.47</td></tr><tr><td>no</td><td>light</td><td>43.84</td><td>9.91</td></tr><tr><td>no</td><td>heavier</td><td>47.14</td><td>11.03</td></tr></table>",
+        "bbox": [
+            191,
+            185,
+            810,
+            880
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The results in Table 3 clearly support the conclusion presented in Section 2.3: data augmentation alone resists better the lack of training data compared to explicit regularizers. Already with $80 \\%$ and ",
+        "bbox": [
+            174,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/68addeb3be44f546b7f1e3931b821eb752e0ad52ed9a1556e8b081ce0bba0445.jpg",
+        "table_caption": [
+            "Table 4: Test accuracy of the shallower and deeper versions of the All-CNN network on CIFAR-10 and CIFAR-100. Results in parentheses show the difference with respect to the original model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>Explicit Reg.</td><td>Aug. scheme</td><td>Test CIFAR-10</td><td>Test CIFAR-100</td></tr><tr><td rowspan=\"5\">All-CNN shallower</td><td>yes</td><td>no</td><td>76.45 (-13.59)</td><td>51.31 (-9.23)</td></tr><tr><td>yes</td><td>light</td><td>82.02 (-11.24)</td><td>56.81 (-8.76)</td></tr><tr><td>yes</td><td>heavier</td><td>86.66 (-6.42)</td><td>58.64 (-9.98)</td></tr><tr><td>no</td><td>no</td><td>85.22 (+0.69)</td><td>58.95 (+0.96)</td></tr><tr><td>no</td><td>light</td><td>90.02 (-3.24)</td><td>65.51 (-3.75)</td></tr><tr><td rowspan=\"6\">All-CNN deeper</td><td>no</td><td>heavier</td><td>90.34 (-3.21)</td><td>65.87 (-5.38)</td></tr><tr><td>yes</td><td>no</td><td>86.26 (-3.78)</td><td>49.06 (-11.48)</td></tr><tr><td>yes</td><td>light</td><td>85.04 (-8.22)</td><td>52.03 (-13.54)</td></tr><tr><td>yes</td><td>heavier</td><td>88.46 (-4.62)</td><td>51.78 (-16.84)</td></tr><tr><td>no</td><td>no</td><td>83.30 (-1.23)</td><td>54.22 (-3.77)</td></tr><tr><td>no no</td><td>light heavier</td><td>93.46 (+0.20) 94.19 (+0.64)</td><td>72.16 (+2.90) 73.30 (+2.35)</td></tr></table>",
+        "bbox": [
+            194,
+            145,
+            805,
+            332
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$50 \\%$ of the data better results are obtained in some cases, but the differences become much bigger when training with only $10 \\%$ and $1 \\%$ of the available data. It seems that explicit regularization prevents the model from both fitting the data and generalizing well, whereas data augmentation provides useful transformed examples. Interestingly, with only $1 \\%$ of the data, even without data augmentation the models without explicit regularization perform better. ",
+        "bbox": [
+            174,
+            359,
+            825,
+            431
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The same effect can be observed in Table 4, where both the shallower and deeper versions of AllCNN perform much worse when trained with explicit regularization, even when trained without data augmentation. This is another piece of evidence that explicit regularization needs to be used very carefully, it requires a proper tuning of the hyperparameters and is not always benefitial. ",
+        "bbox": [
+            174,
+            436,
+            825,
+            493
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B NORM OF THE WEIGHT MATRIX",
+        "text_level": 1,
+        "bbox": [
+            176,
+            513,
+            470,
+            530
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/0634c94c05211770e1d8b32f96639e213238e498443b7ef8a7ef3bf509e319ce.jpg",
+        "table_caption": [
+            "Table 5: Frobenius norm of the weight matrices learned by the networks All-CNN and WRN on CIFAR-10 and CIFAR-100, trained with and without explicit regularizers and the different augmentation schemes. Norms within brackets correspond to the models without batch normalization "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>WD</td><td>Dropout</td><td>Aug. scheme</td><td>Norm CIFAR-10</td><td>Norm( CIFAR-100</td></tr><tr><td rowspan=\"9\">All-CNN</td><td>yes</td><td>yes</td><td>no</td><td>48.7 (64.9)</td><td>76.5 (97.9)</td></tr><tr><td>yes</td><td>yes</td><td>light</td><td>52.7 (63.2)</td><td>77.6 (86.8)</td></tr><tr><td>yes</td><td>yes</td><td>heavier</td><td>57.6 (62.8)</td><td>78.1 (83.1)</td></tr><tr><td>no</td><td>yes</td><td>no</td><td>52.4 (70.5)</td><td>79.7 (103.3)</td></tr><tr><td>no</td><td>yes</td><td>light</td><td>57.0 (67.9)</td><td>83.6 (93.0)</td></tr><tr><td>no</td><td>yes</td><td>heavier</td><td>62.8 (67.5)</td><td>84.0 (88.0)</td></tr><tr><td>no</td><td>no</td><td>no</td><td>37.3 (63.7)</td><td>47.6 (102.7)</td></tr><tr><td>no</td><td>no</td><td>light</td><td>47.0 (69.5)</td><td>80.0 (108.9)</td></tr><tr><td>no</td><td>no</td><td>heavier</td><td>62.0 (71.7)</td><td>91.7 (91.7)</td></tr><tr><td rowspan=\"8\">WRN</td><td>yes</td><td>yes</td><td>no</td><td>101.4 (122.6)</td><td>134.8 (126.5)</td></tr><tr><td>yes</td><td>yes</td><td>light</td><td>106.1 (123.9)</td><td>140.8 (129.3)</td></tr><tr><td>yes</td><td>yes</td><td>heavier</td><td>119.3 (125.3)</td><td>164.2 (132.5)</td></tr><tr><td>no</td><td>yes</td><td>no</td><td>153.3 (122.5)</td><td>185.1 (126.5)</td></tr><tr><td>no</td><td>yes</td><td>light</td><td>160.6 (123.9)</td><td>199.0 (129.4)</td></tr><tr><td>no</td><td>yes</td><td>heavier</td><td>175.1 (125.2)</td><td>225.4 (132.5)</td></tr><tr><td>no</td><td>no</td><td>no</td><td>139.0 (120.4)</td><td>157.9 (122.0)</td></tr><tr><td>no</td><td>no</td><td>light</td><td>153.6 (123.2)</td><td>187.0 (127.2)</td></tr><tr><td></td><td>no</td><td>no</td><td>heavier</td><td>170.4 (125.4)</td><td>217.6 (132.9)</td></tr></table>",
+        "bbox": [
+            209,
+            604,
+            790,
+            875
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "One of the simplest way of getting a rough idea of the complexity of the learned models is computing the norm of the weight matrix. Table 5 shows the Frobenius norm of the weight matrices of the models trained with different levels of explicit regularization and data augmentation. The clearest conclusion is that heavier data augmentation seems to yield solutions with larger norm. This is always true except in some All-CNN models trained without batch normalization. Another observation is that, as expected, weight decay constrains the norm of the learned function. Besides, the models trained without batch normalization exhibit smaller differences between different levels of regularization and augmentation and, in the case of All-CNN, less consistency. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/11a20af136de2fa6f8b40d02330a37ec03853ae0a3a9226d6c9a3a01a7269ad7.jpg",
+        "table_caption": [
+            "Table 6: Frobenius norm of the weight matrices learned by the shallower and deeper versions of the All-CNN network on CIFAR-10 and CIFAR-100. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Network</td><td>Explicit Reg.</td><td>Aug.scheme</td><td>Norm CIFAR-10</td><td>Norm CIFAR-100</td></tr><tr><td rowspan=\"6\">All-CNN shallower</td><td>yes</td><td>no</td><td>47.9</td><td>68.9</td></tr><tr><td>yes</td><td>light</td><td>49.7</td><td>67.1</td></tr><tr><td>yes</td><td>heavier</td><td>51.9</td><td>66.2</td></tr><tr><td>no</td><td>no</td><td>34.8</td><td>64.7</td></tr><tr><td>no</td><td>light</td><td>45.6</td><td>68.8</td></tr><tr><td>no</td><td>heavier</td><td>53.1</td><td>68.3</td></tr><tr><td rowspan=\"6\">All-CNN deeper</td><td>yes</td><td>no</td><td>62.3</td><td>92.1</td></tr><tr><td>yes</td><td>light</td><td>66.5</td><td>95.7</td></tr><tr><td>yes</td><td>heavier</td><td>71.5</td><td>96.9</td></tr><tr><td>no</td><td>no</td><td>45.4</td><td>53.4</td></tr><tr><td>no</td><td>light</td><td>57.3</td><td>77.3</td></tr><tr><td>no</td><td>heavier</td><td>70.7</td><td>97.5</td></tr></table>",
+        "bbox": [
+            183,
+            145,
+            813,
+            332
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            362,
+            825,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "One of the relevant results presented in this paper is the poor performance of the regularized models on the shallower and deeper versions of All-CNN, compared to the models without explicit regularization (see Table 4). One hypothesis is that the amount of regularization is not properly adjusted through the hyperparameters. This could be reflected in the norm of the learned weights, shown in Table 6. However, the norm alone does not seem to fully explain the large performance differences between the different models. Finding the exact reasons why the regularized models not able to generalize well might require a much thourough analysis and we leave it as future work. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 14
+    }
+]

parse/train/HygtHnR5tQ/HygtHnR5tQ.md

@@ -0,0 +1,457 @@
+# GENERATIVE ADVERSARIAL NETWORKS FOR EXTREME LEARNED IMAGE COMPRESSION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We propose a framework for extreme learned image compression based on Generative Adversarial Networks (GANs), obtaining visually pleasing images at significantly lower bitrates than previous methods. This is made possible through our GAN formulation of learned compression combined with a generator/decoder which operates on the full-resolution image and is trained in combination with a multi-scale discriminator. Additionally, if a semantic label map of the original image is available, our method can fully synthesize unimportant regions in the decoded image such as streets and trees from the label map, therefore only requiring the storage of the preserved region and the semantic label map. A user study confirms that for low bitrates, our approach is preferred to state-of-the-art methods, even when they use more than double the bits.
+
+Ours 1567 Bytes
+
+![](images/5173aa3f65ede063b4a256c582b6c785e4544ba0fff9e06d377b3308a497b717.jpg)  
+BPG 3573 Bytes $12 8 \%$ larger JPEG 13959B $+ 7 9 0 \%$ WebP 9437B $+ 5 0 2 \%$   
+Figure 1: Visual comparison of our result to that obtained by other codecs. Note that even when using more than twice the number of bytes, all other codecs are outperformed by our method visually.
+
+# 1 INTRODUCTION
+
+Image compression systems based on deep neural networks (DNNs), or deep compression systems for short, have become an active area of research recently. These systems (e.g. (Theis et al., 2017; Balle et al., 2016b; Rippel & Bourdev, 2017; Ball ´ e et al., 2018; Mentzer et al., 2018)) are often com- ´ petitive with modern engineered codecs such as WebP (WebP), JPEG2000 (Taubman & Marcellin, 2001) and even BPG(Bellard) (the state-of-the-art engineered codec). Besides achieving competitive compression rates on natural images, they can be easily adapted to specific target domains such as stereo or medical images, and promise efficient processing and indexing directly from compressed representations (Torfason et al., 2018). However, deep compression systems are typically optimized for traditional distortion metrics such as peak signal-to-noise ratio (PSNR) or multi-scale structural similarity (MS-SSIM) (Wang et al., 2003). For very low bitrates (below 0.1 bits per pixel (bpp)), where preserving the full image content becomes impossible, these distortion metrics lose significance as they favor pixel-wise preservation of local (high-entropy) structure over preserving texture and global structure. To further advance deep image compression it is therefore of great importance to develop new training objectives beyond PSNR and MS-SSIM. A promising candidate towards this goal are adversarial losses (Goodfellow et al., 2014) which were shown recently to capture global semantic information and local texture, yielding powerful generators that produce visually appealing high-resolution images from semantic label maps (Isola et al., 2017; Wang et al., 2018).
+
+In this paper, we propose and study a generative adversarial network (GAN)-based framework for extreme image compression, targeting bitrates below 0.1 bpp. We rely on a principled GAN formulation for deep image compression that allows for different degrees of content generation. In contrast to prior works on deep image compression which applied adversarial losses to image patches for artifact suppression (Rippel & Bourdev, 2017; Galteri et al., 2017), generation of texture details (Ledig et al., 2017), or representation learning for thumbnail images (Santurkar et al., 2017), our generator/decoder operates on the full-resolution image and is trained with a multi-scale discriminator (Wang et al., 2018).
+
+We consider two modes of operation (corresponding to unconditional and conditional GANs (Goodfellow et al., 2014; Mirza & Osindero, 2014)), namely
+
+• generative compression $( G C )$ , preserving the overall image content while generating structure of different scales such as leaves of trees or windows in the facade of buildings, and • selective generative compression $( S C )$ , completely generating parts of the image from a semantic label map while preserving user-defined regions with a high degree of detail.
+
+We emphasize that GC does not require semantic label maps (neither for training, nor for deployment). A typical use case for GC are bandwidth constrained scenarios, where one wants to preserve the full image as well as possible, while falling back to synthesized content instead of blocky/blurry blobs for regions for which not sufficient bits are available to store the original pixels. SC could be applied in a video call scenario where one wants to fully preserve people in the video stream, but a visually pleasing synthesized background serves the purpose as well as the true background. In the GC operation mode the image is transformed into a bitstream and encoded using arithmetic coding. SC requires a semantic/instance label map of the original image which can be obtained using off-the-shelf semantic/instance segmentation networks, e.g., PSPNet (Zhao et al., 2017) and Mask R-CNN (He et al., 2017), and which is stored as a vector graphic. This amounts to a small, image dimension-independent overhead in terms of coding cost. On the other hand, the size of the compressed image is reduced proportionally to the area which is generated from the semantic label map, typically leading to a significant overall reduction in storage cost.
+
+For GC, a comprehensive user study shows that our compression system yields visually considerably more appealing results than BPG (Bellard) (the current state-of-the-art engineered compression algorithm) and the recently proposed autoencoder-based deep compression (AEDC) system (Mentzer et al., 2018). In particular, our GC models trained for compression of general natural images are preferred to BPG when BPG uses up to $9 5 \%$ and $12 4 \%$ more bits than those produced by our models on the Kodak (Kodak) and RAISE1K (Dang-Nguyen et al., 2015) data set, respectively. When constraining the target domain to the street scene images of the Cityscapes data set (Cordts et al., 2016), the reconstructions of our GC models are preferred to BPG even when the latter uses up to $181 \%$ more bits. To the best of our knowledge, these are the first results showing that a deep compression method outperforms BPG on the Kodak data set in a user study—and by large margins. In the SC operation mode, our system seamlessly combines preserved image content with synthesized content, even for regions that cross multiple object boundaries, while faithfully preserving the image semantics. By partially generating image content we achieve bitrate reductions of over $50 \%$ without notably degrading image quality.
+
+# 2 RELATED WORK
+
+Deep image compression has recently emerged as an active area of research. The most popular DNN architectures for this task are to date auto-encoders (Theis et al., 2017; Balle et al., 2016b; Agustsson ´ et al., 2017; Li et al., 2017; Torfason et al., 2018; Minnen et al., 2018) and recurrent neural networks (RNNs) (Toderici et al., 2015; 2016). These DNNs transform the input image into a bit-stream, which is in turn losslessly compressed using entropy coding methods such as Huffman coding or arithmetic coding. To reduce coding rates, many deep compression systems rely on context models to capture the distribution of the bit stream (Balle et al., 2016b; Toderici et al., 2016; Li et al., 2017; ´ Rippel & Bourdev, 2017; Mentzer et al., 2018). Common loss functions to measure the distortion between the original and decompressed images are the mean-squared error (MSE) (Theis et al., 2017; Balle et al., 2016b; Agustsson et al., 2017; Li et al., 2017; Ball ´ e et al., 2018; Torfason et al., ´ 2018), or perceptual metrics such as MS-SSIM (Toderici et al., 2016; Rippel & Bourdev, 2017; Balle et al., 2018; Mentzer et al., 2018). Some authors rely on advanced techniques including multi- ´ scale decompositions (Rippel & Bourdev, 2017), progressive encoding/decoding strategies (Toderici et al., 2015; 2016), and generalized divisive normalization (GDN) layers (Balle et al., 2016b;a). ´
+
+Generative adversarial networks (GANs) (Goodfellow et al., 2014) have emerged as a popular technique for learning generative models for intractable distributions in an unsupervised manner. Despite stability issues (Salimans et al., 2016; Arjovsky & Bottou, 2017; Arjovsky et al., 2017; Mao et al., 2017), they were shown to be capable of generating more realistic and sharper images than prior approaches and to scale to resolutions of $1 0 2 4 \times 1 0 2 4 \mathrm { p x }$ (Zhang et al., 2017; Karras et al., 2017) for some datasets. Another direction that has shown great progress are conditional GANs (Goodfellow et al., 2014; Mirza & Osindero, 2014), obtaining impressive results for image-to-image translation (Isola et al., 2017; Wang et al., 2018; Zhu et al., 2017; Liu et al., 2017) on various datasets (e.g. maps to satellite images), reaching resolutions as high as $1 0 2 4 \times 2 0 4 8 \mathrm { p x }$ (Wang et al., 2018).
+
+Arguably the most closely related work to ours is (Rippel & Bourdev, 2017), which uses an adversarial loss term to train a deep compression system. However, this loss term is applied to small image patches and its purpose is to suppress artifacts rather than to generate image content. Furthermore, it uses a non-standard GAN formulation that does not (to the best of our knowledge) have an interpretation in terms of divergences between probability distributions, as in (Goodfellow et al., 2014; Nowozin et al., 2016). We refer to Sec. 6.1 and Appendix A for a more detailed discussion. Santurkar et al. (2017) use a GAN framework to learn a generative model over thumbnail images, which is then used as a decoder for thumbnail image compression. Other works use adversarial training for compression artifact removal (for engineered codecs) (Galteri et al., 2017) and single image super-resolution (Ledig et al., 2017). Finally, related to our SC mode, spatially allocating bitrate based on saliency of image content has a long history in the context of engineered compression algorithms, see, e.g.,, (Stella & Lisin, 2009; Guo & Zhang, 2010; Gupta et al., 2013).
+
+# 3 BACKGROUND
+
+Generative Adversarial Networks: Given a data set $\mathcal { X }$ , Generative Adversarial Networks (GANs) can learn to approximate its (unknown) distribution $p _ { { \pmb x } }$ through a generator $G ( z )$ that tries to map samples $_ z$ from a fixed prior distribution $p _ { z }$ to the distribution $p _ { \pmb { x } }$ . The generator $G$ is trained in parallel with a discriminator $D$ by searching (using stochastic gradient descent (SGD)) for a saddle point of a mini-max objective
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } ~ \mathbb { E } [ f ( D ( \pmb { x } ) ) ] + \mathbb { E } [ g ( D ( G ( z ) ) ) ] ,
+$$
+
+where $G$ and $D$ are DNNs and $f$ and $g$ are scalar functions. The original paper (Goodfellow et al., 2014) uses the “Vanilla GAN” objective with $f ( y ) = \log ( y )$ and $g ( y ) = \log ( 1 - y )$ . This corresponds to $G$ minimizing the Jensen-Shannon (JS) Divergence between the (empirical) distribution of $_ { \textbf { \em x } }$ and $G ( z )$ . The JS Divergence is a member of a more generic family of $f$ -divergences, and Nowozin et al. (2016) show that for suitable choices of $f$ and $g$ , all such divergences can be minimized with (1). In particular, if one uses $f ( y ) = ( y - 1 ) ^ { 2 }$ and $g ( y ) = y ^ { 2 }$ , one obtains the LeastSquares GAN (Mao et al., 2017) (which corresponds to the Pearson $\chi ^ { 2 }$ divergence), which we adopt in this paper. We refer to the divergence minimized over $G$ as
+
+$$
+\mathcal { L } _ { \mathrm { G A N } } : = \operatorname* { m a x } _ { D } ~ \mathbb { E } [ f ( D ( \pmb { x } ) ) ] + \mathbb { E } [ g ( D ( G ( z ) ) ) ] .
+$$
+
+Conditional Generative Adversarial Networks: For conditional GANs (cGANs) (Goodfellow et al., 2014; Mirza $\&$ Osindero, 2014), each data point $_ { \textbf { \em x } }$ is associated with additional information $\pmb { s }$ , where $( { \pmb x } , { \pmb s } )$ have an unknown joint distribution $p _ { { \pmb x } , { \pmb s } }$ . We now assume that $\pmb { s }$ is given and that we want to use the GAN to model the conditional distribution $p _ { { \pmb x } | { \pmb s } }$ . In this case, both the generator $G ( z , s )$ and discriminator $D ( z , s )$ have access to the side information $\pmb { s }$ , leading to the divergence
+
+$$
+\mathcal { L } _ { \mathrm { c G A N } } : = \operatorname* { m a x } _ { D } ~ \mathbb { E } [ f ( D ( \pmb { x } , \pmb { s } ) ) ] + \mathbb { E } [ g ( D ( G ( z , \pmb { s } ) , \pmb { s } ) ) ] .
+$$
+
+Deep Image Compression: To compress an image $\mathbf { \boldsymbol { x } } \in \mathcal { X }$ , we follow the formulation of (Agustsson et al., 2017; Mentzer et al., 2018) where one learns an encoder $E$ , a decoder $G$ , and a finite quantizer $q$ . The encoder $E$ maps the image to a latent feature map $\pmb { w }$ , whose values are then quantized to $L$ levels $\{ c _ { 1 } , \ldots , c _ { L } \} \subset \mathbf { \bar { \mathbb { R } } }$ to obtain a representation $\pmb { \hat { w } } = q ( E ( \pmb { x } ) )$ that can be encoded to a bitstream. The decoder then tries to recover the image by forming a reconstruction $\hat { \pmb x } = G ( \hat { \pmb w } )$ . To be able to backpropagate through the non-differentiable $q$ , one can use a differentiable relaxation of $q$ , as in (Mentzer et al., 2018).
+
+The average number of bits needed to encode $\hat { \pmb { w } }$ is measured by the entropy $H ( \hat { \pmb w } )$ , which can be modeled with a prior (Agustsson et al., 2017) or a conditional probability model (Mentzer et al., 2018). The trade-off between reconstruction quality and bitrate to be optimized is then
+
+$$
+\mathbb { E } [ d ( { \pmb x } , \hat { \pmb x } ) ] + \beta H ( \hat { \pmb w } ) .
+$$
+
+where $d$ is a loss that measures how perceptually similar $\hat { \pmb x }$ is to $_ { \textbf { \em x } }$ . Given a differentiable estimator of the entropy $H ( \hat { \pmb w } )$ , the weight $\beta$ controls the bitrate of the model (large $\beta$ pushes the bitrate down). However, since the number of dimensions $\mathrm { d i m } ( \hat { \pmb w } )$ and the number of levels $L$ are finite, the entropy is bounded by (see, e.g., (Cover $\&$ Thomas, 2012))
+
+$$
+H ( \hat { w } ) \leq \dim ( \hat { w } ) \log _ { 2 } ( L ) .
+$$
+
+It is therefore also valid to set $\beta = 0$ and control the maximum bitrate through the bound (5) (i.e., adjusting $L$ and/or $\mathrm { d i m } ( \hat { \pmb w } )$ through the architecture of $E$ ). While potentially leading to suboptimal bitrates, this avoids to model the entropy explicitly as a loss term.wˆ
+
+# 4 GANS FOR EXTREME IMAGE COMPRESSIONxˆ
+
+# 4.1 GENERATIVE COMPRESSION
+
+The proposed GAN framework for extreme image compression can be viewed as a combination of (conditional) GANs and learned compression. With an encoder $E$ and quantizer $q$ , we encode the image $_ { \textbf { \em x } }$ to a compressed representation $\hat { \pmb { w } } = q ( E ( \pmb { x } ) )$ . This representation is optionally concatenated with noise $\pmb { v }$ drawn from a fixed prior $p _ { v }$ , to form the latent vector $_ z$ . The decoder/generator $G$ then tries to generate an image ${ \hat { \pmb x } } = G ( { \pmb z } )$ that is consistent with the image distribution $p _ { \pmb { x } }$ while also recovering the specific encoded image $_ { \textbf { \em x } }$ to a certain degree (see inset Fig.). Using $z = [ \hat { \pmb w } , \pmb v ]$ , this can be expressed by our saddle-point objective for (unconditional) generative compression,
+
+![](images/2ab1882a97494ceb3a6c05df910eb556040a36b933d7f90bad634cb1b1fc3ea7.jpg)
+
+$$
+\operatorname* { m i n } _ { E , G } \operatorname* { m a x } _ { D } \mathbb { E } [ f ( D ( \pmb { x } ) ) ] + \mathbb { E } [ g ( D ( G ( z ) ) ] + \lambda \mathbb { E } [ d ( \pmb { x } , G ( z ) ) ] + \beta H ( \pmb { \hat { w } } ) ,
+$$
+
+where $\lambda > 0$ balances the distortion term against the GAN loss and entropy terms. Using this formulation, we need to encode a real image, ${ \hat { \pmb w } } = E ( \pmb { x } )$ , to be able to sample from $p _ { \hat { \mathbf { \alpha } } \hat { \mathbf { \beta } } }$ . However, this is not a limitation as our goal is to compress real images and not to generate completely new ones.
+
+Since the last two terms of (6) do not depend on the discriminator $D$ , they do not affect its optimization directly. This means that the discriminator still computes the same $f$ divergence $\mathcal { L } _ { \mathrm { G A N } }$ as in (2), so we can write (6) as
+
+$$
+\operatorname* { m i n } _ { E , G } \quad \mathcal { L } _ { \mathrm { G A N } } + \lambda \mathbb { E } [ d ( \pmb { x } , G ( z ) ) ] + \beta H ( \pmb { \hat { w } } ) .
+$$
+
+We note that equation (6) has completely different dynamics than a normal GAN, because the latent space $_ { z }$ contains $\hat { \pmb { w } }$ , which stores information about a real image $_ { \textbf { \em x } }$ . A crucial ingredient is the bitrate limitation on $H ( \hat { \pmb w } )$ . If we allow $\hat { \pmb { w } }$ to contain arbitrarily many bits by setting $\beta = 0$ and letting $L$ and $\mathrm { d i m } ( \hat { \pmb w } )$ be large enough, $E$ and $G$ could learn to near-losslessly recover $_ { \textbf { \em x } }$ from $G ( z ) \bar { = } \ G ( q ( E ( \bar { \bf x } ) ) )$ , such that the distortion term would vanish. In this case, the divergence between $p _ { { \pmb x } }$ and $p _ { G ( z ) }$ would also vanish and the GAN loss would have no effect.
+
+By constraining the entropy of $\hat { \pmb { w } }$ , $E$ and $G$ will never be able to make $d$ fully vanish. In this case, $E , G$ need to balance the GAN objective ${ \mathcal { L } } _ { \mathrm { G A N } }$ and the distortion term $\lambda \mathbb { E } [ d ( \dot { \pmb { x } } , G ( \pmb { z } ) ) ]$ , which leads to $G ( z )$ on one hand looking “realistic”, and on the other hand preserving the original image. For example, if there is a tree for which $E$ cannot afford to store the exact texture (and make $d$ small) $G$ can synthesize it to satisfy $\mathcal { L } _ { \mathrm { G A N } }$ , instead of showing a blurry green blob.
+
+In the extreme case where the bitrate becomes zero (i.e., $H ( \hat { \pmb w } )  0$ , e.g., by setting $\beta = \infty$ or $\mathrm { d i m } ( \hat { \pmb w } ) = 0 ,$ ), $\hat { \pmb { w } }$ becomes deterministic. In this setting, $_ z$ is random and independent of $_ { \textbf { \em x } }$ (through the $\textbf {  { v } }$ component) and the objective reduces to a standard GAN plus the distortion term, which acts as a regularizer.
+
+We refer to the setting in (6) as generative compression (GC), where $E , G$ balance reconstruction and generation automatically over the image. As for the conditional GANs described in Sec. 3, we can easily extend GC to a conditional case. Here, we also consider this setting, where the additional information $\pmb { s }$ for an image $_ { \textbf { \em x } }$ is a semantic label map of the scene, but with a twist: Instead of feeding the semantics to $E , G$ and $D$ , we only give them to the discriminator $D$ during training.1 This means that no semantics are needed to encode or decode images with the trained models (since $E , G$ do not depend on $\pmb { s }$ ). We refer to this setting as GC $( D ^ { + } )$ .
+
+# 4.2 SELECTIVE GENERATIVE COMPRESSION
+
+For GC and its conditional variant described in the previous section, $E , G$ automatically navigate the trade-off between generation and preservation over the entire image, without any guidance. Here, we consider a different setting, where we guide the network in terms of which regions should be preserved and which regions should be synthesized. We refer to this setting as selective generative compression (SC) (an overview of the network structure is given in Fig. 8 in Appendix C).
+
+For simplicity, we consider a binary setting, where we construct a single-channel binary heatmap $_ { m }$ of the same spatial dimensions as $\hat { \pmb { w } }$ . Regions of zeros correspond to regions that should be fully synthesized, whereas regions of ones correspond to regions that should be preserved. However, since our task is compression, we constrain the fully synthesized regions to have the same semantics $\pmb { s }$ as the original image $_ { \textbf { \em x } }$ . We assume the semantics $\pmb { s }$ are separately stored, and thus feed them through a feature extractor $F$ before feeding them to the generator $G$ . To guide the network with the semantics, we mask the (pixel-wise) distortion $d$ , such that it is only computed over the region to be preserved. Additionally, we zero out the compressed representation $\hat { \pmb w }$ in the regions that should be synthesized. Provided that the heatmap $_ { \mathbf { \nabla } } \mathbf { m }$ is also stored, we then only encode the entries of $\hat { \pmb { w } }$ corresponding to the preserved regions, greatly reducing the bitrate needed to store it.
+
+At bitrates where $\hat { \pmb { w } }$ is normally much larger than the storage cost for $\pmb { s }$ and $_ { m }$ (about 2kB per image when encoded as a vector graphic), this approach can result in large bitrate savings.
+
+We consider two different training modes: Random instance (RI) which randomly selects $2 5 \%$ of the instances in the semantic label map and preserves these, and random box (RB) which picks an image location uniformly at random and preserves a box of random dimensions. While the RI mode is appropriate for most use cases, the RB can create more challenging situations for the generator as it needs to integrate the preserved box seamlessly into the generated content.
+
+# 5 EXPERIMENTS
+
+5.1 ARCHITECTURE, LOSSES, AND HYPERPARAMETERS
+
+The architecture for our encoder $E$ and generator $G$ is based on the global generator network proposed in (Wang et al., 2018), which in turn is based on the architecture of (Johnson et al., 2016). We present details in Appendix C.
+
+For the entropy term $\beta H ( \hat { \pmb w } )$ , we adopt the simplified approach described in Sec. 3, where we set $\beta = 0$ , use $L = 5$ centers $\mathcal { C } = \{ - 2 , 1 , 0 , 1 , 2 \}$ , and control the bitrate through the upper bound $H ( \pmb { \hat { w } } ) \leq \dim ( \pmb { \hat { w } } ) \log _ { 2 } ( L )$ . For example, for GC, with $C = 2$ channels, we obtain 0.0181bpp.2 We note that this is an upper bound; the actual entropy of $H ( \hat { \pmb w } )$ is generally smaller, since the learned distribution will neither be uniform nor i.i.d, which would be required for the bound to hold with equality. When encoding the channels of $\hat { \pmb { w } }$ to a bit-stream, we use an arithmetic encoder where frequencies are stored for each channel separately and then encode them in a static (non-adaptive)
+
+![](images/565864cac4a50657bb0b2ac56f32614f4ee0b5fe46b2cff12deb8b5a5bc4b7d9.jpg)  
+Figure 2: Visual example of images produced by our GC network with $C = 4$ along with the corresponding results for BPG, and a baseline model with the same architecture $C = 4$ ) but trained for MSE only (MSE bl.), on Cityscapes. The reconstruction of our GC network is sharper and has more realistic texture than those of BPG and the MSE baseline, even though the latter two have higher PSNR (indicated in dB for each image) than our GC network. In particular, the MSE baseline produces blurry reconstructions even though it was trained on the Cityscapes data set, demonstrating that domain-specific training alone is not enough to obtain sharp reconstructions at low bitrates.
+
+manner, similar to Agustsson et al. (2017). In our experiments, this leads to $8 . 8 \%$ smaller bitrates compared to the upper bound.
+
+By using a context model and adaptive arithmetic encoding, we could reduce the bitrate further, either in a post processing step (as in (Rippel & Bourdev, 2017; Balle et al., 2016b)), or jointly ´ during training (as in (Mentzer et al., 2018; Minnen et al., 2018))—which led to $\approx 1 0 \%$ savings in these prior works.
+
+For the distortion term we adopt $d ( \pmb { x } , \hat { \pmb { x } } ) = \mathbf { M S E } ( \pmb { x } , \hat { \pmb { x } } )$ with coefficient $\lambda = 1 0$ . Furthermore, we adopt the feature matching and VGG perceptual losses, $\mathcal { L } _ { \mathrm { F M } }$ and ${ \mathcal { L } } _ { \mathrm { V G G } }$ , as proposed in (Wang et al., 2018) with the same weights, which improved the quality for images synthesized from semantic label maps. These losses can be viewed as a part of $d ( { \pmb x } , \hat { { \pmb x } } )$ . However, we do not mask them in SC, since they also help to stabilize the GAN in this operation mode (as in (Wang et al., 2018)). We refer to Appendix D for training details.
+
+# 5.2 EVALUATION
+
+Data sets: We train GC models (without semantic label maps) for compression of diverse natural images using $1 8 8 \mathrm { k }$ images from the Open Images data set (Krasin et al., 2017) and evaluate them on the widely used Kodak image compression data set (Kodak) as well as 20 randomly selected images from the RAISE1K data set (Dang-Nguyen et al., 2015). To investigate the benefits of having a somewhat constrained application domain and semantic information at training time, we also train GC models with semantic label maps on the Cityscapes data set (Cordts et al., 2016), using 20 randomly selected images from the validation set for evaluation. To evaluate the proposed SC method (which requires semantic label maps for training and deployment) we again rely on the Cityscapes data set. Cityscapes was previously used to generate images form semantic label maps using GANs (Isola et al., 2017; Zhu et al., 2017).
+
+Baselines: We compare our method to the HEVC-based image compression algorithm BPG (Bellard) (in the 4:2:2 chroma format) and to the AEDC network from (Mentzer et al., 2018). BPG is the current state-of-the-art engineered image compression codec and outperforms other recent codecs such as JPEG2000 and WebP on different data sets in terms of PSNR (see, e.g. (Balle et al., 2018)). ´ We train the AEDC network (with bottleneck depth $C = 4$ ) on Cityscapes exactly following the procedure in (Mentzer et al., 2018) except that we use early stopping to prevent overfitting (note that Cityscapes is much smaller than the ImageNet dataset used in (Mentzer et al., 2018)). The so-obtained model has a bitrate of 0.07 bpp and gets a slightly better MS-SSIM than BPG at the same bpp on the validation set. To investigate the effect of the GAN term in our total loss, we train a baseline model with an MSE loss only (with the same architecture as GC and the same training parameters, see Sec. D in the Appendix), referred to as “MSE baseline”.
+
+![](images/e57dca17551513a6348cc5177ed9b7c0a7a7d1daf648423dbe854a67c2606396.jpg)  
+Figure 3: Visual example of images from RAISE1k produced by our GC network with $C = 4$ along with the corresponding results for BPG.
+
+User study: In the extreme compression regime realized by our GC models, where texture and sometimes even more abstract image content is synthesized, common reconstruction quality measures such as PSNR and MS-SSIM arguably lose significance as they penalize changes in local structure rather than assessing preservation of the global image content (this also becomes apparent by comparing reconstructions produced by our GC model with those obtained by the MSE baseline and BPG, see Fig. 2). Indeed, measuring PSNR between synthesized and real texture patches essentially quantifies the variance of the texture rather than the visual quality of the synthesized texture.
+
+To quantitatively evaluate the perceptual quality of our GC models in comparison with BPG and AEDC (for Cityscapes) we therefore conduct a user study using Amazon Mechanical Turk (AMT).3 We consider two GC models with $C = 4 , 8$ trained on Open Images, three GC $( D ^ { + } )$ models with $C = 2 , 4 , 8$ trained on Cityscapes, and BPG at rates ranging from 0.045 to 0.12 bpp. Questionnaires are composed by combining the reconstructions produced by the selected GC model for all testing images with the corresponding reconstruction produced by the competing baseline model side-byside (presenting the reconstructions in random order). The original image is shown along with the reconstructions, and the pairwise comparisons are interleaved with 3 probing comparisons of an additional uncompressed image from the respective testing set with an obviously JPEG-compressed version of that image. 20 randomly selected unique users are asked to indicate their preference for each pair of reconstructions in the questionnaire, resulting in a total of 480 ratings per pairing of methods for Kodak, and 400 ratings for RAISE1K and Cityscapes. For each pairing of methods, we report the mean preference score as well as the standard error (SE) of the per-user mean preference percentages. Only users correctly identifying the original image in all probing comparisons are taken into account for the mean preference percentage computation. To facilitate comparisons for future works, we will release all images used in the user studies.
+
+Semantic quality of SC models: The issues with PSNR and MS-SSIM for evaluating the quality of generated content described in the previous paragraph become even more severe for SC models as a large fraction of the image content is generated from a semantic label map. Following image translation works Isola et al. (2017); Wang et al. (2018), we therefore measure the capacity of our SC models to preserve the image semantics in the synthesized regions and plausibly blend them with the preserved regions—the objective SC models are actually trained for. Specifically, we use PSPNet (Zhao et al., 2016) and compute the mean intersection-over-union (IoU) between the label map obtained for the decompressed validation images and the ground truth label map. For reference we also report this metric for baselines that do not use semantic label maps for training and/or deployment.
+
+# 6 RESULTS
+
+# 6.1 GENERATIVE COMPRESSION
+
+Fig. 4 shows the mean preference percentage obtained by our GC models compared to BPG at different rates, on the Kodak and the RAISE1K data set. In addition, we report the mean preference percentage for GC models compared to BPG and AEDC on Cityscapes. Example validation images for side-by-side comparison of our method with BPG for images from the Kodak, RAISE1K, and Cityscapes data set can be found in Figs. 1, 3, and 2, respectively. Furthermore, we perform extensive visual comparisons of all our methods and the baselines, presented in Appendix F.
+
+![](images/d0a2c7c8cc9ed8ee23a8276b6963b1a6e4b0b07a69a4c6303372b5b91ed9c2b8.jpg)  
+Figure 4: User study results evaluating our GC models on Kodak, RAISE1K (top) and Cityscapes GC C=4 preferred GC C=8 preferredGC C=4 preferred GC C=8 preferred(bottom). For Kodak and RAISE1K, we use GC models trained on Open Images, without any semantic label maps. For Cityscapes, we used GC $( D ^ { + } )$ , using semantic label maps only for $D$ and BPG 123% larger, andGC C=4 still preferredrger, andpreferred BPG 40% larger, andGC C=8 still preferredBPG 123% larger, andGC C=4 still preferredger, andreferred BPG 40% larger, andGC C=8 still preferredonly during training. The standard error is computed over per-user mean preference percentages. [bpp][bpp] 0.069AEDC[bpp][bpp] 0.069AEDCThe blue arrows visualize how many more bits BPG uses when $> 5 0 \%$ users still prefer our result.
+
+Our GC models with $C = 4$ are preferred to BPG even when images produced by BPG use $9 5 \%$ and $12 4 \%$ more bits than those produced by our models for Kodak and RAISE1K, respectively. Notably this is achieved even though there is a distribution shift between the training and testing set (recall 2ch 8chthat these GC models are trained on Open Images). The gains of domain-specificity and semantic BPG X 0.04974 0.059375 BPG X 0.079155 0.09909 0.12171BPG Preflabel maps (for training) becomes apparent from the results on Cityscapes: Our GC models with $C = 2$ CVPR X 0.069 CVPR X are preferred to BPG even when the latter uses $181 \%$ 069 more bits. For $C = 4$ the gains on CVPR Pref 47.8125% CVPR Pref 83%Cityscapes are comparable to those obtained for GC on RAISE1K. For all three data sets, BPG BPG STDDEV 0.0requires between 21 and $49 \%$ 0.056529 BPG STDDEV 0.029273  more bits than our GC models with $C = 8$ 7 .
+
+ours 0.01767578125 ours Discussion: The GC models produce images with much finer detail than BPG, which suffers from smoothed patches and blocking artifacts. In particular, the GC models convincingly reconstruct BPG X 0.079155 0.09909 0.12171 BPG X 0.079155 0.09909 0.12171BPG X 0.059375 0.079155 0.09909texture in natural objects such as trees, water, and sky, and is most challenged with scenes involving 52.6667% 38%   BPG Pref 63.125% 53.5714% 44.7222% 52.6667% 38%   BPG Pref 63.125% 53.5714% 44.7222%   BPG Pref 62.0588% 50.3571% 32.6667%humans. AEDC and the MSE baseline both produce blurry images.
+
+47.8125% CVPR Pref 83%47.8125% CVPR Pref 83%CVPR Pref 79.3333%We see that the gains of our models are maximal at extreme bitrates, with BPG needing $9 5 \mathrm { - } 1 8 1 \%$   
+0.071782 0.0565290.071782 0.056529BPG STDDEV more bits for the $C = 2 , 4$ STDDEV 0.029273 0.038276 0.024047TDDEV 0.029273 0.038276 0.0240470.036931models on the three datasets. For $C = 8$ gains are smaller but still   
+0.072210.07221 ours 0.035592651very large (BPG needing $2 1 \mathrm { - } 4 9 \%$ 0.070944976810.07094497681 more bits). This is expected, since as the bitrate increases the   
+.81401988950276 gain 1.3967162222827481401988950276 gain 1.39671622228274gain 2.22391412140534classical compression measures (PSNR/MS-SSIM) become more meaningful—and our system does   
+4ch4chnot employ the full complexity of current state-of-the-art systems, as discussed next.
+
+0.059375 0.079155 0.0990962.0588% 50.3571% 32.6667%0.059375 0.079155 0.0990962.0588% 50.3571% 32.6667%State-of-the-art on Kodak: We give an overview of relevant recent learned compression methods 0.0690.069and their differences to our GC method and BPG in Table 1 in the Appendix. Rippel & Bourdev 79.3333%79.3333%(2017) also used GANs (albeit a different formulation) and were state-of-the-art in MS-SSIM in 0.07221 0.072212017, while the concurrent work of Minnen et al. (2018) is the current state-of-the-art in image 0.035592651370.03559265137compression in terms of classical metrics (PSNR and MS-SSIM) when measured on the Kodak .2239141214053422391412140534dataset (Kodak). Notably, all methods except ours (BPG, Rippel et al., and Minnen et al.) employ adaptive arithmetic coding using context models for improved compression performance. Such models could also be implemented for our system, and have led to additional savings of $10 \%$ in Mentzer et al. (2018). Since Rippel et al. and Minnen et al. have only released a selection of their decoded images (for 3 and 4, respectively, out of the 24 Kodak images), and at significantly higher bitrates, a comparison with a user study is not meaningful. Instead, we try to qualitatively put our results into context with theirs.
+
+In Figs. 12–14 in the Appendix, we compare qualitatively to Rippel & Bourdev (2017). We can observe that even though Rippel & Bourdev (2017) use $2 9 - 1 7 9 \%$ more bits, our models produce images of comparable or better quality.
+
+![](images/ddefc9cd4593952ee7e8d6d911c7970a6ba436a472c8707c3680ec2e08ce9de0.jpg)
+
+In Figs. 15–18, we show a qualitative comparison of our results to the images provided by the concurrent work of Minnen et al. (2018), as well as to BPG (Bellard) on those images. First, we see that BPG is still visually competitive with the current state-of-the-art, which is consistent with moderate $8 . 4 1 \%$ bitrate savings being reported by Minnen et al. (2018) in terms of PSNR. Second, even though we use much fewer bits compared to the example images available from Minnen et al. (2018), for some of them (Figs. 15 and 16) our method can still produce images of comparable visual quality.
+
+Given the dramatic bitrate savings we achieve according to the user study (BPG needing $2 1 \mathrm { - } 1 8 1 \%$ more bits), and the competitiveness of BPG to the most recent state-of-the-art (Minnen et al., 2018), we conclude that our proposed system presents a significant step forward for visually pleasing compression at extreme bitrates.
+
+Sampling the compressed representations: In Fig. 5 we explore the representation learned by our GC models (with $C = 4$ ), by sampling the (discrete) latent space of $\hat { \pmb { w } }$ . When we sample uniformly, and decode with our GC model into images, we obtain a “soup of image patches” which reflects the domain the models were trained on (e.g. street sign and building patches on Cityscapes). Note that we should not expect these outputs to look like normal images, since nothing forces the encoder output $\hat { \pmb w }$ to be uniformly distributed over the discrete latent space.
+
+However, given the low dimensionality of $\hat { \pmb { w } }$ $3 2 \times 6 4 \times 4$ for $5 1 2 \times 1 0 2 4 \mathrm { p x }$ Cityscape images), it would be interesting to try to learn the true distribution. To this end, we perform a simple experiment and train an improved Wasserstein GAN (WGAN-GP) (Gulrajani et al., 2017) on $\hat { \pmb { w } }$ extracted from Cityscapes, using default parameters and a ResNet architecture.4 By feeding our GC model with samples from the WGAN-GP generator, we easily obtain a powerful generative model, which generates sharp $1 0 2 4 \times 5 1 2 \mathrm { p x }$ images from scratch. We think this could be a promising direction for building high-resolution generative models. In Figs. 19–21 in the Appendix, we show more samples, and samples obtained by feeding the MSE baseline with uniform and learned code samples. The latter yields noisier “patch soups” and much blurrier image samples than our GC network.
+
+![](images/b76dbd8e2726ab9f6204105553d4e5d3c58f1806ec36f87729b1abd197327192.jpg)  
+Figure 6: Mean IoU as a function of bpp on the Cityscapes validation set for our GC and SC networks, and for the MSE baseline. We show both SC modes: RI (inst.), RB (box). $D ^ { + }$ annotates models where instance semantic label maps are fed to the discriminator (only during training); $E D G ^ { + }$ indicates that semantic label maps are used both for training and deployment. The pix2pixHD baseline (Wang et al., 2018) was trained from scratch for 50 epochs, using the same downsampled $1 0 2 4 \times 5 1 2 \mathrm { p x }$ training images as for our method.
+
+Fig. 6 shows the mean IoU on the Cityscapes validation set as a function of bpp for SC networks with $C = 2 , 4 , 8$ , along with the values obtained for the baselines. Additionally, we plot mean IoU for GC with semantic label maps fed to the discriminator $( D ^ { + } )$ , and the MSE baseline.
+
+![](images/9519fa3fdfa6fa120c6920b47a34bd5e30035c1b01013951ddb822f22cea35a7.jpg)  
+Figure 7: Synthesizing different classes using our SC network with $C = 8$ . In each image except for no synthesis, we additionally synthesize the classes vegetation, sky, sidewalk, ego vehicle, wall. The heatmaps in the lower left corners show the synthesized parts in gray. We show the bpp of each image as well as the relative savings due to the selective generation.
+
+In Fig. 7 we present example Cityscapes validation images produced by the SC network trained in the RI mode with $C = 8$ , where different semantic classes are preserved. More visual results for the SC networks trained on Cityscapes can be found in Appendix F.7, including results obtained for the RB operation mode and by using semantic label maps estimated from the input image via PSPNet (Zhao et al., 2017).
+
+Discussion: The quantitative evaluation of the semantic preservation capacity (Fig. 6) reveals that the SC networks preserve the semantics somewhat better than pix2pixHD, indicating that the SC networks faithfully generate texture from the label maps and plausibly combine generated with preserved image content. The mIoU of BPG, AEDC, and the MSE baseline is considerably lower than that obtained by our SC and GC models, which can arguably be attributed to blurring and blocking artifacts. However, it is not surprising as these baseline methods do not use label maps during training and prediction.
+
+In the SC operation mode, our networks manage to seamlessly merge preserved and generated image content both when preserving object instances and boxes crossing object boundaries (see Appendix F.7). Further, our networks lead to reductions in bpp of $50 \%$ and more compared to the same networks without synthesis, while leaving the visual quality essentially unimpaired, when objects with repetitive structure are synthesized (such as trees, streets, and sky). In some cases, the visual quality is even better than that of BPG at the same bitrate. The visual quality of more complex synthesized objects (e.g. buildings, people) is worse. However, this is a limitation of current GAN technology rather than our approach. As the visual quality of GANs improves further, SC networks will as well. Notably, the SC networks can generate entire images from the semantic label map only.
+
+Finally, the semantic label map, which requires 0.036 bpp on average for the downscaled $1 0 2 4 \times$ 512px Cityscapes images, represents a relatively large overhead compared to the storage cost of the preserved image parts. This cost vanishes as the image size increases, since the semantic mask can be stored as an image dimension-independent vector graphic.
+
+# 7 CONCLUSION
+
+We proposed and evaluated a GAN-based framework for learned compression that significantly outperforms prior works for low bitrates in terms of visual quality, for compression of natural images. Furthermore, we demonstrated that constraining the application domain to street scene images leads to additional storage savings, and we explored combining synthesized with preserved image content with the potential to achieve even larger savings. Interesting directions for future work are to develop a mechanism for controlling spatial allocation of bits for GC (e.g. to achieve better preservation of faces; possibly using semantic label maps), and to combine SC with saliency information to determine what regions to preserve. In addition, the sampling experiments presented in Sec. 6.1 indicate that combining our GC compression approach with GANs to (unconditionally) generate compressed representations is a promising avenue to learn high-resolution generative models.
+
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+
+A COMPARISON WITH STATE-OF-THE-ART   
+Table 1: Overview of differences between (Minnen et al., 2018) (s.o.t.a. in MS-SSIM and PSNR), to BPG (previous s.o.t.a. in PSNR) and (Rippel & Bourdev, 2017) (s.o.t.a. in MS-SSIM in 2017, also used GANs).   
+
+<table><tr><td></td><td>BPG</td><td>Rippel et al. (2017)</td><td>Minnen et al. (2018)</td><td>Ours (GC)</td></tr><tr><td>Learned</td><td>No</td><td>Yes</td><td>Yes</td><td>Yes</td></tr><tr><td>Arithmetic encoding</td><td>Adaptive</td><td>Adaptive</td><td>Adaptive</td><td>Static</td></tr><tr><td>Context model</td><td>CABAC</td><td>Autoregressive</td><td>Autoregressive</td><td>None</td></tr><tr><td>Visualized bitrates [bpp]5</td><td>al16</td><td>0.08-</td><td>0.12-</td><td>0.033-0.066</td></tr><tr><td>GAN</td><td>No</td><td>Non-standard</td><td>No</td><td>f-div. based</td></tr><tr><td>S.o.t.a. in MS-SSIM</td><td>No</td><td>No</td><td>Yes</td><td>No</td></tr><tr><td>S.o.t.a. in PSNR</td><td>No</td><td>No</td><td>Yes</td><td>No</td></tr><tr><td>Savings to BPG in PSNR</td><td></td><td></td><td>8.41%</td><td></td></tr><tr><td>Savings to BPG in User Study</td><td></td><td></td><td></td><td>17.2-48.7%</td></tr></table>
+
+# B COMPRESSION DETAILS
+
+When encoding the channels of $\hat { \pmb { w } }$ to a bit-stream, we use an arithmetic encoder where frequencies are stored for each channel separately and then encode them in a static (non-adaptive) manner, similar to Agustsson et al. (2017). In our experiments, this leads to $8 . 8 \%$ smaller bitrates compared to the upper bound.
+
+We compress the semantic label map for SC by quantizing the coordinates in the vector graphic to the image grid and encoding coordinates relative to preceding coordinates when traversing object boundaries (rather than relative to the image frame). The so-obtained bitstream is then compressed using arithmetic coding.
+
+To ensure fair comparison, we do not count header sizes for any of the baseline methods throughout.
+
+# C ARCHITECTURE DETAILS
+
+For the GC, the encoder $E$ convolutionally processes the image $_ { \textbf { \em x } }$ and optionally the label map $\pmb { s }$ , with spatial dimension $W \times H$ , into a feature map of size $W / 1 6 \times H / 1 6 \times 9 6 0$ (with 6 layers, of which four have 2-strided convolutions), which is then projected down to $C$ channels (where $C \in \{ 2 , 4 , 8 \}$ is much smaller than 960). This results in a feature map $\pmb { w }$ of dimension $W \big / 1 6 \times H \big / 1 6 \times C$ , which is quantized over $L$ centers to obtain the discrete $\hat { \pmb { w } }$ . The generator $G$ projects $\hat { \pmb { w } }$ up to 960 channels, processes these with 9 residual units (He et al., 2016) at dimension $W / 1 6 \times H / 1 6 \times 9 6 0$ , and then mirrors $E$ by convolutionally processing the features back to spatial dimensions $W \times H$ (with transposed convolutions instead of strided ones).
+
+Similar to $E$ , the feature extractor $F$ for SC processes the semantic map $\pmb { s }$ down to the spatial dimension of $\hat { \pmb w }$ , which is then concatenated to $\hat { \pmb { w } }$ for generation. In this case, we consider slightly higher bitrates and downscale by $8 \times$ instead of $1 6 \times$ in the encoder $E$ , such that $\mathrm { d i m } ( \hat { \pmb w } ) = \bar { W } / 8 \stackrel {  } { \times }$ $H / _ { 8 } \times C$ . The generator then first processes $\hat { \pmb { w } }$ down to $W / 1 6 \times H / 1 6 \times 9 6 0$ and then proceeds as for GC.
+
+For both GC and SC, we use the multi-scale architecture of (Wang et al., 2018) for the discriminator $D$ , which measures the divergence between $p _ { \pmb { x } }$ and $p _ { G ( z ) }$ both locally and globally.
+
+We adopt the notation from (Wang et al., 2018) to describe our encoder and generator/decoder architectures and additionally use $\mathrm { \Delta q }$ to denote the quantization layer (see Sec. 3 for details). The output of $\mathrm { \Delta q }$ is encoded and stored.
+
+# • Encoders SC:
+
+– Semantic label map encoder: c7s1-60,d120,d240,d480,d960  
+– Image encoder: c7s1-60,d120,d240,d480,c3s1-C,q,c3s1-480,d960
+
+The outputs of the semantic label map encoder and the image encoder are concatenated and fed to the generator/decoder.
+
+• Generator/decoder: c3s1-960,R960,R960,R960,R960,R960,R960,R960, R960,R960,u480,u240,u120,u60,c7s1-3
+
+![](images/9ce5cc68e7a015d39c955a9a98a717ad55ef0993ba611b3f31ea977bfcb9f19f.jpg)  
+Figure 8: Structure of the proposed SC network. $E$ is the encoder for the image $_ { \textbf { \em x } }$ and the semantic label map s. $q$ quantizes the latent code $\pmb { w }$ to $\hat { \pmb w }$ . The subsampled heatmap multiplies $\hat { \pmb { w } }$ (pointwise) for spatial bit allocation. $G$ is the generator/decoder, producing the decompressed image $\hat { \textbf { \textit { x } } }$ , and $D$ is the discriminator used for adversarial training. $F$ extracts features from $\pmb { s }$ .
+
+# D TRAINING DETAILS
+
+We employ the ADAM optimizer (Kingma & Ba, 2014) with a learning rate of 0.0002 and set the mini-batch size to 1. Our networks are trained for 150000 iterations on Cityscapes and for 280000 iterations on Open Images. For normalization we used instance normalization (Ulyanov et al., 2016), except in the second half of the Open Images training, we train the generator/decoder with fixed batch statistics (as implemented in the test mode of batch normalization (Ioffe & Szegedy, 2015)), since we found this reduced artifacts and color shift.
+
+# E DATASET AND PREPROCESSING DETAILS
+
+To train GC models (which do not require semantic label maps, neither during training nor for deployment) for compression of diverse natural images, we use $2 0 0 \mathrm { k }$ images sampled randomly from the Open Images data set (Krasin et al., 2017) (9M images). The training images are rescaled so that the longer side has length 768px, and images for which rescaling does not result in at least $1 . 2 5 \times$ downscaling as well as high saturation images (average $\mathrm { \ S > 0 . 9 }$ or $\mathrm { \Delta V > 0 . 8 }$ in HSV color space) are discarded (resulting in an effective training set size of 188k). We evaluate these models on the Kodak image compression dataset (Kodak) (24 images, $7 6 8 \times 5 1 2 \mathrm { p x } ,$ ), which has a long tradition in the image compression literature and is still the most frequently used dataset for comparisons of learned image compression methods. Additionally, we evaluate our GC models on 20 randomly selected images from the RAISE1K data set (Dang-Nguyen et al., 2015), a real-world image dataset consisting of 8156 high-resolution RAW images (we rescale the images such that the longer side has length 768px). To investigate the benefits of having a somewhat constrained application domain and semantic labels at training time, we also train GC models with semantic label maps on the Cityscapes data set (Cordts et al., 2016) (2975 training and 500 validation images, 34 classes, $2 0 4 8 \times 1 0 2 4 \mathrm { p x }$ resolution) consisting of street scene images and evaluate it on 20 randomly selected validation images (without semantic labels). Both training and validation images are rescaled to $1 0 2 4 \times 5 1 2 \mathrm { p x }$ resolution.
+
+To evaluate the proposed SC method (which requires semantic label maps for training and deployment) we again rely on the Cityscapes data set. Cityscapes was previously used to generate images form semantic label maps using GANs (Isola et al., 2017; Zhu et al., 2017). The preprocessing for SC is the same as for GC.
+
+# F VISUALS
+
+In the following Sections, F.1, F.2, F.3, we show the first five images of each of the three datasets we used for the user study, next to the outputs of BPG at similar bitrates.
+
+Secs. F.4 and F.5 provide visual comparisons of our GC models with Rippel & Bourdev (2017) and Minnen et al. (2018), respectively, on a subset of images form the Kodak data set.
+
+In Section F.6, we show visualizations of the latent representation of our GC models.
+
+Finally, Section F.7 presents additional visual results for SC.
+
+# F.1 GENERATIVE COMPRESSION ON KODAK
+
+![](images/ddb31ec4c90a1d70abe26ddceafc8e59f1102fe9f55d04f9c3c9ca87b35a1a48.jpg)  
+Figure 9: First 5 images of the Kodak data set, produced by our GC model with $C = 4$ and BPG.
+
+# F.2 GENERATIVE COMPRESSION ON RAISE1K
+
+![](images/db5dbd2293c00429b19fdb3b0c1feccbeca2086b1c670111d07b7aeb286a023b.jpg)  
+Figure 10: First 5 images of RAISE1k, produced by our GC model with $C = 4$ and BPG.
+
+# F.3 GENERATIVE COMPRESSION ON CITYSCAPES
+
+![](images/1dfe212047041153b7f36ba39976cfadc57a8c84d6e3ff8c9906b744dec89b4e.jpg)  
+Figure 11: First 5 images of Cityscapes, produced by our GC model with $C = 4$ and BPG.
+
+# F.4 COMPARISON TO RIPPEL & BOURDEV (2017)
+
+![](images/65361a07066777ca349d52b325c50239aab31aeecfeedd91392e453b23856152.jpg)  
+Figure 12: Our model loses more texture but has less artifacts on the knob. Overall, it looks comparable to the output of Rippel & Bourdev (2017), using significantly fewer bits.
+
+![](images/63640453b1f4d3c694340f8a12565f7c0d80d3b9972a6bfe7fa34a897a03f87f.jpg)  
+Rippel et al., 0.0840bpp $( + 2 9 \% )$
+
+![](images/51683fb9f2185f9aaf9183fc305533a8b296f8c506dd24d61b7721e55d20a9c5.jpg)  
+Figure 13: Notice that compared to Rippel & Bourdev (2017), our model produces smoother lines at the jaw and a smoother hat, but proides a worse reconstruction of the eye.
+
+![](images/171d9efb0b2ca2d0541fef42237f74d0f7fb34ae78e7e837346a7547389739b1.jpg)
+
+![](images/8a3fa8c437f0e1c1d10111308350798ec8481f9dc7a54c858a3c7d5edefdc98a.jpg)  
+Rippel et al., 0.0928bpp $( + 3 9 \% )$
+
+Figure 14: Notice that our model produces much better sky and grass textures than Rippel & Bourdev (2017), and also preserves the texture of the light tower more faithfully.
+
+# F.5 COMPARISON TO MINNEN ET AL. (2018)
+
+![](images/68f698b015f6d6433c894410ffa2cb33a5d6a4ceffa5da93485e3e4519417f2e.jpg)  
+Figure 15: Notice that our model yields sharper grass and sky, but a worse reconstruction of the fence and the lighthouse compared to Minnen et al. (2018). Compared to BPG, Minnen et al. produces blurrier grass, sky and lighthouse but BPG suffers from ringing artifacts on the roof of the second building and the top of the lighthouse.
+
+![](images/43932acbb967867713b396c8aa364b0665ee7ffd56f810c01eef2b77864453b3.jpg)  
+BPG, 0.164bpp
+
+Minnen et al., 0.155bpp, $12 7 \%$ larger
+
+![](images/0f28a5bf2a2b71c20d76a21333c8a8038a78beb1af3e9c13731f96f499270fb1.jpg)  
+Figure 16: Our model produces an overall sharper face compared to Minnen et al. (2018), but the texture on the cloth deviates more from the original. Compared to BPG, Minnen et al. has a less blurry face and fewer artifacts on the cheek.   
+Figure 17: Here we obtain a significantly worse reconstruction than Minnen et al. (2018) and BPG, but use only a fraction of the bits. Between BPG and Minnen et al., it is hard to see any differences.
+
+![](images/eeed297b72cd6c7736a7cf46958c7328bc5abcaa08a6b8b709265bda25027a97.jpg)  
+Original
+
+![](images/466563c9255c96b6c70010f25c5af072ebe57226c804db65a51e20d03a944528.jpg)  
+Ours, 0.03418bpp
+
+![](images/b7d2cadf47be0e976acfe1a4fab334577bb4d1fecbf81876e5994b71414538a3.jpg)  
+BPG, 0.119bpp
+
+![](images/984fe010795455574602696c40e223554d531c4c3f117dd4ab43e94d1b9205cf.jpg)  
+Figure 18: Here we obtain a significantly worse reconstruction compared to Minnen et al. (2018) and BPG, but use only a fraction of the bits. Compared to BPG, Minnen et al.has a smoother background but less texture on the birds.
+
+Minnen et al., 0.123bpp, $2 5 9 \%$ larger,
+
+![](images/adc0af3fa74b7970bf9f705a73a19f8d8d85ac9e362a1c86c23238a534107937.jpg)  
+Figure 19: We uniformly sample codes from the (discrete) latent space $\hat { \pmb w }$ of our generative compression models (GC with $C = 4$ ) trained on Cityscapes and Open Images. The Cityscapes model outputs domain specific patches (street signs, buildings, trees, road), whereas the Open Images samples are more colorful and consist of more generic visual patches.
+
+![](images/d19da1b658c9e19fffb0ecf9e5333fca079e778ab90428600e4f117298fdd8a5.jpg)  
+Figure 20: We train the same architecture with $C = 4$ for MSE and for generative compression on Cityscapes. When uniformly sampling the (discrete) latent space $\hat { \pmb { w } }$ of the models, we see stark differences between the decoded images $G ( \hat { \textbf { \em w } } )$ . The GC model produces patches that resemble parts of Cityscapes images (street signs, buildings, etc.), whereas the MSE model outputs looks like low-frequency noise.
+
+GC model with $C = 4$
+
+![](images/bef5c537f130170ec182afc8ddf92aedbc011561e71c74e44af18009adb44f7a.jpg)  
+MSE baseline model with $C = 4$   
+Figure 21: We experiment with learning the distribution of $\hat { \pmb { w } } = E ( \pmb { x } )$ by training an improved Wasserstein GAN (Gulrajani et al., 2017). When sampling form the decoder/generator $G$ of our model by feeding it with samples from the improved WGAN generator, we obtain much sharper images than when we do the same with an MSE model.
+
+![](images/6ef18ebd46c87657e1601ddc3a728d2384abf1afb5f5ebae22b335b498b2ba19.jpg)  
+Figure 22: Synthesizing different classes for two different images from Cityscapes, using our SC network with $C = 4$ . In each image except for no synthesis, we additionally synthesize the classes vegetation, sky, sidewalk, ego vehicle, wall.
+
+![](images/bfc13c3c7d00a4c37a26dfa879c944c4572c4da1b0863444ca5c1b05795d6d77.jpg)  
+Figure 23: Example images obtained by our SC network $C = 8$ ) preserving a box and synthesizing the rest of the image, on Cityscapes. The SC network seamlessly merges preserved and generated image content even in places where the box crosses object boundaries.
+
+![](images/6d816b956091b46916535b16833e64d256592b6af653a86337efe3bf6c51d539.jpg)  
+Figure 24: Reconstructions obtained by our SC network using semantic label maps estimated from the input image via PSPNet (Zhao et al., 2017).
+
+We collect here additional results for the discussion with the reviewers, so that they are easily found.   
+We will integrate these results into the paper.
+
+In Table 2 we compute the PSNR on the Cityscapes test set, when varying the entropy constraint (i.e. changing $C$ ), and the two extremes (a) when MSE is only optimized and (b) when the GAN loss is only optimized. The first three rows shows as the entropy constraint is relaxed, the network can more easily optimize the distortion term leading to a higher PSNR. The fourth row shows that when optimizing for MSE only (see Fig.7 for a qualitative example) we obtain superior PSNR (but at the expense of visual quality with blurry images). The last rows shows that when turning off distortion losses $\lambda = 0$ ), the network does optimize reconstruction at all. Here we observe that the GAN ”collapses” and outputs repetitive textures (see Fig. 25), suggesting the distortion losses are crucial for stability of training.
+
+In Figures 26&27 we show the loss curves when training our GC, $C = 8$ model on OpenImages(Krasin et al., 2017). We note that the loss fluctuates heavily across iterations due to the small batch size (one), but the smoothed losses are stable. For all our experiments, both on Cityscapes and OpenImages, we kept the weights of the losses and ratio between discriminator/generator iterations constant and at point did our (GC and SC) models collapse during training for either dataset.
+
+Table 2: We consider the effect of the GAN loss, the distortion losses and the entropy constraint on the PSNR of the trained model on the Cityscapes dataset.   
+
+<table><tr><td>Setting</td><td>PSNR (dB)</td></tr><tr><td>Our GC,C = 2,H(ω)&lt; 0.018bpp</td><td>21.46</td></tr><tr><td>Our GC,C = 4,H(ω) &lt;0.036</td><td>23.17</td></tr><tr><td>Our GC, C = 8,H(ω) &lt;0.072</td><td>24.93</td></tr><tr><td>MSE bl.,C = 4,H(ω)&lt;0.036</td><td>25.91</td></tr><tr><td>GC,λ=0,C=8,H(ω)&lt;0.072</td><td>11.65</td></tr></table>
+
+![](images/44f1a9660cb510226196a08777117ae44f37946f4f5312bc64b86dfb726f51f6.jpg)  
+Figure 25: When disabling the distortion losses (i.e. $\lambda = 0$ ), such that only $\mathcal { L } _ { \mathrm { G A N } }$ remains, we observe that the training ”collapses” and produces repetitive textures, both for OpenImages and Citycapes.
+
+![](images/67a5a24a796d19b41f715bc08e14cb0f5de36fb4e70be6d432432be8e70a4606.jpg)  
+G_GAN_FeatFigure 26: We show convergence plots for the generator and discriminator losses from training our GC $C = 8$ ) channel model on OpenImages.
+
+![](images/f2f7128db7e4391b5461f0c48383b63ac4b80d6856bd92bf6054c3e1356e27bc.jpg)
+
+![](images/5454d9c0881b2f07be470633b1be3c82da09df801dfe509101f11bfc0fe742c2.jpg)
+
+Ⅲ□
+
+![](images/2837e117b466ee732ae069f5897dd2177aaecdcad5aa9d82a9323283cab33e8c.jpg)  
+Figure 27: We show convergence plots for the distortion losses from training our GC $C = 8$ ) channel model on OpenImages.

parse/train/RYcgfqmAOHh/RYcgfqmAOHh_content_list.json

@@ -0,0 +1,1561 @@
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+    {
+        "type": "text",
+        "text": "Deep Learning with Label Differential Privacy ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Badih Ghazi Google Research badihghazi@goog\u0000e.com ",
+        "bbox": [
+            227,
+            200,
+            397,
+            242
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Noah Golowich⇤ EECS, MIT nzg@mit.edu ",
+        "bbox": [
+            455,
+            200,
+            570,
+            242
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ravi Kumar Google Research ravi.k53@gmai\u0000.com ",
+        "bbox": [
+            625,
+            200,
+            771,
+            242
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Pasin Manurangsi Google Research pasin@goog\u0000e.com ",
+        "bbox": [
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+            263,
+            428,
+            306
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chiyuan Zhang Google Research chiyuan@goog\u0000e.com ",
+        "bbox": [
+            555,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            342,
+            535,
+            358
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The Randomized Response (RR) algorithm $\\left[ \\left[ 9 6 \\right] \\right]$ is a classical technique to improve robustness in survey aggregation, and has been widely adopted in applications with differential privacy guarantees. We propose a novel algorithm, Randomized Response with Prior (RRWithPrior), which can provide more accurate results while maintaining the same level of privacy guaranteed by RR. We then apply RRWithPrior to learn neural networks with label differential privacy (Labe\u0000DP), and show that when only the label needs to be protected, the model performance can be significantly improved over the previous state-of-the-art private baselines. Moreover, we study different ways to obtain priors, which when used with RRWithPrior can additionally improve the model performance, further reducing the accuracy gap between private and non-private models. We complement the empirical results with theoretical analysis showing that Labe\u0000DP is provably easier than protecting both the inputs and labels. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The widespread adoption of machine learning in recent years has increased the concerns about the privacy of individuals whose data is used during the model training. Differential privacy (DP) [34, 33] has emerged as a popular privacy notion that has been the basis of several practical deployments in industry $\\boxed { 1 3 5 } \\boxed { 8 2 } \\boxed { 4 3 } \\boxed { 9 } \\boxed { 2 9 }$ and the U.S. Census [3]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A classical algorithm—that predates DP and was initially designed to eliminate evasive answer biases in survey aggregation—is Randomized Response (RR) $\\dot { \\textcircled { 9 6 } }$ : when the input is an element from a finite alphabet, the output is equal to the input with a certain probability, and is a uniform random other element from the alphabet with the remaining probability. This simple algorithm is shown to satisfy the strong notion of local DP [38, 58], whereby the response of each user is protected, in contrast with the so-called central DP setting where a curator has access to the raw user data, and only the output of the curator is required to be DP. We note that schemes building on RR have been studied in several previous works on DP estimation (e.g., [30, 57]), and have been deployed in practice $\\pmb { \\Vert 8 2 \\Vert }$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Meanwhile, the large error incurred by RR (e.g., [17]) has stimulated significant research aiming to improve its accuracy, mostly by relaxing to weaker privacy models (e.g., [36, 24, 4]). In this work, we use a different approach and seek to improve RR by leveraging available prior information. (A recent work of Liu et al. $[ [ 6 3 ] ]$ also used priors to improve accuracy, but in the context of the DP multiplicative weights algorithm, which applies in the central DP setting.) The prior information can consist of domain-specific knowledge, (models trained on) publicly available data, or historical runs of a training algorithm. Our algorithm is presented in Section 3 (Algorithm 2). At a high level, given a prior distribution $\\mathbf { p }$ on the alphabet, the algorithm uses $\\mathbf { p }$ to prune the alphabet. If the prior is reliable and even if the alphabet is only minimally pruned, the probability that the output equals the input is larger than when RR is applied to the entire alphabet. On the other hand, if the prior is uniform over the entire alphabet, then our algorithm recovers the classical RR. To implement the above recipe, one needs to specify how to effect the pruning using p. It turns out that the magnitude of pruning can itself vary depending on $\\mathbf { p }$ , but we can obtain a closed-form formula for determining this. Interestingly, by studying a suitable linear program, we show that the resulting RRWithPrior strategy is optimal in that among all $\\varepsilon$ -DP algorithms, it maximizes the probability that the output equals the input when the latter is sampled from p (Theorem $3 )$ ",
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+    {
+        "type": "text",
+        "text": "1.1 Applications to Learning with Label Differential Privacy ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "There have been a great number of papers over the last decade that developed DP machine learning algorithms (e.g., [19, 102, 88, 83–85, 78]). In the case of deep learning, the seminal work of Abadi et al. $\\left[ \\left[ 2 \\right] \\right]$ introduced a DP training framework (DP-SGD) that was integrated into TensorFlow $[ \\textcircled { 8 0 } ]$ and PyTorch $\\pmb { \\Vert 9 2 \\Vert }$ . Despite numerous followup works, including, e.g., [75–77, 67, 98, 71, 20, 93], and extensive efforts, the accuracy of models trained with DP-SGD remains significantly lower than that of models trained without DP constraints. Notably, for the widely considered CIFAR-10 dataset, the highest reported accuracy for DP models is $6 9 . { \\dot { 3 } } \\%$ $\\underline { { \\| 9 3 \\| } }$ , which strikingly relies on handcrafted visual features despite that in non-private scenarios learned features long been shown to be superior. Even using pre-training with external (CIFAR-100) data, the best reported DP accuracy, $7 3 \\%$ i s still far below the non-private baselines $( > 9 5 \\% )$ ). The performance gap becomes a roadblocker for many real-world applications to adopt DP. In this paper, we focus on a more restricted, but important, special case where the DP guarantee is only required to hold with respect to the labels, as described next. ",
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+    {
+        "type": "text",
+        "text": "In the label differential privacy (Labe\u0000DP) setting, the labels are considered sensitive, and their privacy needs to be protected, while the input points are not sensitive. This notion has been studied in the PAC setting [18, 13] and for the particular case of sparse linear regression $\\textcircled { 9 4 } \\textcircled { 1 }$ , and it captures several practical scenarios. Examples include: (i) computational advertising where the impressions are known to the Ad Tech3, and thus considered non-sensitive, while the conversions reveal user interest and are thus private (see, e.g., Nalpas and Dutton $\\mathbb { \\ m }$ and $\\pmb { \\| \\bigtriangledown \\| }$ ), (ii) recommendation systems where the choices are known, e.g., to the streaming service provider, but the user ratings are considered sensitive, and (iii) user surveys and analytics where demographic information (e.g., age, gender) is non-sensitive but income is sensitive—in fact, this was the motivating reason for Warner $\\boxed { \\boxed { 9 6 } }$ t o propose RR many decades ago! We present a novel multi-stage algorithm (LP-MST) for training deep neural networks with Labe\u0000DP that builds on top of RRWithPrior (see Section 3 and Algorithm 3), and we benchmark its empirical performance (Section $\\textcircled { 5 }$ on multiple datasets, domains, and architectures, including the following. ",
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+    {
+        "type": "text",
+        "text": "• On CIFAR-10, we show that it achieves $2 0 \\%$ higher accuracy than DP-SGD4.   \n• On the more challenging CIFAR-100, we present the first non-trivial DP learning results.   \n• On MovieLens, which consists of user ratings of movies, we show improvements via LP-MST. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In some applications, domain specific algorithms can be used to obtain priors directly without going through multi-stage training. For image classification problems, we demonstrate how priors computed from a (non-private) self-supervised learning [22, 23, 44, 53, 16] phase on the input images can be used to achieve higher accuracy with a Labe\u0000DP guarantee with extremely small privacy budgets $\\varepsilon \\le 0 . 1$ , see Section $5 . 2$ for details). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We note that due to the requirement of DP-SGD to compute and clip per-instance gradient, it remains technically challenging to scale to larger models or mini-batch sizes, despite numerous attempts to minigate this problem [42, 5, 26, 90]. On the other hand, our formulation allows us to use state-ofthe-art deep learning architectures such as ResNet [51]. We also stress that our LP-MST algorithm goes beyond deep learning methods that are robust to label noise. (See $ { \\mathbb { R } } 7  { \\mathbb { I } }$ for a survey of the latter.) ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our empirical results suggest that protecting the privacy of labels can be significantly easier than protecting the privacy of both inputs and labels. We find further evidence to this by showing that for the special case of stochastic convex optimization (SCO), the sample complexity of algorithms privatizing the labels is much smaller than that of algorithms privatizing both labels and inputs; specifically, we achieve dimension-independent bounds for Labe\u0000DP (Section $\\textcircled { 6 }$ . We also show that a good prior can ensure smaller population error for non-convex loss. (Details are in the Supplementary Material.) ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For any positive integer $K$ , let $[ K ] : = \\{ 1 , \\dots , K \\}$ . Randomized response (RR) $\\left[ \\left[ 9 6 \\right] \\right]$ is the following: let $\\varepsilon \\geq 0$ be a parameter and let $y \\in [ K ]$ be the true value known to $\\mathsf { R R } _ { \\varepsilon }$ . When an observer queries the value of $y$ , $\\mathsf { R R } _ { \\varepsilon }$ responds with a random draw $\\tilde { y }$ from the following probability distribution: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/061173c004f5aa1656f1c25d6134686581bf85daa286713d6287377babc4f1bd.jpg",
+        "text": "$$\n\\operatorname* { P r } [ \\tilde { y } = \\hat { y } ] = \\left\\{ \\begin{array} { l l } { \\frac { e ^ { \\varepsilon } } { e ^ { \\varepsilon } + K - 1 } } & { \\mathrm { ~ f o r ~ } \\hat { y } = y , } \\\\ { \\frac { 1 } { e ^ { \\varepsilon } + K - 1 } } & { \\mathrm { ~ o t h e r w i s e . } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we focus on the application of learning with label differential privacy. We recall the definition of differential privacy (DP), which is applicable to any notion of neighboring datasets. For a textbook reference, we refer the reader to Dwork and Roth $\\pmb { \\mathbb { B 2 } }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.1 (Differential Privacy (DP) $\\textcircled { 1 3 3 } , \\textcircled { 3 4 } $ ). Let $\\varepsilon , \\delta \\in \\mathbb { R } _ { \\geq 0 }$ . A randomized algorithm A taking as input a dataset is said to be $( \\varepsilon , \\delta )$ -differentially private ( $( \\varepsilon , { \\bar { \\delta } } )$ -DP) if for any two neighboring datasets $\\mathbf { D }$ and $\\mathbf { D ^ { \\prime } }$ , and for any subset $S$ of outputs of A, it is the case that $\\mathrm { P r } [ \\mathsf { A } ( \\mathbf { D } ) ] \\ \\in \\ S ] \\ \\bar { \\leq }$ $e ^ { \\varepsilon } \\cdot \\operatorname* { P r } [ \\mathsf { A } ( \\mathbf { D } ^ { \\prime } ) \\in S ] + \\delta$ . If $\\delta = 0$ , then A is said to be $\\varepsilon$ -differentially private ( $\\varepsilon$ -DP). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "When applied to machine learning methods in general and deep learning in particular, DP is usually enforced on the weights of the trained model [see, e.g., 19, 59, 2]. In this work, we focus on the notion of label differential privacy. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2.2 (Label Differential Privacy). Let $\\varepsilon , \\delta \\in \\mathbb { R } _ { \\geq 0 }$ . A randomized training algorithm A taking as input a dataset is said to be $( \\varepsilon , \\delta )$ -label differentially private $( ( \\varepsilon , \\delta )$ -Labe\u0000DP) if for any two training datasets $\\mathbf { D }$ and $\\mathbf { D ^ { \\prime } }$ that differ in the label of a single example, and for any subset $S$ of outputs of A, it is the case that $\\operatorname* { P r } [ \\mathsf { A } ( \\mathbf { D } ) \\in S ] \\leq e ^ { \\varepsilon } \\cdot \\operatorname* { P r } [ \\mathsf { A } ( \\mathbf { D } ^ { \\prime } ) \\in \\bar { S } ] + \\delta$ . If $\\delta = 0$ , then A is said to be $\\varepsilon$ -label differentially private ( $\\dot { \\varepsilon }$ -Labe\u0000DP). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "All proofs skipped in the main body are given in the Supplementary Material. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Randomized Response with Prior ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In many real world applications, a prior distribution about the labels could be publicly obtained from domain knowledge and help the learning process. In particular, we consider a setting where for each (private) label $y$ in the training set, there is an associated prior $\\mathbf { p } = \\left( p _ { 1 } , \\ldots , p _ { K } \\right)$ . The goal is to output a randomized label $\\tilde { y }$ that maximizes the probability that the output is correct (or equivalently maximizes the signal-to-noise ratio), i.e., $\\operatorname* { P r } [ y = \\tilde { y } ]$ . The privacy constraint here is that the algorithm should be $\\varepsilon$ -DP with respect to $y$ . (It need not be private with respect to the prior $\\mathbf { p }$ .) ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We first describe our algorithm RRWithPrior by assuming access to such priors. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Algorithm: RRWithPrior ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We build our RRWithPrior algorithm with a subroutine called RRTop- $k$ , as shown in Algorithm 1, which is a modification of randomized response where we only consider the set of $k$ labels $i$ with largest $p _ { i }$ . Then, if the input label $y$ belongs to this set, we use standard randomized response on this set. Otherwise, we output a label from this set uniformly at random. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The main idea behind RRWithPrior is to dynamically estimate an optimal $k ^ { * }$ based on the prior $\\mathbf { p }$ and run RRTop- $k$ with $k ^ { * }$ . Specifically, we choose $k ^ { * }$ by maximizing $\\operatorname* { P r } [ \\mathsf { R R T o p } \\ – k ( y ) = y ]$ . It is not ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 RRTop- $k$ ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Input: A label $y \\in [ K ]$ ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Parameters: $k \\in [ K ]$ , prior $\\mathbf { p } = \\left( p _ { 1 } , \\ldots , p _ { K } \\right)$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. Let $Y _ { k }$ be the set of $k$ labels with maximum prior probability (with ties broken arbritrarily).   \n2. If $y \\in Y _ { k }$ , then output $y$ with probabilit y e\"e\"+k 1 and output y0 2 Yk \\ {y} with probability $\\frac { 1 } { e ^ { \\varepsilon } + k - 1 }$ .   \n3. If $y \\notin Y _ { k }$ , output an element from $Y _ { k }$ uniformly at random. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "hard to see that this expression is exactly equal to $\\begin{array} { r } { \\frac { e ^ { \\varepsilon } } { e ^ { \\varepsilon } + k - 1 } \\cdot \\left( \\sum _ { \\tilde { y } \\in Y _ { k } } p _ { \\tilde { y } } \\right) } \\end{array}$ if $y \\sim \\mathbf { p }$ . RRWithPrior is presented in Algorithm 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 RRWithPrior ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Input: A label $y \\in [ K ]$ Parameters: prior $\\mathbf { p } = \\left( p _ { 1 } , \\ldots , p _ { K } \\right)$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. For $k \\in [ K ]$ : (a) Compute $\\begin{array} { r } { w _ { k } : = \\frac { e ^ { \\varepsilon } } { e ^ { \\varepsilon } + k - 1 } \\cdot \\left( \\sum _ { \\tilde { y } \\in Y _ { k } } p _ { \\tilde { y } } \\right) } \\end{array}$ , where $Y _ { k }$ is the set of $k$ labels with maximum prior probability (ties broken arbritrarily). ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2. Let $k ^ { * } = \\arg \\operatorname* { m a x } _ { k \\in [ K ] } w _ { k }$ .   \n3. Return an output of RRTop- $k$ $( y )$ with $k = k ^ { * }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1.1 Privacy Analysis ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It is not hard to show that RRTop- $k$ is $\\varepsilon$ -DP. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. RRTop- $k$ is $\\varepsilon$ -DP. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The privacy guarantee of RRWithPrior follows immediately from that of RRTop- $k$ (Lemma 1) since our choice of $k$ does not depend on the label $y$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Corollary 2. RRWithPrior is $\\varepsilon$ -DP. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For learning with a Labe\u0000DP guarantee, we first use RRWithPrior to query a randomized label for each example of the training set, and then apply a general learning algorithm that is robust to random label noise to this dataset. Note that unlike DP-SGD $\\left[ \\left[ 2 \\right] \\right]$ that makes new queries on the gradients in every training epoch, we query the randomized label once and reuse it in all the training epochs. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Optimality of RRWithPrior ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section we will prove the optimality of RRWithPrior. For this, we will need additional notation. For any algorithm R that takes as input a label $y$ and outputs a randomized label $\\tilde { y }$ , we let $\\mathrm { O b j } _ { \\mathbf { p } _ { \\mathrm { . } } } ( \\mathsf { R } )$ denote the probability that the output label is equal to the input label $y$ when $y$ is distributed as $\\mathbf { p }$ ; i.e., $\\mathrm { O b j } _ { \\mathbf { p } } ( \\mathsf { R } ) \\stackrel { - } { = } \\operatorname* { P r } _ { y \\sim \\mathbf { p } } [ \\mathsf { \\bar { R } } ( y ) = y ]$ , where the distribution of $y \\sim \\mathbf { p }$ is $\\operatorname* { P r } [ y = i ] = p _ { i }$ for all $i \\in [ K ]$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The main result of this section is that, among all $\\varepsilon$ -DP algorithms, RRWithPrior maximizes $\\mathrm { O b j } _ { \\mathbf { p } } ( \\mathsf { R } )$ , as stated more formally next. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3. Let $\\mathbf { p }$ be any probability distribution on $[ K ]$ and R be any $\\varepsilon$ -DP algorithm that randomizes the input label given the prior $\\mathbf { p }$ . We have that ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7543dc38ce36f778f1775e646eebc83e74bcc4ddc1e213e47f783b1ac505fdba.jpg",
+        "text": "$$\n\\mathrm { O b j } _ { \\mathbf { p } } ( \\mathsf { R R W i t h P r i o r } ) \\geq \\mathrm { O b j } _ { \\mathbf { p } } ( \\mathsf { R } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Before we proceed to the proof, we remark that our proof employs a linear program (LP) to characterize the optimal mechanisms; a generic form of such LPs has been used before in $\\mathbb { H } \\ 8 \\mathbb { H } \\mathbb { H }$ . However, these works focus on different problems (linear queries) and their results do not apply here. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proof of Theorem 3. Consider any $\\varepsilon$ -DP algorithm R, and let $q _ { \\tilde { y } | y }$ denote $\\mathrm { P r } [ \\mathsf { R } ( y ) = \\tilde { y } ]$ . Observe that Objp(R) = P y [k] py · qy|y. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since $q . | _ { y }$ is a probability distribution, we must have that ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2359107543662090d0ec0a55a8e4bbab32ae117d196d91ecabaa3f05e1b950bc.jpg",
+        "text": "$$\n\\sum _ { \\tilde { y } \\in [ K ] } q _ { \\tilde { y } | y } = 1 , \\forall y \\in [ K ] , \\mathrm { ~ a n d ~ } q _ { \\tilde { y } | y } \\geq 0 , \\forall \\tilde { y } , y \\in [ K ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, the $\\varepsilon$ -DP guarantee of R implies that ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b9da90e4a70f7edd58a4a335e14286ddb529b5ebc6502e7382dfef1ada8da4eb.jpg",
+        "text": "$$\nq _ { \\tilde { y } | y } \\leq e ^ { \\varepsilon } \\cdot q _ { \\tilde { y } | y ^ { \\prime } } \\qquad \\forall \\tilde { y } , y , y ^ { \\prime } \\in [ K ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Combining the above, $\\mathrm { O b j } _ { \\mathbf { p } } ( \\mathsf { R } )$ is upper-bounded by the optimum of the following linear program (LP), which we refer to as LP1: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "max ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "s.t. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8ad646f155ed70bdfa51ebbdfcdf95829e7709c6b69e79f67819dddcfb26fc6d.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\displaystyle \\sum _ { y \\in [ k ] } p _ { y } \\cdot q _ { y | y } } & { } & \\\\ { \\displaystyle q _ { \\tilde { y } | y } \\leq e ^ { \\varepsilon } \\cdot q _ { \\tilde { y } | y ^ { \\prime } } } & { } & { \\qquad \\forall \\tilde { y } , y , y ^ { \\prime } \\in [ K ] , } \\\\ { \\displaystyle \\sum _ { \\tilde { y } \\in [ K ] } q _ { \\tilde { y } | y } = 1 } & { } & { \\qquad \\forall y \\in [ K ] , } \\\\ { \\displaystyle q _ { \\tilde { y } | y } \\geq 0 } & { } & { \\qquad \\forall \\tilde { y } , y \\in [ K ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Notice that constraints $\\textcircled{2}$ and $( 3 )$ together imply that: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/92d777a9acbfb5fc42b58cde32672c31fabda4854d4b7a985746b01fe1690bad.jpg",
+        "text": "$$\nq _ { y | y } + e ^ { - \\varepsilon } \\cdot \\sum _ { \\tilde { y } \\in [ K ] \\backslash \\{ y \\} } q _ { \\tilde { y } | \\tilde { y } } \\leq 1\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/6c1b0b05b95ed63e67f51671046241d85764ca7bdb2af0831c4efb0106f294aa.jpg",
+        "text": "$$\n\\forall y \\in [ K ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In other words, the optimum of LP1 is at most the optimum of the following LP that we call LP2: ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "max ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "s.t. ",
+        "bbox": [
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+            482,
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+            494
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e526a0bcadfbde050aac50471b0f3cbe6ba2ea1c771a9d56233f9d672b3fd33.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\displaystyle { \\sum _ { y \\in [ k ] } p _ { y } \\cdot q _ { y | y } } } \\\\ { \\displaystyle { q _ { y | y } + e ^ { - \\varepsilon } \\cdot \\sum _ { \\tilde { y } \\in [ K ] \\setminus \\{ y \\} } q _ { \\tilde { y } | \\tilde { y } } \\leq 1 } } & { } & { \\forall y \\in [ K ] , } \\\\ { \\displaystyle { q _ { y | y } \\geq 0 } } & { } & { \\forall y \\in [ K ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            438,
+            740,
+            534
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "An optimal solution to LP2 must be a vertex (aka extreme point) of the polytope defined by and $( \\hat { 5 } )$ . Recall that an extreme point of a $K$ -dimensional polytope must satisfy $K$ independent constraints with equality. In our case, this means that one of the following occurs: ",
+        "bbox": [
+            176,
+            542,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "• Inequality $( 5 )$ is satisfied with equality for all $y \\in [ K ]$ resulting in the all-zero solution (whose objective is zero), or, • For some non-empty subset $Y \\subseteq [ K ]$ , inequality $\\textcircled{4}$ is satisfied with equality for all $y \\in Y$ , and inequality $\\textcircled{5}$ is satisfied with equality for all $y \\in [ K ] \\setminus Y$ . This results in ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/405f0bc11cce9a9d19f3090091f61c38aea15f294c51d60980b589d522c7f191.jpg",
+        "text": "$$\nq _ { y | y } = { \\left\\{ \\begin{array} { l l } { { \\frac { e ^ { \\varepsilon } } { e ^ { \\varepsilon } + | Y | - 1 } } } & { { \\mathrm { ~ i f ~ } } y \\in Y , } \\\\ { 0 } & { { \\mathrm { ~ i f ~ } } y \\not \\in Y . } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            652,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This yields an objective value of $\\frac { e ^ { \\varepsilon } } { e ^ { \\varepsilon } + | Y | - 1 } \\cdot \\sum _ { y \\in Y } p _ { y }$ ",
+        "bbox": [
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+            700,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In conclusion, we have that ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/492e95e1dbd37564f290414a1cd68ef8a4733664da5c67826c5755b1b35d9afc.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\mathrm { O b j } _ { \\bf p } ( { \\mathsf { R } } ) \\leq \\displaystyle \\operatorname* { m a x } _ { \\theta \\neq Y \\subseteq C [ K ] } \\displaystyle \\frac { e ^ { \\varepsilon } } { e ^ { \\varepsilon } + | Y | - 1 } \\cdot \\sum _ { y \\in Y } p _ { y } } } \\\\ { { \\ = \\displaystyle \\operatorname* { m a x } _ { k \\in [ K ] } \\displaystyle \\frac { e ^ { \\varepsilon } } { e ^ { \\varepsilon } + k - 1 } \\cdot \\operatorname* { m a x } _ { Y \\subseteq [ K ] , | Y | = k } \\sum _ { y \\in Y } p _ { y } } } \\\\ { { \\ = \\displaystyle \\operatorname* { m a x } _ { k \\in [ K ] } \\displaystyle \\frac { e ^ { \\varepsilon } } { e ^ { \\varepsilon } + k - 1 } \\cdot \\sum _ { y \\in Y _ { k } } p _ { y } \\ = \\ \\operatorname* { m a x } _ { k \\in [ K ] } w _ { k } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            746,
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+            866
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the last two equalities follow from our definitions of $Y _ { k }$ and $w _ { k }$ . Notice that $\\operatorname { O b j } _ { \\mathbf { p } } \\left( { \\mathsf { R R W i t h P r i o r } } \\right) = \\operatorname* { m a x } _ { k \\in [ K ] } w _ { k }$ . Thus, we get that $\\mathrm { O b j } _ { \\mathbf { p } }$ (RRWithPrior) $\\geq \\mathrm { O b j } _ { \\mathbf { p } } ( \\mathsf { R } )$ as desired. □ ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Application of RRWithPrior: Multi-Stage Training ",
+        "text_level": 1,
+        "bbox": [
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+            88,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our RRWithPrior algorithm requires publicly available priors, which could usually be obtained from domain specific knowledge. In this section, we describe a training framework that bootstraps from a uniform prior, and progressively learns refined priors via multi-stage training. This general framework can be applied to arbitrary domains even when no public prior distributions are available. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Specifically, we assume that we have a training algorithm A that outputs a probabilistic classifier which, on a given unlabeled sample $\\mathbf { x }$ , can assign a probability $p _ { y }$ to each class $y \\in [ K ]$ . We partition our dataset into subsets $S ^ { ( 1 ) } , \\ldots , S ^ { ( T ) }$ , and we start with a trivial model $M ^ { ( 0 ) }$ that outputs equal probabilities for all classes. At each stage $t \\in [ T ]$ , we use the most recent model $M ^ { ( t - 1 ) }$ to assign the probabilities $\\left( p _ { 1 } , \\ldots , p _ { K } \\right)$ for each sample $\\mathbf { x } _ { i }$ from $S ^ { ( t ) }$ . Applying RRWithPrior with this prior on the true label $y _ { i }$ , we get a randomized label $\\tilde { y } _ { i }$ for $\\mathbf { x } _ { i }$ . We then use all the samples with randomized labels obtained so far to train the model $M ^ { ( t ) }$ . ",
+        "bbox": [
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+            183,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The full description of our LP-MST (Label Privacy Multi-Stage Training) method is presented in Algorithm $\\textcircled { 3 }$ We remark here that the partition $S ^ { ( 1 ) } , \\ldots , S ^ { ( T ) }$ can be arbitrarily chosen, as long as it does not depend on the labels $y _ { 1 } , \\ldots , y _ { n }$ . We also stress that the training algorithm A need not be private. We use LP-1ST to denote our algorithm with one stage, ${ \\mathsf { L P - } } 2 { \\mathsf { S } } { \\mathsf { T } }$ to denote our algorithm with two stages, and so on. We also note that LP-1ST is equivalent to using vanilla RR. The tth stage of a multi-stage algorithm is denoted stage- $\\cdot t$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 3 Mu\u0000ti-Stage Training (LP-MST) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            395,
+            485,
+            410
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Input: Dataset $S = \\{ ( \\mathbf { x } _ { 1 } , y _ { 1 } ) , \\dotsc , ( \\mathbf { x } _ { n } , y _ { n } ) \\}$ ",
+        "bbox": [
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+            412,
+            446,
+            426
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Parameters: Number $T$ of stages, training algorithm A ",
+        "bbox": [
+            176,
+            426,
+            495,
+            438
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1. Partition $S$ into $S ^ { ( 1 ) } , \\ldots , S ^ { ( T ) }$ ",
+        "bbox": [
+            178,
+            440,
+            380,
+            454
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "2. Let $M ^ { ( 0 ) }$ be the trivial model that always assigns equal probability to each class. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For $t = 1$ to $T$ :   \n(a) Let $\\tilde { S } ^ { ( t ) } = \\emptyset$ .   \n(b) For each $( \\mathbf { x } _ { i } , y _ { i } ) \\in S ^ { ( t ) }$ : i. Let $\\mathbf { p } = \\left( p _ { 1 } , \\ldots , p _ { K } \\right)$ be the probabilities predicted by $M ^ { ( t ) }$ on $\\mathbf { x } _ { i }$ . ii. Let $\\tilde { y } _ { i } = \\mathsf { R R W i t h P r i o r } _ { \\mathbf { p } } ( y _ { i } )$ . iii. Add $\\left( \\mathbf { x } _ { i } , \\tilde { y } _ { i } \\right)$ to $\\tilde { S } ^ { ( t ) }$ .   \n(c) Let $M ^ { ( t ) }$ be the model resulting from training on $\\tilde { S } ^ { ( 1 ) } \\cup \\cdots \\cup \\tilde { S } ^ { ( t ) }$ using A. ",
+        "bbox": [
+            189,
+            473,
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+            565
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4. Output $M ^ { ( T ) }$ ",
+        "bbox": [
+            176,
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+            281,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The privacy guarantee of LP-MST is given by the following: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Observation 4. For any $\\varepsilon > 0$ , $i f$ RRWithPrior is $\\varepsilon$ -DP, then LP-MST is $\\varepsilon$ -Labe\u0000DP. ",
+        "bbox": [
+            174,
+            619,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof. We will in fact prove a stronger statement that the algorithm is $\\varepsilon$ -DP even when we output all the $T$ models $M ^ { ( 1 ) } , \\ldots , M ^ { ( T ) }$ together with all the randomized labels $\\tilde { y } _ { 1 } , \\ldots , \\tilde { y } _ { n }$ . For any possible output models $m ^ { ( 1 ) } , \\ldots , m ^ { ( T ) }$ and output labels $z _ { 1 } , \\ldots , z _ { n }$ , we have ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4a8c00ef26006eb3a0e67e047a2166c52bcbc2646106a03c1fa8f1cf205310be.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\mathrm { P r } [ { \\cal M } ^ { ( 1 ) } = m ^ { ( 1 ) } , \\ldots , { \\cal M } ^ { ( T ) } = m ^ { ( T ) } , \\tilde { y } _ { 1 } = z _ { 1 } , \\ldots , \\tilde { y } _ { n } = z _ { n } ] } } \\\\ { { \\displaystyle = \\prod _ { t = 1 } ^ { T } \\left( \\mathrm { P r } \\left[ { \\cal M } ^ { ( t ) } = m ^ { ( t ) } \\left| \\prod _ { i \\in S ^ { ( 1 ) } \\cup \\ldots \\cup S ^ { ( t ) } } \\tilde { y } _ { i } = z _ { i } \\right. \\right] : \\prod _ { i \\in S ^ { ( t ) } } \\mathrm { P r } \\left[ \\tilde { y } _ { i } = z _ { i } \\Big | { \\cal M } ^ { ( t - 1 ) } = m ^ { ( t - 1 ) } \\right] \\right) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            705,
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+            779
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Consider any two datasets $\\mathbf { D } , \\mathbf { D ^ { \\prime } }$ that differ on a single user’s label; suppose this user is $j$ and that the user belongs to partition $\\ell \\in [ T ]$ . Then, the above expression for $\\mathbf { D }$ and that for $\\mathbf { D ^ { \\prime } }$ are the same in all but one term: $\\operatorname* { P r } \\left[ \\tilde { y } _ { j } = z _ { j } \\middle | M ^ { ( \\ell - 1 ) } = m ^ { ( \\ell - 1 ) } \\right]$ , which is the probability that RRWithPrior $\\underset { . } { m } ^ { ( \\ell - 1 ) } \\bigl ( \\underset { . } { x } _ { i } \\bigr )$ outputs $z _ { i }$ . Since RRWithPrior is $\\varepsilon$ -DP, we can conclude that the ratio between the two probabilities is at most $e ^ { \\varepsilon }$ as desired. □ ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We stress that this observation holds because each sensitive label $y _ { i }$ is only used once in Line $\\boxed { 3 ( \\mathrm { b } ) \\ddot { \\mathrm { u } } }$ of Algorithm $\\bigtriangledown$ as the dataset $S$ is partitioned at the beginning of the algorithm. As a result, since each stage is $\\varepsilon$ -Labe\u0000DP, the entire algorithm is also $\\varepsilon$ -Labe\u0000DP. This is known as (an adaptive version of a) parallel composition $\\pmb { \\Vert 6 8 \\Vert }$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/81f354f90f370c01611a7fd48798d18fd141bdcc2111fa238f11ff59228485f9.jpg",
+        "table_caption": [
+            "Table 1: Test accuracy $( \\% )$ on CIFAR-10. The baseline performances taken from previously published results correspond to $( \\varepsilon , \\delta )$ -DP with $\\delta = 1 0 ^ { - 5 }$ . The star? indicates the use of CIFAR-100 pre-trained representations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm</td><td>ε=1</td><td>ε=2</td><td>ε=3</td><td>ε=4</td><td>m=6</td><td>m=8</td><td>m=8</td></tr><tr><td>DP-SGD w/ pre-train* [2] DP-SGD □</td><td></td><td>67</td><td></td><td>70</td><td></td><td>73</td><td>80</td></tr><tr><td>Tempered Sigmoid [77]</td><td></td><td></td><td></td><td></td><td>66.2(ε=7.53)</td><td>61.6(=7.53)</td><td>76.6</td></tr><tr><td>Yu et al. [98]</td><td></td><td></td><td></td><td></td><td>44.3(ε=6.78)</td><td></td><td></td></tr><tr><td>Nasr et al. [71]</td><td></td><td></td><td>55</td><td></td><td></td><td></td><td></td></tr><tr><td>Chen and Lee [ 四0 ScatterNet+CNN [93]</td><td></td><td></td><td>69.3</td><td></td><td></td><td>53</td><td></td></tr><tr><td>LP-1ST</td><td>59.96</td><td>82.38</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>LP-2ST</td><td>63.67</td><td>86.05</td><td>89.89 92.19</td><td>92.58 93.37</td><td>93.58 94.26</td><td>94.70 94.52</td><td>94.96</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td></tr><tr><td>LP-1ST w/ pre-train* LP-2ST w/ pre-train*</td><td>67.64 70.16</td><td>83.99 87.22</td><td>90.24 92.12</td><td>92.83 93.53</td><td>94.02 94.41</td><td>94.96 94.59</td><td>95.25</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/21cad9f7c698afc0b168d511fb9e96c3fcde9254d01e9bdb4c3856ed30fbd3ec.jpg",
+        "table_caption": [
+            "Table 2: Experiments on CIFAR-100. The non-private baseline $\\varepsilon = \\infty$ ) is $7 6 . 3 8 \\%$ test accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm </td><td>ε=3</td><td>ε=4</td><td>ε=5</td><td>ε=6</td><td>6ε=8</td></tr><tr><td>LP-1ST</td><td>20.96</td><td>46.28</td><td>61.38</td><td>68.34</td><td>73.59</td></tr><tr><td>LP-2ST</td><td>28.74</td><td>50.15</td><td>63.51</td><td>70.58</td><td>74.14</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we point out that the running time of our RRWithPrior algorithm is quasi-linear in $K$ (the time needed to sort the prior). This is essentially optimal within multistage training, since $O ( K )$ time will be required to write down the prior after each stage. Moreover, for reasonable values of $K$ , the running time will be dominated by back-propagation for gradient estimation. Moreover, the focus of the current work is on small to modest label spaces (i.e., values of $K$ ). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Empirical Evaluation ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluate RRWithPrior on standard benchmark datasets that have been widely used in previous works on private machine learning. Specifically, in the first part, we study our general multi-stage training algorithm that boostraps from a uniform prior. We evaluate it on image classification and collaborative filtering tasks. In the second part, we focus on image classification only and use domainspecific techniques to obtain priors for RRWithPrior. We use modern neural network architectures (e.g., ResNets ${ \\| \\dot { \\ 5 } \\| } ,$ ) and the mixup $\\mathbb { I m }$ regularization for learning with noisy labels. Please see the Supplementary Material for full details on the datasets and the experimental setup. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 Evaluation with Multi-Stage Training ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "CIFAR-10 $\\pmb { \\| 6 0 \\| }$ is a 10-class image classification benchmark dataset. We evaluate our algorithm and compare it to previously reported DP baselines in Table $\\nsupseteq$ Due to scalability issues, previous DP algorithms could only use simplified architectures with non-private accuracy significantly below the state-of-the-art. Moreover, even when compared to those weaker non-private baselines, a large performance drop is observed in the private models. In contrast, we use ResNet18 with $9 5 \\%$ nonprivate accuracy. Overall, our algorithms improve the previous state-of-the-art by a margin of $20 \\%$ across all $\\varepsilon$ ’s. Abadi et al. $\\left[ \\left[ 2 \\right] \\right]$ treated CIFAR-100 as public data and use it to pre-train a representation to boost the performance of DP-SGD. We also observe performance improvements with CIFAR-100 pre-training (Table $\\mathbb { L }$ bottom 2 rows). But even without pre-training, our results are significantly better than DP-SGD even with pre-training. ",
+        "bbox": [
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+            772,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/fd601420150712d171f64cb414fa009162b83407329b5d740881e5e33cca8efd.jpg",
+        "table_caption": [
+            "Table 3: Experiments on MovieLens-1M. The numbers show the test RMSE. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm</td><td>ε=1</td><td>ε=2</td><td>ε=3</td><td>ε=4</td><td>m=8</td><td>m=8</td></tr><tr><td>LP-1ST</td><td>1.122</td><td>0.981</td><td>0.902</td><td>0.877</td><td>0.867</td><td>0.868</td></tr><tr><td>LP-2ST</td><td>1.034</td><td>0.928</td><td>0.891</td><td>0.874</td><td>0.865</td><td></td></tr><tr><td>Gaussian DP 园</td><td></td><td></td><td></td><td></td><td colspan=\"2\">0.915 (ε ≥ 10)</td></tr></table>",
+        "bbox": [
+            233,
+            109,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Table $\\bigtriangledown$ we also show results on CIFAR-100, which is a more challenging variant with $1 0 \\times$ more classes. To the best of our knowledge, these are the first non-trivial reported results on CIFAR-100 for DP learning. For $\\varepsilon = 8$ , our algorithm is only $2 \\%$ below the non-private baseline. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In addition, we also evaluate on MovieLens-1M [49], which contains 1 million anonymous ratings of approximately 3, 900 movies, made by 6,040 MovieLens users. Following $[ [ 1 5 ] ]$ , we randomly split the data into $8 0 \\%$ train and $2 0 \\%$ test, and show the test Root Mean Square Error (RMSE) in Table 3. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on MNIST $\\mathbb { \\left[ 6 1 \\right] }$ , Fashion MNIST $\\boldsymbol { \\mathbb { P 7 } }$ , and KMNIST $\\mathbb { \\left[ \\left. 2 5 \\right] \\right. }$ , and comparison to more baselines can be found in the Supplementary Material. In all the datasets we evaluated, our algorithms not only significantly outperform the previous methods, but also greatly shrink the performance gap between private and non-private models. The latter is critical for applications of deep learning systems in real-world tasks with privacy concerns. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Beyond Two Stages. In Figure $^ { 1 ( \\mathrm { a ) } }$ , we report results on LP-MST with $T > 2$ . For the cases we tried, we consistently observe $1 - 2 \\%$ improvements on test accuracy when going from ${ \\mathsf { L P - } } 2 { \\mathsf { S } } { \\mathsf { T } }$ to $\\mathsf { L P - } 3 \\mathsf { S } \\mathsf { T }$ . In our preliminary experiments, going beyond $T > 4$ stages leads to diminishing returns on some datasets. ",
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+        "type": "text",
+        "text": "5.2 Evaluation with Domain-Specific Priors ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The multi-stage training framework evaluated in the previous section is a general domain-agnostic algorithm that bootstraps itself from uniform priors. In some cases, domain-specific priors can be obtained to further improve the learning performance. In this section, we focus on image classification applications, where new advances in self-supervised learning (SSL) [22, 23, 44, 53, 16] show that high-quality image representations could be learned on large image datasets without using the class labels. In the setting of Labe\u0000DP, the unlabeled images are considered public data, so we design an algorithm to use SSL to obtain priors, which is then fed to RRWithPrior for discriminative learning. ",
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+        "type": "text",
+        "text": "Specifically, we partition the training examples into groups by clustering using their representations extracted from SSL models. We then query a histogram of labels for each group via discrete Laplace mechanism (aka Geometric Mechanism) $\\dot { \\left[ \\left| 4 1 \\right| \\right] }$ . If the groups are largely homogeneous, consisting of mostly examples from the same class, then we can make the histogram queries with minimum privacy budget. The queried histograms are used as label priors for all the points in the group. Figure 1(b) shows the results on two different SSL representations: BYOL $\\pm \\ddagger { 4 } \\rVert$ , trained on unlabeled CIFAR-10 images and DINO [16], trained on ImageNet $\\mathbb { \\left[ \\left[ 2 7 \\right] \\right] }$ images. Comparing to the baseline, the SSL-based priors significantly improves the model performance with small privacy budgets. Note that since the SSL priors are not true priors, with large privacy budget $\\varepsilon = 8$ ), it actually underperforms the uniform prior. But in most real world applications, small $\\varepsilon$ ’s are generally more useful. ",
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+        "type": "text",
+        "text": "6 Theoretical Analysis ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Previous works have shown that Labe\u0000DP can be provably easier than DP in certain settings; specifically, in the PAC learning setting, Beimel et al. $\\mathbb { \\lVert 1 3 \\rVert }$ proved that finite VC dimension implies learnability by Labe\u0000DP algorithms, whereas it is known that this is not sufficient for DP algorithms [7]. ",
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+        "type": "text",
+        "text": "We extend the theoretical understanding of this phenomenon to the stochastic convex optimization (SCO) setting. Specifically, we show that, by applying RR on the labels and running SGD on top of the resulting noisy dataset with an appropriate debiasing of the noise, one can arrive at the following dimension-independent excess population loss. ",
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+        "img_path": "images/31860415881e03a443d1231af647a7c9d7eb696e423ef0c2e5147301c4173b03.jpg",
+        "image_caption": [
+            "Figure 1: (a) Test accuracy $( \\% )$ on various datasets with LP-MST for $T > 2$ . The curve “CIFAR-10 w/ pre-train” is using CIFAR-100 as public data to pre-train the model. (b) RRWithPrior with priors obtained from histogram query based on clustering in various SSL representations. We also plot recent results (ClusterRR) from Esfandiari et al. $\\pmb { \\Vert 3 7 } \\Vert$ , which is a clustering based Labe\u0000DP algorithm. "
+        ],
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+        "type": "text",
+        "text": "Theorem 5 (Informal). For any $\\varepsilon \\in ( 0 , 1 )$ , there is an $\\varepsilon$ -Labe\u0000DP algorithm for stochastic convex optimization with excess population loss ${ \\tilde { O } } \\left( D L \\cdot { \\frac { K } { \\varepsilon { \\sqrt { n } } } } \\right)$ where $D$ denotes the diameter of the parameter space and L denotes the Lipschitz constant of the loss function. ",
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+        "type": "text",
+        "text": "The above excess population loss can be compared to that of Bassily et al. $\\mathbb { \\lVert 1 2 \\rVert }$ , who gave an $( \\varepsilon , \\delta )$ - DP algorithm with excess population loss $\\begin{array} { r } { O _ { D , L } \\left( \\frac { 1 } { \\sqrt { n } } + \\frac { \\sqrt { p } } { \\varepsilon n } \\right) } \\end{array}$ , where $p$ denote the dimension of the parameter space; this bound is also known to be tight in the standard DP setting. The main advantage of our guarantee in Theorem $5$ is that it is independent of the dimension $p$ . Furthermore, we show that our bound is tight up to polylogarithmic factors and the dependency on the number of classes $K$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The above result provides theoretical evidence that running RR on the labels and then training on this noisy dataset can be effective. We can further extend this to the setting where, instead of running RR, we run RRTop- $k$ before running the aforementioned (debiased) SGD, although—perhaps as expected—our bound on the population loss now depends on the quality of the priors. ",
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+    {
+        "type": "text",
+        "text": "Corollary 6 (Informal). Suppose that we are given a prior $\\mathbf { p } _ { x }$ for every $x$ and let $Y _ { k } ^ { x }$ denote the set of top- $k$ labels with respect to $\\mathbf { p } _ { x }$ . Then, for any $\\varepsilon \\in ( 0 , 1 )$ , there is an $\\varepsilon$ -Labe\u0000DP algorithm for stochastic convex optimization with excess population loss $\\begin{array} { r } { \\tilde { O } \\left( D L \\cdot \\left( \\frac { k } { \\varepsilon \\sqrt { n } } + \\operatorname* { P r } _ { ( x , y ) \\sim \\mathcal { D } } [ y \\notin Y _ { k } ^ { x } ] \\right) \\right) } \\end{array}$ where $D , L$ are as defined in Theorem 5 and $\\mathcal { D }$ is the data distribution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When our top- $k$ set is perfect (i.e., $y$ always belongs to $Y _ { k } ^ { x }$ ), the bound reduces to that of Theorem 5, but with the smaller $k$ instead of $K$ . Moreover, the second term is, in some sense, a penalty we pay in the excess population loss for the inaccuracy of the top- $k$ prior. We defer the formal treatment and the proofs to the Supplementary Material, in which we also present additional generalization results for non-convex settings. ",
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+    {
+        "type": "text",
+        "text": "Note that Corollary $6$ is not in the exact setup we run in experiments, where we dynamically calculate an optimal $k$ for each $x$ given generic priors (via RRWithPrior), and for which the utility is much more complicated to analyze mathematically. Nonetheless, the above corollary corroborates the intuition that a good prior helps with training. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusions and Future Directions ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this work, we introduced a novel algorithm RRWithPrior (which can be used to improve on the traditional RR mechanism), and applied it to Labe\u0000DP problems. We showed that prior information can be incorporated to the randomized label querying framework while maintaining privacy constraints. We demonstrated two frameworks to apply RRWithPrior: (i) a general multi-stage training algorithm LP-MST that bootstraps from uniform priors and (ii) an algorithm that build priors from clustering with ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "SSL-based representations. The former is general purpose and can be applied to tasks even when no domain-specific priors are available, while the latter uses a domain-specific algorithm to extract priors and performs well even with very small privacy budget. As summarized by the figure on the right, in both cases, by focusing on Labe\u0000DP, our RRWithPrior significantly improved the model performance of previous state-of-the-art DP models that aimed to protect both the inputs and outputs. We note that, following up on our work, additional results on deep learning with Labe\u0000DP were obtained $[ 6 6 , 1 0 0 ]$ . The narrowed performance gap between private and non-private models is vital for adding DP to real world deep learning models. We nevertheless stress that our algorithms only protect the labels but not the input points, which might not constitute a sufficient privacy protection in all settings. ",
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+        "type": "text",
+        "text": "Our work opens up several interesting questions. Firstly, note that our multi-stage training procedure uses very different ingredients than those of Abadi et al. $[ \\left[ 2 \\right] ]$ (which employ DP-SGD, privacy amplification by subsampling, and Renyi accounting); can these tools be used to further improve Labe\u0000DP? Secondly, while our procedure can be implemented in the most stringent local DP setting5 $\\left[ \\left[ 5 8 \\right] \\right]$ , can it be improved in the weaker central (aka trusted curator) DP model, assuming the curator knows the prior? Thirdly, while our algorithm achieves pure DP (i.e., $\\delta = 0$ ), is higher accuracy possible for approximate DP (i.e., $\\delta > 0$ )? ",
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+        "img_path": "images/168b4980e4d7e07c7fbf0f08bf53429186563ee69a6a3c7bc455362d4393b995.jpg",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Sami Torbey for very helpful feedback on an early version of this work. At MIT, Noah Golowich was supported by a Fannie and John Hertz Foundation Fellowship and an NSF Graduate Fellowship. ",
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+    {
+        "type": "text",
+        "text": "References ",
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+        "type": "text",
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parse/train/Skltqh4KvB/Skltqh4KvB.md

@@ -0,0 +1,216 @@
+# ARE THERE ANY ‘OBJECT DETECTORS’ IN THE HIDDEN LAYERS OF CNNS TRAINED TO IDENTIFY OBJECTS OR SCENES?
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Various methods of measuring unit selectivity have been developed with the aim of better understanding how neural networks work. But the different measures provide divergent estimates of selectivity, and this has led to different conclusions regarding the conditions in which selective object representations are learned and the functional relevance of these representations. In an attempt to better characterize object selectivity, we undertake a comparison of various selectivity measures on a large set of units in AlexNet, including localist selectivity (Bowers et al., 2014), precision (Zhou et al., 2015), class-conditional mean activity selectivity (CCMAS) (Morcos et al., 2018), network dissection (Zhou et al., 2018a), the human interpretation of activation maximization (AM) images, and standard signaldetection measures. We find that the different measures provide different estimates of object selectivity, with precision and CCMAS measures providing misleadingly high estimates. Indeed, the most selective units had a poor hit-rate or a high falsealarm rate (or both) in object classification, making them poor object detectors. We fail to find any units that are even remotely as selective as the ‘grandmother cell’ units reported in recurrent neural networks. In order to generalize these results, we compared selectivity measures on a few units in VGG-16 and GoogLeNet trained on the ImageNet or Places-365 datasets that have been described as ‘object detectors’. Again, we find poor hit-rates and high false-alarm rates for object classification.
+
+# 1 INTRODUCTION
+
+There have been recent attempts to understand how neural networks (NNs) work by analyzing hidden units one-at-a-time using various measures such as localist selectivity (Bowers et al., 2014), class-conditional mean activity selectivity (CCMAS) (Morcos et al., 2018), precision (Zhou et al., 2015), network dissection (Zhou et al., 2018a), and activation maximization (AM) (Erhan et al., 2009). These measures are all taken to provide evidence that some units respond highly selectively to categories of objects under some conditions. Not only are these findings surprising given the widespread assumption that NNs only learn highly distributed and entangled representations, they raise a host of questions, including the functional importance of these selective representations (Zhou et al., 2018b), the conditions in which they are learned (e.g., Morcos et al., 2018), and the relation between these representations and the selective neurons observed in cortex (Bowers, 2009).
+
+To answer these question, it is necessary to have a better understanding of what these metrics actually measure, and how they relate to one another. Accordingly, we directly compare these measures of selectivity on the same set of units as well as adopt standard signal-detection measures in an attempt to provide better measures of single-unit selectivity to object category. In addition, to provide a more intuitive assessment of selectivity, we report jitterplots for a few of the most selective units that visually display how the unit responds to the different image categories. We focus on AlexNet (Krizhevsky et al., 2012) trained on ImageNet (Deng et al., 2009) because many authors have studied the selectivity of single hidden units in this model using a range of quantitative (Zhou et al., 2018a; 2015) and qualitative (Nguyen et al., 2017; Yosinski et al., 2015; Simonyan et al., 2013) methods. But we also compare different selectivity measures on specific units in VGG-16 (Simonyan and Zisserman, 2014) and GoogLeNet (Szegedy et al., 2015) trained on the the ImageNet and Places-365 datasets that were characterized by Zhou et al. (2018a) as “object detectors” based on their Network Dissection method (Zhou et al., 2018a). Our main findings are:
+
+1. The precision and CCMAS measures are misleading with near-maximum selectivity scores associated with units that strongly respond to many different image categories. By contrast, the signal-detection measures more closely capture the level of selectivity displayed in the jitterplots (Sec. 3.1).   
+2. Units with interpretable AM images do not correspond to highly selective representations (Sec. 3.2).   
+3. The Network Dissection method also provides a misleading measure for “object detectors” (Sec. 3.3).
+
+In one line of research, Bowers et al. (2014; 2016) assessed the selectivity of single hidden units in recurrent neural networks (RNNs) designed to model human short-term memory. They reported many ‘localist’ or ‘grandmother cell’ units that were $100 \%$ selective for specific letters or words, where all members of the selective category were more active than and disjoint from all non-members, as can be shown in jitterplots (Berkeley et al., 1995) (see Fig. 1 for a unit selective to the letter ‘j’). The authors argued that the network learned these representations in order to co-activate multiple letters or words at the same time in short-term memory without producing ambiguous blends of overlapping distributed patterns (the so-called ‘superposition catastrophe’). Consistent with this hypothesis, localist units did not emerge when the model was trained on letters or words one-at-a-time (Bowers et al., 2014) (see Fig. 1 for an example of a non-selective unit).
+
+In parallel, researchers have reported selective units in the hidden layers of various CNNs trained to classify images into one of multiple categories (Zhou et al., 2015; Morcos et al., 2018; Zeiler and Fergus, 2014; Erhan et al., 2009), for a review see Bowers (2017). For example, Zhou et al. (2015) assessed the selectivity of units in the pool5 layer of two CNNs trained to classify images into 1000 objects and 205 scene categories, respectively. They reported many highly selective units that they characterized as ‘object detectors’ in both networks. Similarly, Morcos et al. (2018) reported that CNNs trained on CIFAR-10 and ImageNet learned many highly selective hidden units, with CCMAS scores approaching the maximum of 1.0. These later findings appear to be inconsistent with Bowers et al. (2016) who failed to observe selective representations in fully connected NNs trained on stimuli one-at-a-time (see Fig. 1), but the measures of selectivity that have been applied across studies are different, and accordingly, it is difficult to directly compare results.
+
+A better understanding of the relation between selectivity measures is vital given that different measures are frequently used to address similar issues. For example, both the human interpretability of generated images (Le, 2013) and localist selectivity (Bowers et al., 2014) have been used to make claims about ‘grandmother cells’, but it is not clear whether they provide similar insights into unit selectivity. Similarly, based on their precision metric, Zhou et al. (2015) claim that the object detectors learned in CNNs play an important role in identifying specific objects, whereas Morcos et al. (2018) challenge this conclusion based on their finding that units with high CCMAS measures were not especially important in the performance of their CNNs and concluded: “...it implies that methods for understanding neural networks based on analyzing highly selective single units, or finding optimal inputs for single units, such as activation maximization (Erhan et al., 2009) may be misleading". This makes a direct comparison between selectivity measures all the more important.
+
+In order to directly compare and have a better understanding of the different selectivity measures we assessed (1) localist, (2) precision, and (3) CCMAS selectivity of the conv5, fc6, and fc7 of AlexNet trained on ImageNet, and in addition, we employed a range of signal detection methods on these units, namely, (4) recall with $100 \%$ and $9 5 \%$ precision, (5) maximum informedness, (6) specificity at maximum informedness , and (7) recall (also called sensitivity) at maximum informedness, and false alarm rates at maximum informedness (described in Sec. 2). We also assessed the selectivity of a few units in VGG-16 and GoogLeNet models trained on the ImageNet and Places-365 dataset that were highly selective according to the Network Dissection method (Zhou et al., 2018a). We show that the precision and CCMAS measures often provide misleadingly high estimates of object selectivity compared to other measures, and we do not find any units that can be reasonably described as ‘object detectors’ given that the most selective units show a low hit-rate or a high false-alarm rate (or both) when classifying images. At best, the most selective units in CNNs are sensitive to some unknown feature that is weakly associated with the class in question.
+
+![](images/79eb40ad904e0968407b2502f5d29a63932ec8a65908516624ad8dd72655124e.jpg)  
+Figure 1: Examples of selectivity measures used. Top left: jitterplot of unit 113 in an RNN (under the superposition constraint) selective to the letter ‘j’ (Bowers et al., 2016). Top middle: jitterplot of a non-selective unit 160 found in an RNN trained on words one-at-a-time from (Bowers et al., 2016). Top right: Activation maximization image of unit conv59 AlexNet that resembles a lighthouse (Nguyen et al., 2016). Bottom: highest-activation images for a ‘lamp’ detector with $84 \%$ precision in the layer conv5 of AlexNet; from (Zhou et al., 2015).
+
+In addition to these quantitative measures and jitterplots we assessed selectivity with a common qualitative measure, namely, human interpretation of images generated by a state-of-the-art activation maximization (AM) method (Nguyen et al., 2017). AM images are generated to strongly activate individual units, and some of them are interpretable by humans (e.g., a generated image that looks like a lighthouse, see Fig. 1). For the first time, we systematically evaluated the interpretability of the AM images and compare these ratings with the selectivity measures for corresponding units. We show that the few hidden units with interpretable AM images are not highly selective.
+
+# 2 METHODS
+
+Network and Dataset All ${ \sim } 1 . 3 \mathbf { M }$ photos from the ImageNet ILSVRC 2012 dataset (Deng et al., 2009) were cropped to $2 7 7 \times 2 7 7$ pixels and classified by the pre-trained AlexNet CNN (Krizhevsky et al., 2012) shipped with Caffe (Jia et al., 2014), resulting in 721,536 correctly classified images. Once classified, the images are not re-cropped nor subject to any changes. We analyzed the fully connected (fc) layers: fc6 and fc7 (4096 units), and the top convolutional layer conv5 which has 256 filters. We only recorded the activations of correctly classified images. The activation files are stored in .h5 format and will be available at http://anonymizedForReview. We randomly selected 233 conv5, 2738 fc6, 2239 fc7 units for analysis.
+
+Localist selectivity Following Bowers et al. (2014), we define a unit to be localist for class $A$ if the set of activations for class $A$ was higher and disjoint with those of $\neg A$ . Localist selectivity is easily depicted with jitterplots (Berkeley et al., 1995) in which a scatter plot for each unit is generated (see Figs. 1 and 3). Each point in a plot corresponds to a unit’s activation in response to a single image, and only correctly classified images are plotted. The level of activation is coded along the $x$ -axis, and an arbitrary value is assigned to each point on the $y$ -axis.
+
+Precision Precision refers to the proportion of items above some threshold from a given class. The precision method of finding object detectors involves identifying a small subset of images that most strongly activate a unit and then identifying the critical part of these images that are responsible for driving the unit. Zhou et al. (2015) took the 60 images that activated a unit the most strongly and asked independent raters to interpret the critical image patches (e.g., if 50 of the 60 images were labeled as ‘lamp’, the unit would have a precision index of 50/60 or $83 \%$ ; see Fig. 1). Object detectors were defined as units with a precision score $> 7 5 \%$ : they reported multiple such detectors. Here, we approximate this approach by considering the 60 images that most strongly activate a given unit and assess the highest percentage of images from a given output class.
+
+CCMAS Morcos et al. (2018) introduced a selectivity index called the Class-conditional Mean Activation Selectivity (CCMAS). The CCMAS for class $A$ compares the mean activation of all images in class $A$ , $\mu _ { A }$ , with the mean activation of all images not in class $A$ , $\mu { \neg } A$ , and is given by: $\left( \mu _ { A } ^ { \setminus } - \mu _ { \neg A } \right) / \left( \mu _ { A } + \mu _ { \neg A } \right)$ . Here, we assessed class selectivity for the highest mean activation class.
+
+Activation Maximization We harnessed an activation maximization method called Plug & Play Generative Networks (Nguyen et al., 2017) in which an image generator network was used to generate images (AM images) that highly activate a unit in a target network. We used the public code released by Nguyen et al. (2017) and their default hyperparameters.1 We generated 100 separate images that maximally activated each unit in the conv5, fc6, and fc8 layers of AlexNet and asked participants to judge whether they could identify any repeating objects, animals, or places in images in a behavioral experiment (Sec. 3.2). Readers can test themselves at: https://research.sc/ participant/login/dynamic/63907FB2-3CB9-45A9-B4AC-EFFD4C4A95D5
+
+Recall with perfect and $95 \%$ precision Recall with perfect and $9 5 \%$ precision are related to localist selectivity except that they provide a continuous rather than discrete measure. For recall with perfect precision we identified the image that activated a given unit the most and counted the number of images from the same class that were more active than all images from all other classes. We then divided this result by the total number of correctly identified images from this class. A recall with a perfect precision score of 1 is equivalent to a localist representation. Recall with a $9 5 \%$ precision allows $5 \%$ false alarms.
+
+Maximum informedness Maximum informedness identifies the class and threshold where the highest proportion of images above the threshold and the lowest proportion of images below the threshold are from that class (Powers, 2011). The informedness is computed for each class at each threshold, with the highest value selected. Informedness summarises the diagnostic performance of unit for a given class at a certain threshold based on the recall [True Positives / (True Positives $^ +$ False Negatives)] and specificity [True Negatives / (True Negatives $^ +$ False Positives)] in the formula [informedness $=$ recall $^ +$ specificity − 1] (Powers, 2011).
+
+Sensitivity or Recall at Maximum Informedness For the threshold and class selected by Maximum Informedness, recall (or hit-rate) is the proportion of items from the given class that are above the threshold. Also known as true postive rate.
+
+Specificity at Maximum Informedness For the threshold and class selected by Maximum Informedness, the proportion of items that are not from the given class that are below the threshold. Also known as true negative rate.
+
+False Alarm Rate at Maximum Informedness For the threshold and class selected by Maximum Informedness, the proportion of items that are not from the given class that are above the threshold.
+
+Network Dissection To assess the selectivity of a unit in the Network Dissection technique, Zhou et al. (2018a) compute the Intersection over Union (IoU) of an annotated input image $L _ { c }$ , for the set of all ‘concepts’ $c$ and a spatial activation map, $M _ { k }$ , of where a unit $k$ is. A unit $k$ is taken as a detector for concept $c$ if its $\mathrm { I o U } _ { k , c }$ exceeds a pre-defined threshold $T$ . See Zhou et al. (2018a) for more details.
+
+# 3 RESULTS
+
+# 3.1 COMPARISON OF SELECTIVITY MEASURES IN ALEXNET
+
+The results from the various of selectivity measures applied to the conv5, fc6, and fc7 layers of AlexNet are displayed in Fig. 2a–i. We did not plot the localist selectivity as there were no localist ‘grandmother units’. The first point to note is that multiple units in the fc6 and fc7 layers had near $100 \%$ precision scores and multiple units had CCMAS scores approaching 1. For example, in layer fc7, we found 14 units with a precision $> 0 . 9$ , and 1487 units with a $\mathrm { C C M A S } > 0 . 9$ . The second point is that other measures provided much reduced estimates of selectivity. For example, the unit with the highest recall with a perfect precision score was only .08 (unit 255 responding to images of Monarch butterflies), and the unit with the top maximum informedness score (unit 3290 also responding to images of Monarch butterflies with a score of 0.91) had a false alarm rate above its optimal threshold $> 9 9 \%$ (indeed the minimum false alarm rate was 0.96).
+
+To illustrate the contrasting measures of selectivity consider unit $\mathsf { f c 6 } _ { 1 1 9 9 }$ depicted in Fig. 3 that has a precision score of $98 \%$ and a CCMAS score of .92. By Zhou et al.’s criterion, this is a ‘Monarch Butterfly’ detector (its precision score $> 7 5 \%$ ). By contrast, the scatter plot and signal-detection scores show this is a mischaracterisation of this unit given that the false alarm rate at maximum informedness was greater than $9 9 \%$ and the modal response to Monarch butterflies was zero.
+
+![](images/f080f4e7d6eb4f52ab14011f1fc60eded3687247e4ebcd11ddb551530f0343ba.jpg)  
+Figure 2: Different selectivity measures across the conv5, fc6, and fc7 layers of AlexNet. Red-line: median of data, top and bottom of box edges is the $2 5 ^ { \mathrm { t h } }$ and $7 5 ^ { \mathrm { t h } }$ percentile, whiskers extend to extreme edges of distribution not considered outliers and red crosses are outliers. Green points and dashed lines are the means of the distributions with standard errors. The high levels of selectivity observed with the precision and CCMAS measures are in stark contrast with the low levels of selectivity observed with the recall with perfect precision and high false-alarm rates at maximum informedness.
+
+What level of selectivity is required before a unit can be considered an ‘object detector’ for a given category? In the end, this is a terminological point. On an extreme view, one might limit the term to the ‘grandmother units’ that categorize objects with perfect recall and specificity, or alternatively, it might seem reasonable to describe a unit as a detector for a specific object category if there is some threshold of activation that supports more hits than misses (the unit is strongly activated by the majority of images from a given category), and at the same time, supports more hits than false alarms (the unit is strongly activated by items from the given category more often than by items from other categories). Or perhaps a lower standard could be defended, but in our view, the term "object detector" suggests a higher level of selectivity than $8 \%$ recall at perfect precision. That said, our results show that some units respond strongly to some (unknown) features that are weakly correlated with an object category. For instance, unit $\mathsf { f c } 6 _ { 1 1 9 9 }$ is responding to features that occur more frequently in Monarch Butterflies than other categories. This can also be seen in a recent ablation study in which removing the most selective units tended to impair the CNN’s performance in identifying the corresponding object categories more than other categories (Zhou et al., 2018b). But again, the pattern of performance is not consistent with the units being labeled ‘object detectors’.
+
+![](images/304a6ae45b0d36961bc7a3ece31444019196ee73cb8388a7c33d438ebf85af5d.jpg)  
+Figure 3: Data for unit $\mathsf { f c } 6 _ { 1 1 9 9 }$ . Left: activation jitterplot, black diamonds: Monarch butterfly images; grey circles: all other classes; white dashed line: threshold for the butterfly class maximum informedness; blue solid line: threshold for top 60 activations. Middle: histogram of activations of Monarch butterflies; red dashed line: threshold for the butterfly class maximum informedness; black solid line: threshold for top 60 activations. Inset: zoomed-in histogram of all activations across all ImageNet classes of unit $\mathsf { f c 6 } _ { 1 1 9 9 }$ (N.B. this plot shows only the highest 121,586 activations; there are 596,734 activations at 0). There are Monarch butterfly images covering the whole range of values, with 72 images $5 . 8 \%$ of the total) having an activation of 0. Right: example ImageNet images with activations of 0 (top), the mean, $3 9 . 2 { \pm } 0 . 6 $ , (middle), and the maximum, 95, (bottom) of the range. Although the high precision score suggests that this unit is a butterfly detector this is misleading given there are butterfly images over the entire activation range (including 0).
+
+# 3.2 HUMAN INTERPRETATION OF ACTIVATION MAXIMIZATION IMAGES FOR ALEXNET UNITS
+
+Activation Maximization is one of the most commonly used interpretability methods for explaining what a single unit has learned in many artificial CNNs and even biological neural networks (see Nguyen et al. (2019) for a survey). Our behavioral experiment provides the first quantitative assessment of AM images and compares AM interpretability to other selectivity measures.
+
+Table 1: Human judgements of whether AM images look like familiar objects in layers conv5, fc6, and fc8 in AlexNet.   
+
+<table><tr><td rowspan="2">layer</td><td rowspan="2">%‘yes’ responses</td><td rowspan="2">% units ≥ 80% ‘yes’ response</td><td colspan="3">% overlap between humans and:</td></tr><tr><td>humans</td><td>most active</td><td>CCMAS</td></tr><tr><td>conv5</td><td>(a) 21.7% (±1.1%)</td><td>(b) 4.3% (± 1.3%)</td><td>(c) 89.5% (±5.7%)</td><td>object (d) 34.1% (±14.4%)</td><td>class (e) 0%</td></tr><tr><td>fc6</td><td>21.0% (±0.4%)</td><td>3.1% (± 0.4%)</td><td>80.4% (±4.1%)</td><td>23.3% (±5.9%)</td><td>18.9% (±5.9%)</td></tr><tr><td>fc8</td><td>71.2% (±0.6%)</td><td>59.3% (±1.6%)</td><td>96.5% (±0.4%)</td><td>95.4% (±0.6%)</td><td>94.6% (±0.7%)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+We generated 100 AM images images for every unit in the layers conv5, fc6, and fc8 in AlexNet, as in Nguyen et al. (2017), and displayed them as $1 0 \times 1 0$ -image panels. A total of 3,299 image panels were used in the experiment (995 fc8, 256 conv5, and 2048 randomly selected fc6 image panels) and were divided into 64 counterbalanced lists for testing. To assess the interpretability for these units as object detectors, 333 paid volunteers were asked to look at image panels and asked if the images had an object / animal or place in common. If the answer was ‘yes’, they were asked to write down a generic name for that object (e.g. “fish” rather than “goldfish”). Analyses of common responses was done for any units where over $80 \%$ of humans agreed there was an object present.
+
+The results are summarized in Table 1. Not surprisingly, the AM images for output fc8 units are the most human-recognizable as objects across the AlexNet layers $( 7 1 . 2 \%$ ; Table 1a). In addition, when they were given a consistent interpretation, they almost always $( 9 5 . 4 \%$ ; Table 1d) match the corresponding ImageNet category. By contrast, less than $5 \%$ of units in conv5 or fc6 were associated with consistently interpretable images (Table 1b), and the interpretations only weakly matched the category associated with the highest-activation images or CCMAS selectivity (Table 1d–e). Apart from showing that there are few interpretable units in the hidden layers of AlexNet, our findings show that the interpretability of images does not imply a high level of selectivity given the signal-detection results (Fig. 2d–h). See Fig. 4 for an example of the types of images that participants rated as objects or non-objects.
+
+![](images/65641a224bfb7b02c868516d88e7aa46f99affaf3cbcdcc65915c0b9ce1c5028.jpg)  
+Figure 4: Example AM images that were either judged by all participants to contain objects (a–c) or to be uninterpretable as objects (d–f). The human label for unit $\mathsf { c o n v } 5 _ { 1 8 3 }$ (a) was ‘dogs’; the most active image was of a ‘flat-coated retriever’; CCMAS class was ‘monitor’. For $\mathsf { f c } 6 _ { 3 1 9 }$ (b), subjects reported ‘green peppers’ or ‘apples’ (all classified as the same broad class in our analysis); both the most active item and CCMAS class were ‘Granny Smith apples’. For $\mathsf { f c } 8 _ { 9 6 9 }$ (c), humans suggested ‘beverage’ or ‘drink’; both the most active item and CCMAS class were ‘eggnog’.
+
+# 3.3 COMPARING SELECTIVITY MEASURES IN OTHER CNNS
+
+![](images/79c3d69b92c6d6545b4a213df8d41637fad9929156d47df2115d5006595fe5f0.jpg)
+
+![](images/3d9bcbe0d7ce0618a23a8ab24517ac41f8869d7e9784c127c7321419e788621a.jpg)
+
+a. GoogLeNet on ImageNet inception4e494 precision: 0.0 CCMAS:0.52
+
+![](images/8bae5fc8b73350af8466d9f1e3ed096fbd11ddafd16049b57ef8c5a154b9cc4f.jpg)  
+Figure 5: The units with with the highest Network Dissection scores for the category ‘bus’. The scatter plots, precision, and CCMAS scores all indicate a low selectivity for this category. blue squares: ‘school bus’; red pentagons: ‘trolleybus’; green stars: ‘minibus’; grey circles: other classes.
+
+b. GoogLeNet on Places-365 inception4e824 precision: 0.27 CCMAS: 0.55
+
+c. VGG-16 on Places-365   
+conv5_320   
+precision: 0.53   
+CCMAS: 0.82
+
+$$
+\mu _ { A } = 7 2 . 5 0 \mu _ { - , A } = 2 2 . 8 1 \mu _ { A } = 4 0 . 9 9 \mu _ { - , A } = 1 1 . 7 8 \mu _ { A } = 1 5 7 . 6 \mu _ { - , A } = 1 5 . 2 8
+$$
+
+Thus far we have assessed the selectivity of hidden units in AlexNet and shown that no units can reasonably be characterized as object detectors despite the high precision and CCMAS scores of some units. This raises the question as to whether more recent CNNs learn object detector units. In order to address this we display jitterplots for three units that have the highest IoU scores according to the Network Dissection for the category BUS in (a) GoogLeNet trained on ImageNet, (b) GoogLeNet trained on Places-365, and (c) VGG-16 trained on Places-365, respectively (Zhou et al., 2018a). Models trained on the Places-365 dataset learn to categorize images into scenes (e.g., bedrooms, kitchens, etc.) rather than into object categories, and nevertheless, Zhou et al. (2018a) reported more object detectors in the former models. We illustrate the selectivity of the BUS category because it is an output category in ImageNet so we can easily plot the jitterplots for these units.
+
+As was the case with AlexNet, the jitterplots show that the most selective units show some degree of selectivity, with the BUS images more active on average compared to non-Buses, and the percentage of nonzero activations for BUS higher than the non-BUS categories (see tables A3 - A5 in the appendix for summary of more units). But the units are no more selective than the units we observed in AlexNet. Indeed, the precision measure of selectivity for the first units is 0.0, with none of the units having a precision of .75 that was the criterion of object detectors by Zhou et al. (2015), and CCMAS scores for first two units were roughly similar to the mean CCMAS score for AlexNet units in conv 5 (and much lower than the mean in fc6 and fc7). The most selective VGG-16 unit trained on Places-365 has lower precision and CCMAS scores than the Monarch Butterfly unit depicted in Figure 3. So again, different measures of selectivity provide support different conclusions, and even the most selective units are far from the selective units observed in recurrent networks as reported in Figure 1a. See tables A3 - A5 in the appendix for more details about these three units.
+
+# 4 DISCUSSIONS AND CONCLUSIONS
+
+Our central finding is that different measures of single-unit selectivity for objects support very different conclusions when applied to the same units in AlexNet. In contrast with the precision (Zhou et al., 2015) and CCMAS (Morcos et al., 2018) measures that suggest some highly selective units for objects in layers conv5, fc6, and fc7, the recall with perfect precision and false alarm rates at maximum informedness show low levels of selectivity. Indeed, the most selective units have a poor hit-rate or a high false-alarm rate (or both) for identifying an object class. The same outcome was observed with units in VGG-16 and GoogLeNet trained on either ImageNet or the Places-365 dataset.
+
+Not only do the different measures provide very different assessments of selectivity, the precision, CCMAS, and Network Dissection measures provide highly misleading estimates of selectivity that have led to mistaken conclusions. For example, unit $\mathsf { f c } 6 _ { 1 1 9 9 }$ in AlexNet trained on ImageNet is considered an Monarch Butterfly detector according to Zhou et al. (2015) with a precision score of $98 \%$ (and a CCMAS score of .93). But the jitterplot in Fig. 3 and signal detection scores (e.g., high false alarm rate at maximum informedness) show this is a mischaracterisation of this unit. In the same way, the Network Dissection method identified many object detectors in VGG-16 and GoogLeNet CNNs, but the jitterplots in Fig. 5 (and precision scores) show that this conclusion is unjustified. For additional problems with the CCMAS score see Figure 5 in Appendix C. Similarly, the images generated by Activation Maximization also provided a misleading estimate of selectivity given that interpretable images were associated with very low selectivity scores. This has led to confusions that have delayed theoretical progress. For example, describing single units in CNNs as “object detectors” in response to high precision measures (Zhou et al.) suggests similar types of representations are learned in CNNs and RNNs. Indeed, we are not aware of anyone in the machine learning community who has even considered the hypothesis that selectivity is reduced in CNNs compared RNNs. Our findings highlight the contrasting results.
+
+What should be made of the finding that localist representations are sometimes learned in RNNs (units with perfect specificity and recall), but not in AlexNet and related CNNs? The failure to observe localist units in the hidden layers of these CNNs is consistent with Bowers et al. (2014)’s claim that these units emerge in order to support the co-activation of multiple items at the same time in short-term memory. That is, localist representations may be the solution to the superposition catastrophe, and these CNNs only have to identify one image at a time. The pressure to learn highly selective representations in response to the superposition constraint may help explain the reports of highly selective neurons in cortex given that the cortex needs to co-activate multiple items at the same time in order to support short-term memory (Bowers et al., 2016).
+
+Note, the RNNs that learned localist units were very small in scale compared to CNNs we have studied here, and accordingly, it is possible that the contrasting results reflect the size of the networks rather than the superposition catastrophe per se. Relevant to this issue a number of authors have reported the existence of selective units in larger RNNs with long-short term memory (LSTM) units (Karpathy et al., 2016; Radford et al., 2017; Lakretz et al., 2019; Na et al., 2019). Indeed, Lakretz et al. (2019) use the term ‘grandmother cell’ to describe the units they observed. It will be interesting to apply our measures of selectivity to these larger RNNs and see whether these units are indeed ‘grandmother units’.
+
+It should also be noted that there are recent reports of impressively selective representations in Generative Adversarial Networks (Bau et al., 2019) and Variational Autoencoders (Burgess et al., 2018) where the superposition catastrophe is not an issue. Again, it will be interesting to assess the selectivity of these units according to signal detection measures in order to see whether there are additional computational pressures to learn highly selective or even grandmother cells. We will be exploring these issues in future work.
+
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+
+# APPENDIX
+
+# A METHODOLOGICAL DETAILS FOR THE BEHAVIORAL EXPERIMENT
+
+One hundred generated images were made for every unit in layers conv5, fc6 and fc8 in AlexNet, as in Nguyen et al. (2017), and displayed as $1 0 \mathrm { x } 1 0$ image panels (figures A4 and Figures A2 and A3). A total of 3,299 image panels were used in the experiment (995 fc8, 256 conv5, and 2048 randomly selected fc6 image panels) and were divided into 64 counterbalanced lists of 51 or 52 (4 conv5, 15 or $1 6 \mathsf { f c } 8$ and 32 fc6). 51 of the lists were assigned to 5 participants and 13 lists were assigned to 6 participants.
+
+To test the interpretability of these units, paid volunteers were asked to look at image panels and asked if the images had an object / animal or place in common. If the answer was ‘yes’, they were asked to name that object simply (i.e. fish rather than goldfish). Analyses of common responses was carried out for any units where over $80 \%$ of humans agreed there was an object present, by reading the human responses and comparing them to both each other and to the output classes. Agreement was taken if the object was the same rough class. For example, ‘beer’, ‘glass’, and ‘drink’ were all considered to be in agreement in the general object of ‘drink’, and in agreement with both the classes of ‘wine glass’ and ‘beer’ as these classes were also general drink classes (this is an actual example, most responses were more obvious and required far less interpretation than that). Participants were given six practice trials, each with panels of 20 images before starting the main experiment. Practice trials included images that varied in their interpretability.
+
+![](images/1a017687051cbc18f9a0a5a59b15c224ba0a7883a3aa8fe5700a6c6bba0d47a1.jpg)  
+Figure A1: Example screen from the identification task shown to participants as part of the instructions. The images included on this practice trial are ImageNet2012 images, not AM images.
+
+Some examples of the $1 0 \mathrm { x } 1 0$ grids of activation maximisation images that were presented to participants are shown in Figures A2, A3 and A4. Figure A2 shows an example from conv5 that human participants agreed had no obvious object in common (although there are repeated shape motifs, the participants were specifically asked for objects, and not abstract concepts like shape or color. Figure A3 is also from the conv5 and was judged by participants as some images containing ‘dogs’.
+
+![](images/86890cf8b17928ab35c92142a67999a22d2b41dcd29aa3e79dc1617600be0818.jpg)  
+Figure A4 is the AM images for the supposed ‘butterfly detector’ unit example discussed in the paper.   
+Figure A2: Example activation maximisation images for unit conv5.65. These images were judged by humans to not contain any interpretable objects in common (although the reader may agree that there are some shape and colour similarities in the images).
+
+![](images/e3976498e1d5e03e9eda7017306d8b6048a0f3a2d4ccc174532d3143bb3d0631.jpg)  
+Figure A3: Example activation maximisation images for unit conv5.183. These images were judged by humans to contain some interpretable images, in this case, of the type ‘dogs’.
+
+![](images/53b865e27c047b3029870a705da9f697a5bdda0957ce95f1cc0927947dbd92ed.jpg)  
+Figure A4: Example activation maximisation images for unit fc6.1199. Whilst there are some butterfly wing shapes in these images, there are not obvious butterflies. N.B. the second highest activating class for this unit is ladybirds, and there are some orange round shapes that could conceivably be ladybug-alikes.
+
+# B FURTHER DATA ON THE SELECTIVITY MEASURES ACROSS ALEXNET
+
+Table A1 gives the highest values of CCMAS and precision for each layer in AlexNet. It is worth noting that the highest CCMAS score of all hidden units was .94 (fc7.31), which at first glance suggests that this unit is close to ‘perfect’ selectivity. However, this unit only has low a precision score of $11 \%$ . In other words, although the mean activation for the given class is very high relative to the mean of all other activations (high CCMAS), the proportion of items from that class in the 100 most active items is low (low precision). See appendix Sec. C for discussion of how this occurs and Fig. A5(a) for an illustrative example.
+
+Table A1: The units with the highest CCMAS and precision scores in AlexNet. Unit fc6.1199 was displayed in Fig. 3.   
+
+<table><tr><td>LAYER.UNIT CCMAS</td><td>Precision</td></tr><tr><td>Top CCMAS units</td><td></td></tr><tr><td>output.322 0.991</td><td>100%</td></tr><tr><td>fc7.31 0.94</td><td>11%</td></tr><tr><td>fc6.582 0.93</td><td>1%</td></tr><tr><td>conv5.78 0.75</td><td>5%</td></tr><tr><td></td><td></td></tr><tr><td>Top precision units</td><td></td></tr><tr><td>output.0 0.99</td><td>100%</td></tr><tr><td>fc7.255 0.90</td><td>97%</td></tr><tr><td>fc6.1199 0.92</td><td>95%</td></tr><tr><td>conv5.0 0.55</td><td>77%</td></tr></table>
+
+Table A2 shows positive correlations between four of the selectivity measures used. There are moderate positive correlations between precision and CCMAS; and precision and Recall at $9 5 \%$ precision. The other correlations between selectivity measures have weak positive correlations. All four selectivity measures are negatively correlated with the number of classes present in the $1 0 0 \mathrm { m o s t }$ active items, that is, the more selective the unit, the fewer classes will be represented in the most active 100 items.
+
+Table A2: The correlations between the different measures. (All $p$ ’s $< . 0 0 1$ )   
+
+<table><tr><td></td><td>CCMAS</td><td>recalll0.95</td><td>Max. Inf.</td><td>No.classes in top100</td></tr><tr><td>precision</td><td>0.38</td><td>0.30</td><td>0.15</td><td>-0.68</td></tr><tr><td>CCMAS</td><td></td><td>0.09</td><td>0.14</td><td>-0.47</td></tr><tr><td>recall0.95</td><td></td><td></td><td>0.10</td><td>-0.19</td></tr><tr><td>Max. Inf.</td><td></td><td></td><td></td><td>-0.22</td></tr></table>
+
+# C FURTHER ISSUES WITH THE CCMAS MEASURE
+
+The CCMAS measure is based on comparing the mean activation of a category with the mean activation for all other items, and this is problematic for a few reasons. First, in many units a large proportion of images do not activate a unit at all. For instance, our butterfly ‘detector’ unit fc6.1199 has a high proportion of images with an activation of 0.0 (see figure 3). Indeed, the inset on the middle figure shows that the distribution can be better described by exponential-derived fits rather than a Gaussian. This means that the CCMAS selectivity is heavily influenced by the the proportion of images that have an activation value of zero (or close to zero). This can lead to very different estimates of selectivity for CCMAS and precision or localist selectivity, which are driven by the most highly activated items. In A5 we generate example data to highlight ways in which CCMAS score may be non-intuitive. In subplot (a) we demonstrate that a unit can have a CCMAS score of of 1.0 despite only a single item activating the unit. The point that we wish to emphasise is that a high CCMAS score does not necessarily imply selectivity for a given class, but might in fact relate to selectivity for a small subset of items from a given class, and this is especially true when a unit’s activation is sparse (many items do not activate the unit). However, the reverse can also be true. In subplot (c) we demonstrate that a unit can have a very low CCMAS score of .06 despite all of the most active items being from the same class.
+
+In addition, if the CCMAS provided a good measure of a unit’s class selectivity, then one should expect that a high measure of selectivity for one class would imply that the unit is not highly selective for other classes. However, the CCMAS score for the most selective category and the second most selective category (CCMAS2) were similar across the conv5, fc6 and fc7. layers, with the mean CCMAS scores .491, .844, and .848, and the CCMAS2 scores .464, .821, .831. For example, unit fc7.0 has a CCMAS of .813 for the class ‘maypole’, and a CCMAS2 score of .808 for ‘chainsaw’ (with neither of these categories corresponding ‘orangutan’ that had the highest precision of score of $14 \%$ ).
+
+![](images/77025fa9e4dcfbee2b67b179b8eba0ebd1ff4a2f835efa476052be3c934048af.jpg)  
+Figure A5: Example of where the CCMAS does not match intuitive understandings of selectivity. Generated example data: (a) If a unit is off to all but a single image from a large class of objects, the CCMAS for that class is 1 (maximum possible selectivity). (b) An archetypal ‘grandmother’ cell (GMC), where the unit is strongly activated to all members of one class and off to everything else. The CCMAS is the same for (b) as for (a) although the precision is vastly different. (c): If a unit has high activations for all classes, but one class (black squares) is 0.1 more than all others (coloured circles), the CCMAS is very low (0.06) despite being $\% 1 0 0$ precision. The generated examples are for 10 classes of 100 items
+
+# D TESTING UNITS IN OTHER MODELS
+
+To investigate units claimed by Zhou et al. (2018a) to be object detectors, we focus on units from a single layer that are reported to be ‘bus detectors’, that is, units with an $\mathrm { I o U } \geq . 0 4$ . We used the first 100 images per class from the ImageNet 2012 dataset as our test data. There are three classes of bus in this dataset: ‘n04146614 school bus’, ‘n04487081 trolleybus, trolley coach, trackless trolley’, ‘n03769881 minibus’, and this corresponded to 300 items out of 100000 images. Data for all bus unit detectors for VGG trained on places 365 are shown in table A3; for GoogLeNet trained on places 365 in table A4; and for GoogLeNet trained on ImageNet are shown in table A5. Note that for all units there are very few busses with activation at zero and that the mean activation for busses is higher than the mean activation for non-busses. However, all precision scores are all below .6, meaning that of the 100 items that most strongly activated the unit, at least 40 of them were not busses. Together these results suggests that whilst these units demonstrate some sensitivity to busses, they show poor specificity for busses (e.g., high false-alarm rate).
+
+<table><tr><td>unit</td><td>IoU</td><td>no.ax&gt;0 x∈A</td><td>no.ax&gt;0 x∈-A</td><td>μA</td><td>μ-A precision</td><td>CCMAS</td></tr><tr><td>conv5_3191</td><td>.15</td><td>99.0%</td><td>63.9%</td><td>131.9 16.1</td><td>.45</td><td>.78</td></tr><tr><td>conv5_320</td><td>.15</td><td>99.0%</td><td>49.1%</td><td>157.6</td><td>15.2 .53</td><td>.82</td></tr><tr><td>conv5_3333</td><td>.08</td><td>99.0%</td><td>71.4%</td><td>101.7</td><td>17.5 .24</td><td>.71</td></tr><tr><td>conv5_3145</td><td>.07</td><td>97.3%</td><td>61.7%</td><td>75.5</td><td>12.5 .19</td><td>.72</td></tr><tr><td>conv5_3113</td><td>.06</td><td>97.4%</td><td>41.0%</td><td>62.8</td><td>9.1 .12</td><td>.75</td></tr><tr><td>conv5_3443</td><td>.04</td><td>95.3%</td><td>38.2%</td><td>59.3</td><td>8.1 .12</td><td>.76</td></tr><tr><td>conv5_3131</td><td>.04</td><td>93.7%</td><td>22.3%</td><td>54.0</td><td>5.86 .08</td><td>.80</td></tr></table>
+
+Table A3: Selectivity measures for VGG-16, trained on Places-365, top convolutional layer units identified by Zhou et al. (2018a) as object detectors. Standard errors not shown for space, but were below $\pm 5$ . The IoU is from Zhou et al. (2018a)’s network dissection method. no. $a _ { x } { > } 0$ and no. $a _ { x } { > } 0$ $x \in A$ refer to the proportion of activations that were greater than zero for busses and non-busses respectively. $\mu _ { A }$ and $\mu { \neg } A$ are the class means for busses and non busses respectively.
+
+<table><tr><td>unit</td><td>IoU</td><td>correct</td><td>no.ax&gt;0 x∈A</td><td>no.ax&gt;0 x∈-A</td><td>μA</td><td>μ-A</td><td>precision</td><td>CCMAS</td></tr><tr><td>incep4e824</td><td>.17</td><td>T</td><td>100.0</td><td>91.4</td><td>41.0</td><td>11.8</td><td>.27</td><td>.55</td></tr><tr><td>incep4e745</td><td>.13</td><td>T</td><td>98.3</td><td>74.8</td><td>34.8</td><td>11.4</td><td>.06</td><td>.51</td></tr><tr><td>incep4e791</td><td>.11</td><td>T</td><td>98.3</td><td>71.4</td><td>32.7</td><td>5.3</td><td>.41</td><td>.72</td></tr><tr><td>incep4e194</td><td>.11</td><td>F</td><td>100.0</td><td>85.3</td><td>26.6</td><td>8.8</td><td>.02</td><td>.51</td></tr><tr><td>incep4e82</td><td>.11</td><td>T</td><td>100.0</td><td>97.3</td><td>26.7</td><td>10.9</td><td>.14</td><td>.42</td></tr><tr><td>incep4e736</td><td>.11</td><td>T</td><td>100.0</td><td>78.8</td><td>38.7</td><td>9.9</td><td>.05</td><td>.59</td></tr><tr><td>incep4e663</td><td>.10</td><td>F</td><td>96.0</td><td>38.0</td><td>33.4</td><td>3.7</td><td>.15</td><td>.80</td></tr><tr><td>incep4eg4</td><td>.10</td><td>T</td><td>100.0</td><td>91.6</td><td>38.3</td><td>9.5</td><td>.35</td><td>.60</td></tr><tr><td>incep4e772</td><td>.08</td><td>F</td><td>97.3</td><td>54.6</td><td>21.7</td><td>5.2</td><td>.00</td><td>.61</td></tr><tr><td>incep4e113</td><td>.08</td><td>F</td><td>100.0</td><td>88.0</td><td>24.9</td><td>9.2</td><td>.02</td><td>.46</td></tr><tr><td>incep4e708</td><td>.06</td><td>F</td><td>100.0</td><td>85.1</td><td>29.7</td><td>9.1</td><td>.02</td><td>.53</td></tr><tr><td>incep4eg01</td><td>.06</td><td>F</td><td>100.0</td><td>64.5</td><td>35.2</td><td>6.4</td><td>.14</td><td>.69</td></tr><tr><td>incep4e199</td><td>.06</td><td>F</td><td>99.7</td><td>92.2</td><td>21.5</td><td>7.7</td><td>.09</td><td>.47</td></tr><tr><td>incep4eg</td><td>.05</td><td>F</td><td>99.7</td><td>83.5</td><td>18.5</td><td>7.3</td><td>.01</td><td>.43</td></tr><tr><td>incep4e121</td><td>.05</td><td>F</td><td>100.0</td><td>90.4</td><td>17.9</td><td>8.9</td><td>.01</td><td>.34</td></tr><tr><td>incep4e622</td><td>.05</td><td>T</td><td>96.0</td><td>65.0</td><td>27.5</td><td>6.4</td><td>.20</td><td>.62</td></tr><tr><td>incep4e97</td><td>.04</td><td>T</td><td>99.3</td><td>86.4</td><td>21.1</td><td>9.3</td><td>.04</td><td>.39</td></tr></table>
+
+Table A4: Selectivity measures for GoogLeNet, trained on Places-365, layer inception4e units identified by Zhou et al. (2018a) as object detectors. Standard errors not shown for space, but were below $\pm 2$ . The IoU is from Zhou et al. (2018a)’s network dissection method. A units is marked as correct if there was a single bus in the 4 example pictures on the website (http://netdissect. csail.mit.edu/dissect/googlenet_places365/), and false if not. Where units are False, this might suggest that the units were responding to ’bus like’ features in none bus objects. no. $a _ { x } { > } 0$ and no $. a _ { x } { > } 0 \ x \in A$ refer to the proportion of activations that were greater than zero for busses and non-busses respectively. $\mu _ { A }$ and $\mu { \_ } A$ are the class means for busses and non busses respectively.
+
+Table A5: Selectivity measures for GoogLeNet, trained on ImageNet, layer inception4e units units identified by Zhou et al. (2018a) as object detectors. Standard errors not shown for space, but were below $\pm 2$ . A units is marked as correct if there was a single bus in the 4 example pictures on the website (http://netdissect.csail.mit.edu/dissect/googlenet_imagenet/), and false if not. This might suggest that the units were responding to ’bus like’ features in none bus objects.   
+
+<table><tr><td>unit</td><td>IoU</td><td>correct</td><td>no.ax&gt;0 x∈A</td><td>no.ax&gt;0 x∈-A</td><td>μA</td><td>μ-A</td><td>precision</td><td>CCMAS</td></tr><tr><td>incep4e494</td><td>.11</td><td>F</td><td>99.0</td><td>82.4</td><td>72.5</td><td>22.8</td><td>.00</td><td>.52</td></tr><tr><td>incep4e828</td><td>.10</td><td>T</td><td>100.0</td><td>72.6</td><td>109.4</td><td>17.6</td><td>.45</td><td>.72</td></tr><tr><td>incep4e569</td><td>.10</td><td>T</td><td>99.7</td><td>85.9</td><td>74.9</td><td>20.0</td><td>.05</td><td>.58</td></tr><tr><td>incep4e384</td><td>.10</td><td>T</td><td>100.0</td><td>71.6</td><td>67.0</td><td>18.5</td><td>.00</td><td>.57</td></tr><tr><td>incep4e455</td><td>.09</td><td>T</td><td>99.7</td><td>89.6</td><td>69.1</td><td>14.3</td><td>.3</td><td>.66</td></tr><tr><td>incep4e579</td><td>.09</td><td>T</td><td>100.0</td><td>97.0</td><td>91.5</td><td>26.0</td><td>.23</td><td>.56</td></tr><tr><td>incep4e331</td><td>.08</td><td>T</td><td>98.0</td><td>75.5</td><td>51.0</td><td>11.8</td><td>.12</td><td>.62</td></tr><tr><td>incep4e582</td><td>.08</td><td>T</td><td>100.0</td><td>83.4</td><td>125.7</td><td>21.95</td><td>.58</td><td>.70</td></tr><tr><td>incep4e498</td><td>.07</td><td>T</td><td>97.7</td><td>77.2</td><td>73.5</td><td>15.0</td><td>.52</td><td>.66</td></tr><tr><td>incep4e534</td><td>.07</td><td>F</td><td>99.3</td><td>81.2</td><td>62.7</td><td>19.</td><td>.02</td><td>.53</td></tr><tr><td>incep4e693</td><td>.07</td><td>T</td><td>98.7</td><td>91.2</td><td>75.4</td><td>22.3</td><td>.15</td><td>.54</td></tr><tr><td>incep4e673</td><td>.07</td><td>T</td><td>99.7</td><td>88.4</td><td>88.6</td><td>23.0</td><td>.33</td><td>.59</td></tr><tr><td>incep4e469</td><td>.06</td><td>T</td><td>98.7</td><td>78.1</td><td>34.7</td><td>14.6</td><td>.00</td><td>.41</td></tr><tr><td>incep4e207</td><td>.06</td><td>T</td><td>100.0</td><td>93.5</td><td>76.1</td><td>21.3</td><td>.07</td><td>.56</td></tr><tr><td>incep4e491</td><td>.06</td><td>F</td><td>99.0</td><td>74.5</td><td>41.1</td><td>13.7</td><td>.01</td><td>.50</td></tr><tr><td>incep4e645</td><td>.06</td><td>T</td><td>98.0</td><td>83.9</td><td>59.9</td><td>18.1</td><td>.20</td><td>.54</td></tr><tr><td>incep4e527</td><td>.06</td><td>F</td><td>100.0</td><td>91.5</td><td>58.0</td><td>21.7</td><td>.00</td><td>.46</td></tr><tr><td>incep4e511</td><td>.05</td><td>F</td><td>100.0</td><td>89.4</td><td>53.5</td><td>21.7</td><td>.00</td><td>.42</td></tr><tr><td>incep4e308</td><td>.05</td><td>F</td><td>100.0</td><td>89.4</td><td>53.5</td><td>21.7</td><td>.00</td><td>.42</td></tr><tr><td>incep4e541</td><td>.05</td><td>F</td><td>99.67</td><td>88.7</td><td>44.9</td><td>13.7</td><td>.00</td><td>.53</td></tr><tr><td>incep4e367</td><td>.05</td><td>T</td><td>97.3</td><td>80.3</td><td>37.7</td><td>15.4</td><td>.02</td><td>.42</td></tr><tr><td>incep4e665</td><td>.05</td><td>T</td><td>100.0</td><td>82.45</td><td>107.2</td><td>21.0</td><td>.33</td><td>.67</td></tr><tr><td>incep4e532</td><td>.05</td><td>T</td><td>100.0</td><td>91.5</td><td>52.9</td><td>22.4</td><td>.05</td><td>.41</td></tr><tr><td>incep4e297</td><td>.04</td><td>T</td><td>99.7</td><td>90.2</td><td>48.2</td><td>17.9</td><td>.00</td><td>.46</td></tr><tr><td>incep4e480</td><td>.04</td><td>T</td><td>100.0</td><td>92.9</td><td>69.4</td><td>21.4</td><td>.02</td><td>.53</td></tr></table>

parse/train/Skltqh4KvB/Skltqh4KvB_content_list.json

@@ -0,0 +1,1046 @@
+[
+    {
+        "type": "text",
+        "text": "ARE THERE ANY ‘OBJECT DETECTORS’ IN THE HIDDEN LAYERS OF CNNS TRAINED TO IDENTIFY OBJECTS OR SCENES? ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            821,
+            170
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            195,
+            398,
+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            261,
+            544,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Various methods of measuring unit selectivity have been developed with the aim of better understanding how neural networks work. But the different measures provide divergent estimates of selectivity, and this has led to different conclusions regarding the conditions in which selective object representations are learned and the functional relevance of these representations. In an attempt to better characterize object selectivity, we undertake a comparison of various selectivity measures on a large set of units in AlexNet, including localist selectivity (Bowers et al., 2014), precision (Zhou et al., 2015), class-conditional mean activity selectivity (CCMAS) (Morcos et al., 2018), network dissection (Zhou et al., 2018a), the human interpretation of activation maximization (AM) images, and standard signaldetection measures. We find that the different measures provide different estimates of object selectivity, with precision and CCMAS measures providing misleadingly high estimates. Indeed, the most selective units had a poor hit-rate or a high falsealarm rate (or both) in object classification, making them poor object detectors. We fail to find any units that are even remotely as selective as the ‘grandmother cell’ units reported in recurrent neural networks. In order to generalize these results, we compared selectivity measures on a few units in VGG-16 and GoogLeNet trained on the ImageNet or Places-365 datasets that have been described as ‘object detectors’. Again, we find poor hit-rates and high false-alarm rates for object classification. ",
+        "bbox": [
+            233,
+            290,
+            766,
+            568
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            593,
+            336,
+            609
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There have been recent attempts to understand how neural networks (NNs) work by analyzing hidden units one-at-a-time using various measures such as localist selectivity (Bowers et al., 2014), class-conditional mean activity selectivity (CCMAS) (Morcos et al., 2018), precision (Zhou et al., 2015), network dissection (Zhou et al., 2018a), and activation maximization (AM) (Erhan et al., 2009). These measures are all taken to provide evidence that some units respond highly selectively to categories of objects under some conditions. Not only are these findings surprising given the widespread assumption that NNs only learn highly distributed and entangled representations, they raise a host of questions, including the functional importance of these selective representations (Zhou et al., 2018b), the conditions in which they are learned (e.g., Morcos et al., 2018), and the relation between these representations and the selective neurons observed in cortex (Bowers, 2009). ",
+        "bbox": [
+            174,
+            625,
+            825,
+            763
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To answer these question, it is necessary to have a better understanding of what these metrics actually measure, and how they relate to one another. Accordingly, we directly compare these measures of selectivity on the same set of units as well as adopt standard signal-detection measures in an attempt to provide better measures of single-unit selectivity to object category. In addition, to provide a more intuitive assessment of selectivity, we report jitterplots for a few of the most selective units that visually display how the unit responds to the different image categories. We focus on AlexNet (Krizhevsky et al., 2012) trained on ImageNet (Deng et al., 2009) because many authors have studied the selectivity of single hidden units in this model using a range of quantitative (Zhou et al., 2018a; 2015) and qualitative (Nguyen et al., 2017; Yosinski et al., 2015; Simonyan et al., 2013) methods. But we also compare different selectivity measures on specific units in VGG-16 (Simonyan and Zisserman, 2014) and GoogLeNet (Szegedy et al., 2015) trained on the the ImageNet and Places-365 datasets that were characterized by Zhou et al. (2018a) as “object detectors” based on their Network Dissection method (Zhou et al., 2018a). Our main findings are: ",
+        "bbox": [
+            174,
+            771,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. The precision and CCMAS measures are misleading with near-maximum selectivity scores associated with units that strongly respond to many different image categories. By contrast, the signal-detection measures more closely capture the level of selectivity displayed in the jitterplots (Sec. 3.1).   \n2. Units with interpretable AM images do not correspond to highly selective representations (Sec. 3.2).   \n3. The Network Dissection method also provides a misleading measure for “object detectors” (Sec. 3.3). ",
+        "bbox": [
+            207,
+            143,
+            825,
+            265
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In one line of research, Bowers et al. (2014; 2016) assessed the selectivity of single hidden units in recurrent neural networks (RNNs) designed to model human short-term memory. They reported many ‘localist’ or ‘grandmother cell’ units that were $100 \\%$ selective for specific letters or words, where all members of the selective category were more active than and disjoint from all non-members, as can be shown in jitterplots (Berkeley et al., 1995) (see Fig. 1 for a unit selective to the letter ‘j’). The authors argued that the network learned these representations in order to co-activate multiple letters or words at the same time in short-term memory without producing ambiguous blends of overlapping distributed patterns (the so-called ‘superposition catastrophe’). Consistent with this hypothesis, localist units did not emerge when the model was trained on letters or words one-at-a-time (Bowers et al., 2014) (see Fig. 1 for an example of a non-selective unit). ",
+        "bbox": [
+            173,
+            277,
+            826,
+            416
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In parallel, researchers have reported selective units in the hidden layers of various CNNs trained to classify images into one of multiple categories (Zhou et al., 2015; Morcos et al., 2018; Zeiler and Fergus, 2014; Erhan et al., 2009), for a review see Bowers (2017). For example, Zhou et al. (2015) assessed the selectivity of units in the pool5 layer of two CNNs trained to classify images into 1000 objects and 205 scene categories, respectively. They reported many highly selective units that they characterized as ‘object detectors’ in both networks. Similarly, Morcos et al. (2018) reported that CNNs trained on CIFAR-10 and ImageNet learned many highly selective hidden units, with CCMAS scores approaching the maximum of 1.0. These later findings appear to be inconsistent with Bowers et al. (2016) who failed to observe selective representations in fully connected NNs trained on stimuli one-at-a-time (see Fig. 1), but the measures of selectivity that have been applied across studies are different, and accordingly, it is difficult to directly compare results. ",
+        "bbox": [
+            174,
+            422,
+            825,
+            577
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A better understanding of the relation between selectivity measures is vital given that different measures are frequently used to address similar issues. For example, both the human interpretability of generated images (Le, 2013) and localist selectivity (Bowers et al., 2014) have been used to make claims about ‘grandmother cells’, but it is not clear whether they provide similar insights into unit selectivity. Similarly, based on their precision metric, Zhou et al. (2015) claim that the object detectors learned in CNNs play an important role in identifying specific objects, whereas Morcos et al. (2018) challenge this conclusion based on their finding that units with high CCMAS measures were not especially important in the performance of their CNNs and concluded: “...it implies that methods for understanding neural networks based on analyzing highly selective single units, or finding optimal inputs for single units, such as activation maximization (Erhan et al., 2009) may be misleading\". This makes a direct comparison between selectivity measures all the more important. ",
+        "bbox": [
+            174,
+            583,
+            825,
+            736
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In order to directly compare and have a better understanding of the different selectivity measures we assessed (1) localist, (2) precision, and (3) CCMAS selectivity of the conv5, fc6, and fc7 of AlexNet trained on ImageNet, and in addition, we employed a range of signal detection methods on these units, namely, (4) recall with $100 \\%$ and $9 5 \\%$ precision, (5) maximum informedness, (6) specificity at maximum informedness , and (7) recall (also called sensitivity) at maximum informedness, and false alarm rates at maximum informedness (described in Sec. 2). We also assessed the selectivity of a few units in VGG-16 and GoogLeNet models trained on the ImageNet and Places-365 dataset that were highly selective according to the Network Dissection method (Zhou et al., 2018a). We show that the precision and CCMAS measures often provide misleadingly high estimates of object selectivity compared to other measures, and we do not find any units that can be reasonably described as ‘object detectors’ given that the most selective units show a low hit-rate or a high false-alarm rate (or both) when classifying images. At best, the most selective units in CNNs are sensitive to some unknown feature that is weakly associated with the class in question. ",
+        "bbox": [
+            174,
+            743,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/79eb40ad904e0968407b2502f5d29a63932ec8a65908516624ad8dd72655124e.jpg",
+        "image_caption": [
+            "Figure 1: Examples of selectivity measures used. Top left: jitterplot of unit 113 in an RNN (under the superposition constraint) selective to the letter ‘j’ (Bowers et al., 2016). Top middle: jitterplot of a non-selective unit 160 found in an RNN trained on words one-at-a-time from (Bowers et al., 2016). Top right: Activation maximization image of unit conv59 AlexNet that resembles a lighthouse (Nguyen et al., 2016). Bottom: highest-activation images for a ‘lamp’ detector with $84 \\%$ precision in the layer conv5 of AlexNet; from (Zhou et al., 2015). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            99,
+            813,
+            247
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In addition to these quantitative measures and jitterplots we assessed selectivity with a common qualitative measure, namely, human interpretation of images generated by a state-of-the-art activation maximization (AM) method (Nguyen et al., 2017). AM images are generated to strongly activate individual units, and some of them are interpretable by humans (e.g., a generated image that looks like a lighthouse, see Fig. 1). For the first time, we systematically evaluated the interpretability of the AM images and compare these ratings with the selectivity measures for corresponding units. We show that the few hidden units with interpretable AM images are not highly selective. ",
+        "bbox": [
+            174,
+            371,
+            825,
+            469
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 METHODS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            492,
+            290,
+            508
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Network and Dataset All ${ \\sim } 1 . 3 \\mathbf { M }$ photos from the ImageNet ILSVRC 2012 dataset (Deng et al., 2009) were cropped to $2 7 7 \\times 2 7 7$ pixels and classified by the pre-trained AlexNet CNN (Krizhevsky et al., 2012) shipped with Caffe (Jia et al., 2014), resulting in 721,536 correctly classified images. Once classified, the images are not re-cropped nor subject to any changes. We analyzed the fully connected (fc) layers: fc6 and fc7 (4096 units), and the top convolutional layer conv5 which has 256 filters. We only recorded the activations of correctly classified images. The activation files are stored in .h5 format and will be available at http://anonymizedForReview. We randomly selected 233 conv5, 2738 fc6, 2239 fc7 units for analysis. ",
+        "bbox": [
+            174,
+            526,
+            825,
+            638
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Localist selectivity Following Bowers et al. (2014), we define a unit to be localist for class $A$ if the set of activations for class $A$ was higher and disjoint with those of $\\neg A$ . Localist selectivity is easily depicted with jitterplots (Berkeley et al., 1995) in which a scatter plot for each unit is generated (see Figs. 1 and 3). Each point in a plot corresponds to a unit’s activation in response to a single image, and only correctly classified images are plotted. The level of activation is coded along the $x$ -axis, and an arbitrary value is assigned to each point on the $y$ -axis. ",
+        "bbox": [
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+            645,
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+    {
+        "type": "text",
+        "text": "Precision Precision refers to the proportion of items above some threshold from a given class. The precision method of finding object detectors involves identifying a small subset of images that most strongly activate a unit and then identifying the critical part of these images that are responsible for driving the unit. Zhou et al. (2015) took the 60 images that activated a unit the most strongly and asked independent raters to interpret the critical image patches (e.g., if 50 of the 60 images were labeled as ‘lamp’, the unit would have a precision index of 50/60 or $83 \\%$ ; see Fig. 1). Object detectors were defined as units with a precision score $> 7 5 \\%$ : they reported multiple such detectors. Here, we approximate this approach by considering the 60 images that most strongly activate a given unit and assess the highest percentage of images from a given output class. ",
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+    {
+        "type": "text",
+        "text": "CCMAS Morcos et al. (2018) introduced a selectivity index called the Class-conditional Mean Activation Selectivity (CCMAS). The CCMAS for class $A$ compares the mean activation of all images in class $A$ , $\\mu _ { A }$ , with the mean activation of all images not in class $A$ , $\\mu { \\neg } A$ , and is given by: $\\left( \\mu _ { A } ^ { \\setminus } - \\mu _ { \\neg A } \\right) / \\left( \\mu _ { A } + \\mu _ { \\neg A } \\right)$ . Here, we assessed class selectivity for the highest mean activation class. ",
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+    {
+        "type": "text",
+        "text": "Activation Maximization We harnessed an activation maximization method called Plug & Play Generative Networks (Nguyen et al., 2017) in which an image generator network was used to generate images (AM images) that highly activate a unit in a target network. We used the public code released by Nguyen et al. (2017) and their default hyperparameters.1 We generated 100 separate images that maximally activated each unit in the conv5, fc6, and fc8 layers of AlexNet and asked participants to judge whether they could identify any repeating objects, animals, or places in images in a behavioral experiment (Sec. 3.2). Readers can test themselves at: https://research.sc/ participant/login/dynamic/63907FB2-3CB9-45A9-B4AC-EFFD4C4A95D5 ",
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+    {
+        "type": "text",
+        "text": "Recall with perfect and $95 \\%$ precision Recall with perfect and $9 5 \\%$ precision are related to localist selectivity except that they provide a continuous rather than discrete measure. For recall with perfect precision we identified the image that activated a given unit the most and counted the number of images from the same class that were more active than all images from all other classes. We then divided this result by the total number of correctly identified images from this class. A recall with a perfect precision score of 1 is equivalent to a localist representation. Recall with a $9 5 \\%$ precision allows $5 \\%$ false alarms. ",
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+    {
+        "type": "text",
+        "text": "Maximum informedness Maximum informedness identifies the class and threshold where the highest proportion of images above the threshold and the lowest proportion of images below the threshold are from that class (Powers, 2011). The informedness is computed for each class at each threshold, with the highest value selected. Informedness summarises the diagnostic performance of unit for a given class at a certain threshold based on the recall [True Positives / (True Positives $^ +$ False Negatives)] and specificity [True Negatives / (True Negatives $^ +$ False Positives)] in the formula [informedness $=$ recall $^ +$ specificity − 1] (Powers, 2011). ",
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+    {
+        "type": "text",
+        "text": "Sensitivity or Recall at Maximum Informedness For the threshold and class selected by Maximum Informedness, recall (or hit-rate) is the proportion of items from the given class that are above the threshold. Also known as true postive rate. ",
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+    {
+        "type": "text",
+        "text": "Specificity at Maximum Informedness For the threshold and class selected by Maximum Informedness, the proportion of items that are not from the given class that are below the threshold. Also known as true negative rate. ",
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+    {
+        "type": "text",
+        "text": "False Alarm Rate at Maximum Informedness For the threshold and class selected by Maximum Informedness, the proportion of items that are not from the given class that are above the threshold. ",
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+    {
+        "type": "text",
+        "text": "Network Dissection To assess the selectivity of a unit in the Network Dissection technique, Zhou et al. (2018a) compute the Intersection over Union (IoU) of an annotated input image $L _ { c }$ , for the set of all ‘concepts’ $c$ and a spatial activation map, $M _ { k }$ , of where a unit $k$ is. A unit $k$ is taken as a detector for concept $c$ if its $\\mathrm { I o U } _ { k , c }$ exceeds a pre-defined threshold $T$ . See Zhou et al. (2018a) for more details. ",
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+        "type": "text",
+        "text": "3 RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 COMPARISON OF SELECTIVITY MEASURES IN ALEXNET ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The results from the various of selectivity measures applied to the conv5, fc6, and fc7 layers of AlexNet are displayed in Fig. 2a–i. We did not plot the localist selectivity as there were no localist ‘grandmother units’. The first point to note is that multiple units in the fc6 and fc7 layers had near $100 \\%$ precision scores and multiple units had CCMAS scores approaching 1. For example, in layer fc7, we found 14 units with a precision $> 0 . 9$ , and 1487 units with a $\\mathrm { C C M A S } > 0 . 9$ . The second point is that other measures provided much reduced estimates of selectivity. For example, the unit with the highest recall with a perfect precision score was only .08 (unit 255 responding to images of Monarch butterflies), and the unit with the top maximum informedness score (unit 3290 also responding to images of Monarch butterflies with a score of 0.91) had a false alarm rate above its optimal threshold $> 9 9 \\%$ (indeed the minimum false alarm rate was 0.96). ",
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+    {
+        "type": "text",
+        "text": "To illustrate the contrasting measures of selectivity consider unit $\\mathsf { f c 6 } _ { 1 1 9 9 }$ depicted in Fig. 3 that has a precision score of $98 \\%$ and a CCMAS score of .92. By Zhou et al.’s criterion, this is a ‘Monarch Butterfly’ detector (its precision score $> 7 5 \\%$ ). By contrast, the scatter plot and signal-detection scores show this is a mischaracterisation of this unit given that the false alarm rate at maximum informedness was greater than $9 9 \\%$ and the modal response to Monarch butterflies was zero. ",
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+    {
+        "type": "image",
+        "img_path": "images/f080f4e7d6eb4f52ab14011f1fc60eded3687247e4ebcd11ddb551530f0343ba.jpg",
+        "image_caption": [
+            "Figure 2: Different selectivity measures across the conv5, fc6, and fc7 layers of AlexNet. Red-line: median of data, top and bottom of box edges is the $2 5 ^ { \\mathrm { t h } }$ and $7 5 ^ { \\mathrm { t h } }$ percentile, whiskers extend to extreme edges of distribution not considered outliers and red crosses are outliers. Green points and dashed lines are the means of the distributions with standard errors. The high levels of selectivity observed with the precision and CCMAS measures are in stark contrast with the low levels of selectivity observed with the recall with perfect precision and high false-alarm rates at maximum informedness. "
+        ],
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+    {
+        "type": "text",
+        "text": "What level of selectivity is required before a unit can be considered an ‘object detector’ for a given category? In the end, this is a terminological point. On an extreme view, one might limit the term to the ‘grandmother units’ that categorize objects with perfect recall and specificity, or alternatively, it might seem reasonable to describe a unit as a detector for a specific object category if there is some threshold of activation that supports more hits than misses (the unit is strongly activated by the majority of images from a given category), and at the same time, supports more hits than false alarms (the unit is strongly activated by items from the given category more often than by items from other categories). Or perhaps a lower standard could be defended, but in our view, the term \"object detector\" suggests a higher level of selectivity than $8 \\%$ recall at perfect precision. That said, our results show that some units respond strongly to some (unknown) features that are weakly correlated with an object category. For instance, unit $\\mathsf { f c } 6 _ { 1 1 9 9 }$ is responding to features that occur more frequently in Monarch Butterflies than other categories. This can also be seen in a recent ablation study in which removing the most selective units tended to impair the CNN’s performance in identifying the corresponding object categories more than other categories (Zhou et al., 2018b). But again, the pattern of performance is not consistent with the units being labeled ‘object detectors’. ",
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+    {
+        "type": "image",
+        "img_path": "images/304a6ae45b0d36961bc7a3ece31444019196ee73cb8388a7c33d438ebf85af5d.jpg",
+        "image_caption": [
+            "Figure 3: Data for unit $\\mathsf { f c } 6 _ { 1 1 9 9 }$ . Left: activation jitterplot, black diamonds: Monarch butterfly images; grey circles: all other classes; white dashed line: threshold for the butterfly class maximum informedness; blue solid line: threshold for top 60 activations. Middle: histogram of activations of Monarch butterflies; red dashed line: threshold for the butterfly class maximum informedness; black solid line: threshold for top 60 activations. Inset: zoomed-in histogram of all activations across all ImageNet classes of unit $\\mathsf { f c 6 } _ { 1 1 9 9 }$ (N.B. this plot shows only the highest 121,586 activations; there are 596,734 activations at 0). There are Monarch butterfly images covering the whole range of values, with 72 images $5 . 8 \\%$ of the total) having an activation of 0. Right: example ImageNet images with activations of 0 (top), the mean, $3 9 . 2 { \\pm } 0 . 6 $ , (middle), and the maximum, 95, (bottom) of the range. Although the high precision score suggests that this unit is a butterfly detector this is misleading given there are butterfly images over the entire activation range (including 0). "
+        ],
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+    {
+        "type": "text",
+        "text": "3.2 HUMAN INTERPRETATION OF ACTIVATION MAXIMIZATION IMAGES FOR ALEXNET UNITS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Activation Maximization is one of the most commonly used interpretability methods for explaining what a single unit has learned in many artificial CNNs and even biological neural networks (see Nguyen et al. (2019) for a survey). Our behavioral experiment provides the first quantitative assessment of AM images and compares AM interpretability to other selectivity measures. ",
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+    {
+        "type": "table",
+        "img_path": "images/c073fe7b9d423fa350456cf674e7a781eae1e9e1f2b85ec37e018e946940a6bc.jpg",
+        "table_caption": [
+            "Table 1: Human judgements of whether AM images look like familiar objects in layers conv5, fc6, and fc8 in AlexNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">layer</td><td rowspan=\"2\">%‘yes’ responses</td><td rowspan=\"2\">% units ≥ 80% ‘yes’ response</td><td colspan=\"3\">% overlap between humans and:</td></tr><tr><td>humans</td><td>most active</td><td>CCMAS</td></tr><tr><td>conv5</td><td>(a) 21.7% (±1.1%)</td><td>(b) 4.3% (± 1.3%)</td><td>(c) 89.5% (±5.7%)</td><td>object (d) 34.1% (±14.4%)</td><td>class (e) 0%</td></tr><tr><td>fc6</td><td>21.0% (±0.4%)</td><td>3.1% (± 0.4%)</td><td>80.4% (±4.1%)</td><td>23.3% (±5.9%)</td><td>18.9% (±5.9%)</td></tr><tr><td>fc8</td><td>71.2% (±0.6%)</td><td>59.3% (±1.6%)</td><td>96.5% (±0.4%)</td><td>95.4% (±0.6%)</td><td>94.6% (±0.7%)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "We generated 100 AM images images for every unit in the layers conv5, fc6, and fc8 in AlexNet, as in Nguyen et al. (2017), and displayed them as $1 0 \\times 1 0$ -image panels. A total of 3,299 image panels were used in the experiment (995 fc8, 256 conv5, and 2048 randomly selected fc6 image panels) and were divided into 64 counterbalanced lists for testing. To assess the interpretability for these units as object detectors, 333 paid volunteers were asked to look at image panels and asked if the images had an object / animal or place in common. If the answer was ‘yes’, they were asked to write down a generic name for that object (e.g. “fish” rather than “goldfish”). Analyses of common responses was done for any units where over $80 \\%$ of humans agreed there was an object present. ",
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+        "type": "text",
+        "text": "The results are summarized in Table 1. Not surprisingly, the AM images for output fc8 units are the most human-recognizable as objects across the AlexNet layers $( 7 1 . 2 \\%$ ; Table 1a). In addition, when they were given a consistent interpretation, they almost always $( 9 5 . 4 \\%$ ; Table 1d) match the corresponding ImageNet category. By contrast, less than $5 \\%$ of units in conv5 or fc6 were associated with consistently interpretable images (Table 1b), and the interpretations only weakly matched the category associated with the highest-activation images or CCMAS selectivity (Table 1d–e). Apart from showing that there are few interpretable units in the hidden layers of AlexNet, our findings show that the interpretability of images does not imply a high level of selectivity given the signal-detection results (Fig. 2d–h). See Fig. 4 for an example of the types of images that participants rated as objects or non-objects. ",
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+    {
+        "type": "image",
+        "img_path": "images/65641a224bfb7b02c868516d88e7aa46f99affaf3cbcdcc65915c0b9ce1c5028.jpg",
+        "image_caption": [
+            "Figure 4: Example AM images that were either judged by all participants to contain objects (a–c) or to be uninterpretable as objects (d–f). The human label for unit $\\mathsf { c o n v } 5 _ { 1 8 3 }$ (a) was ‘dogs’; the most active image was of a ‘flat-coated retriever’; CCMAS class was ‘monitor’. For $\\mathsf { f c } 6 _ { 3 1 9 }$ (b), subjects reported ‘green peppers’ or ‘apples’ (all classified as the same broad class in our analysis); both the most active item and CCMAS class were ‘Granny Smith apples’. For $\\mathsf { f c } 8 _ { 9 6 9 }$ (c), humans suggested ‘beverage’ or ‘drink’; both the most active item and CCMAS class were ‘eggnog’. "
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+    {
+        "type": "text",
+        "text": "3.3 COMPARING SELECTIVITY MEASURES IN OTHER CNNS ",
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+    {
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+    {
+        "type": "text",
+        "text": "a. GoogLeNet on ImageNet inception4e494 precision: 0.0 CCMAS:0.52 ",
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+    {
+        "type": "image",
+        "img_path": "images/8bae5fc8b73350af8466d9f1e3ed096fbd11ddafd16049b57ef8c5a154b9cc4f.jpg",
+        "image_caption": [
+            "Figure 5: The units with with the highest Network Dissection scores for the category ‘bus’. The scatter plots, precision, and CCMAS scores all indicate a low selectivity for this category. blue squares: ‘school bus’; red pentagons: ‘trolleybus’; green stars: ‘minibus’; grey circles: other classes. "
+        ],
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+    {
+        "type": "text",
+        "text": "b. GoogLeNet on Places-365 inception4e824 precision: 0.27 CCMAS: 0.55 ",
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+    {
+        "type": "text",
+        "text": "c. VGG-16 on Places-365   \nconv5_320   \nprecision: 0.53   \nCCMAS: 0.82 ",
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+    {
+        "type": "equation",
+        "img_path": "images/90a38e34f4e066eb841fea01836648d01c88385624dc58a420fafc8bf7647d76.jpg",
+        "text": "$$\n\\mu _ { A } = 7 2 . 5 0 \\mu _ { - , A } = 2 2 . 8 1 \\mu _ { A } = 4 0 . 9 9 \\mu _ { - , A } = 1 1 . 7 8 \\mu _ { A } = 1 5 7 . 6 \\mu _ { - , A } = 1 5 . 2 8\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Thus far we have assessed the selectivity of hidden units in AlexNet and shown that no units can reasonably be characterized as object detectors despite the high precision and CCMAS scores of some units. This raises the question as to whether more recent CNNs learn object detector units. In order to address this we display jitterplots for three units that have the highest IoU scores according to the Network Dissection for the category BUS in (a) GoogLeNet trained on ImageNet, (b) GoogLeNet trained on Places-365, and (c) VGG-16 trained on Places-365, respectively (Zhou et al., 2018a). Models trained on the Places-365 dataset learn to categorize images into scenes (e.g., bedrooms, kitchens, etc.) rather than into object categories, and nevertheless, Zhou et al. (2018a) reported more object detectors in the former models. We illustrate the selectivity of the BUS category because it is an output category in ImageNet so we can easily plot the jitterplots for these units. ",
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+    {
+        "type": "text",
+        "text": "As was the case with AlexNet, the jitterplots show that the most selective units show some degree of selectivity, with the BUS images more active on average compared to non-Buses, and the percentage of nonzero activations for BUS higher than the non-BUS categories (see tables A3 - A5 in the appendix for summary of more units). But the units are no more selective than the units we observed in AlexNet. Indeed, the precision measure of selectivity for the first units is 0.0, with none of the units having a precision of .75 that was the criterion of object detectors by Zhou et al. (2015), and CCMAS scores for first two units were roughly similar to the mean CCMAS score for AlexNet units in conv 5 (and much lower than the mean in fc6 and fc7). The most selective VGG-16 unit trained on Places-365 has lower precision and CCMAS scores than the Monarch Butterfly unit depicted in Figure 3. So again, different measures of selectivity provide support different conclusions, and even the most selective units are far from the selective units observed in recurrent networks as reported in Figure 1a. See tables A3 - A5 in the appendix for more details about these three units. ",
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+    {
+        "type": "text",
+        "text": "4 DISCUSSIONS AND CONCLUSIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our central finding is that different measures of single-unit selectivity for objects support very different conclusions when applied to the same units in AlexNet. In contrast with the precision (Zhou et al., 2015) and CCMAS (Morcos et al., 2018) measures that suggest some highly selective units for objects in layers conv5, fc6, and fc7, the recall with perfect precision and false alarm rates at maximum informedness show low levels of selectivity. Indeed, the most selective units have a poor hit-rate or a high false-alarm rate (or both) for identifying an object class. The same outcome was observed with units in VGG-16 and GoogLeNet trained on either ImageNet or the Places-365 dataset. ",
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+    {
+        "type": "text",
+        "text": "Not only do the different measures provide very different assessments of selectivity, the precision, CCMAS, and Network Dissection measures provide highly misleading estimates of selectivity that have led to mistaken conclusions. For example, unit $\\mathsf { f c } 6 _ { 1 1 9 9 }$ in AlexNet trained on ImageNet is considered an Monarch Butterfly detector according to Zhou et al. (2015) with a precision score of $98 \\%$ (and a CCMAS score of .93). But the jitterplot in Fig. 3 and signal detection scores (e.g., high false alarm rate at maximum informedness) show this is a mischaracterisation of this unit. In the same way, the Network Dissection method identified many object detectors in VGG-16 and GoogLeNet CNNs, but the jitterplots in Fig. 5 (and precision scores) show that this conclusion is unjustified. For additional problems with the CCMAS score see Figure 5 in Appendix C. Similarly, the images generated by Activation Maximization also provided a misleading estimate of selectivity given that interpretable images were associated with very low selectivity scores. This has led to confusions that have delayed theoretical progress. For example, describing single units in CNNs as “object detectors” in response to high precision measures (Zhou et al.) suggests similar types of representations are learned in CNNs and RNNs. Indeed, we are not aware of anyone in the machine learning community who has even considered the hypothesis that selectivity is reduced in CNNs compared RNNs. Our findings highlight the contrasting results. ",
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+    {
+        "type": "text",
+        "text": "What should be made of the finding that localist representations are sometimes learned in RNNs (units with perfect specificity and recall), but not in AlexNet and related CNNs? The failure to observe localist units in the hidden layers of these CNNs is consistent with Bowers et al. (2014)’s claim that these units emerge in order to support the co-activation of multiple items at the same time in short-term memory. That is, localist representations may be the solution to the superposition catastrophe, and these CNNs only have to identify one image at a time. The pressure to learn highly selective representations in response to the superposition constraint may help explain the reports of highly selective neurons in cortex given that the cortex needs to co-activate multiple items at the same time in order to support short-term memory (Bowers et al., 2016). ",
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+    {
+        "type": "text",
+        "text": "Note, the RNNs that learned localist units were very small in scale compared to CNNs we have studied here, and accordingly, it is possible that the contrasting results reflect the size of the networks rather than the superposition catastrophe per se. Relevant to this issue a number of authors have reported the existence of selective units in larger RNNs with long-short term memory (LSTM) units (Karpathy et al., 2016; Radford et al., 2017; Lakretz et al., 2019; Na et al., 2019). Indeed, Lakretz et al. (2019) use the term ‘grandmother cell’ to describe the units they observed. It will be interesting to apply our measures of selectivity to these larger RNNs and see whether these units are indeed ‘grandmother units’. ",
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+    {
+        "type": "text",
+        "text": "It should also be noted that there are recent reports of impressively selective representations in Generative Adversarial Networks (Bau et al., 2019) and Variational Autoencoders (Burgess et al., 2018) where the superposition catastrophe is not an issue. Again, it will be interesting to assess the selectivity of these units according to signal detection measures in order to see whether there are additional computational pressures to learn highly selective or even grandmother cells. We will be exploring these issues in future work. ",
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+        "type": "text",
+        "text": "REFERENCES   \nD. Bau, J.-Y. Zhu, H. Strobelt, B. Zhou, J. B. Tenenbaum, W. T. Freeman, and A. Torralba. Visualizing and understanding generative adversarial networks. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id $\\underline { { \\underline { { \\mathbf { \\Pi } } } } } =$ Hyg_X2C5FX.   \nI. S. Berkeley, M. R. Dawson, D. A. Medler, D. P. Schopflocher, and L. Hornsby. Density plots of hidden value unit activations reveal interpretable bands. Connection Science, 7(2):167–187, 1995.   \nJ. S. Bowers. On the biological plausibility of grandmother cells: implications for neural network theories in psychology and neuroscience. Psychological review, 116(1):220, 2009.   \nJ. S. Bowers. Grandmother cells and localist representations: a review of current thinking. Language, Cognition, and Neuroscience, pages 257–273, 2017.   \nJ. S. Bowers, I. I. Vankov, M. F. Damian, and C. J. Davis. Neural networks learn highly selective representations in order to overcome the superposition catastrophe. Psychological review, 121(2): 248–261, 2014.   \nJ. S. Bowers, I. I. Vankov, M. F. Damian, and C. J. Davis. Why do some neurons in cortex respond to information in a selective manner? insights from artificial neural networks. Cognition, 148:47–63, 2016.   \nC. P. Burgess, I. Higgins, A. Pal, L. Matthey, N. Watters, G. Desjardins, and A. Lerchner. Understanding disentangling in $\\beta$ -VAE. arXiv preprint arXiv:1804.03599, 2018.   \nJ. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pages 248–255. IEEE, 2009.   \nD. Erhan, Y. Bengio, A. Courville, and P. Vincent. Visualizing higher-layer features of a deep network. University of Montreal, 1341(3):1, 2009.   \nY. Jia, E. Shelhamer, J. Donahue, S. Karayev, J. Long, R. Girshick, S. Guadarrama, and T. Darrell. Caffe: Convolutional architecture for fast feature embedding. arXiv preprint arXiv:1408.5093, 2014.   \nA. Karpathy, J. Johnson, and L. Fei-Fei. Visualizing and understanding recurrent networks. Workshop Track at International Conference on Learning Representations, 2016.   \nA. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pages 1097–1105, 2012.   \nY. Lakretz, G. Kruszewski, T. Desbordes, D. Hupkes, S. Dehaene, and M. Baroni. The emergence of number and syntax units in lstm language models. arXiv preprint arXiv:1903.07435, 2019.   \nQ. V. Le. Building high-level features using large scale unsupervised learning. In 2013 IEEE international conference on acoustics, speech and signal processing, pages 8595–8598. IEEE, 2013.   \nA. S. Morcos, D. G. Barrett, N. C. Rabinowitz, and M. Botvinick. On the importance of single directions for generalization. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id $=$ r1iuQjxCZ.   \nS. Na, Y. J. Choe, D.-H. Lee, and G. Kim. Discovery of natural language concepts in individual units of cnns. In International Conference on Learning Representations, 2019. URL https: //openreview.net/forum?id $\\underline { { \\underline { { \\mathbf { \\Pi } } } } } =$ S1EERs09YQ.   \nA. Nguyen, A. Dosovitskiy, J. Yosinski, T. Brox, and J. Clune. Synthesizing the preferred inputs for neurons in neural networks via deep generator networks. In Advances in Neural Information Processing Systems, pages 3387–3395, 2016.   \nA. Nguyen, J. Clune, Y. Bengio, A. Dosovitskiy, and J. Yosinski. Plug & play generative networks: Conditional iterative generation of images in latent space. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4467–4477, 2017.   \nA. Nguyen, J. Yosinski, and J. Clune. Understanding neural networks via feature visualization: A survey. arXiv preprint arXiv:1904.08939, 2019.   \nD. M. Powers. Evaluation: from precision, recall and f-measure to roc, informedness, markedness and correlation. 2011.   \nA. Radford, R. Jozefowicz, and I. Sutskever. Learning to generate reviews and discovering sentiment. arXiv preprint arXiv:1704.01444, 2017.   \nK. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.   \nK. Simonyan, A. Vedaldi, and A. Zisserman. Deep inside convolutional networks: Visualising image classification models and saliency maps. arXiv preprint arXiv:1312.6034, 2013.   \nC. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1–9, 2015.   \nJ. Yosinski, J. Clune, A. Nguyen, T. Fuchs, and H. Lipson. Understanding neural networks through deep visualization. arXiv preprint arXiv:1506.06579, 2015.   \nM. D. Zeiler and R. Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pages 818–833. Springer, 2014.   \nB. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba. Object detectors emerge in deep scene CNNs. In International Conference on Learning Representations, 2015.   \nB. Zhou, D. Bau, A. Oliva, and A. Torralba. Interpreting deep visual representations via network dissection. IEEE transactions on pattern analysis and machine intelligence, 2018a.   \nB. Zhou, Y. Sun, D. Bau, and A. Torralba. Revisiting the importance of individual units in CNNs via ablation. arXiv preprint arXiv:1806.02891, 2018b. ",
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "A METHODOLOGICAL DETAILS FOR THE BEHAVIORAL EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "One hundred generated images were made for every unit in layers conv5, fc6 and fc8 in AlexNet, as in Nguyen et al. (2017), and displayed as $1 0 \\mathrm { x } 1 0$ image panels (figures A4 and Figures A2 and A3). A total of 3,299 image panels were used in the experiment (995 fc8, 256 conv5, and 2048 randomly selected fc6 image panels) and were divided into 64 counterbalanced lists of 51 or 52 (4 conv5, 15 or $1 6 \\mathsf { f c } 8$ and 32 fc6). 51 of the lists were assigned to 5 participants and 13 lists were assigned to 6 participants. ",
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+    },
+    {
+        "type": "text",
+        "text": "To test the interpretability of these units, paid volunteers were asked to look at image panels and asked if the images had an object / animal or place in common. If the answer was ‘yes’, they were asked to name that object simply (i.e. fish rather than goldfish). Analyses of common responses was carried out for any units where over $80 \\%$ of humans agreed there was an object present, by reading the human responses and comparing them to both each other and to the output classes. Agreement was taken if the object was the same rough class. For example, ‘beer’, ‘glass’, and ‘drink’ were all considered to be in agreement in the general object of ‘drink’, and in agreement with both the classes of ‘wine glass’ and ‘beer’ as these classes were also general drink classes (this is an actual example, most responses were more obvious and required far less interpretation than that). Participants were given six practice trials, each with panels of 20 images before starting the main experiment. Practice trials included images that varied in their interpretability. ",
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+    {
+        "type": "image",
+        "img_path": "images/1a017687051cbc18f9a0a5a59b15c224ba0a7883a3aa8fe5700a6c6bba0d47a1.jpg",
+        "image_caption": [
+            "Figure A1: Example screen from the identification task shown to participants as part of the instructions. The images included on this practice trial are ImageNet2012 images, not AM images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Some examples of the $1 0 \\mathrm { x } 1 0$ grids of activation maximisation images that were presented to participants are shown in Figures A2, A3 and A4. Figure A2 shows an example from conv5 that human participants agreed had no obvious object in common (although there are repeated shape motifs, the participants were specifically asked for objects, and not abstract concepts like shape or color. Figure A3 is also from the conv5 and was judged by participants as some images containing ‘dogs’. ",
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+    {
+        "type": "image",
+        "img_path": "images/86890cf8b17928ab35c92142a67999a22d2b41dcd29aa3e79dc1617600be0818.jpg",
+        "image_caption": [
+            "Figure A4 is the AM images for the supposed ‘butterfly detector’ unit example discussed in the paper. ",
+            "Figure A2: Example activation maximisation images for unit conv5.65. These images were judged by humans to not contain any interpretable objects in common (although the reader may agree that there are some shape and colour similarities in the images). "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/e3976498e1d5e03e9eda7017306d8b6048a0f3a2d4ccc174532d3143bb3d0631.jpg",
+        "image_caption": [
+            "Figure A3: Example activation maximisation images for unit conv5.183. These images were judged by humans to contain some interpretable images, in this case, of the type ‘dogs’. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/53b865e27c047b3029870a705da9f697a5bdda0957ce95f1cc0927947dbd92ed.jpg",
+        "image_caption": [
+            "Figure A4: Example activation maximisation images for unit fc6.1199. Whilst there are some butterfly wing shapes in these images, there are not obvious butterflies. N.B. the second highest activating class for this unit is ladybirds, and there are some orange round shapes that could conceivably be ladybug-alikes. "
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "B FURTHER DATA ON THE SELECTIVITY MEASURES ACROSS ALEXNET ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table A1 gives the highest values of CCMAS and precision for each layer in AlexNet. It is worth noting that the highest CCMAS score of all hidden units was .94 (fc7.31), which at first glance suggests that this unit is close to ‘perfect’ selectivity. However, this unit only has low a precision score of $11 \\%$ . In other words, although the mean activation for the given class is very high relative to the mean of all other activations (high CCMAS), the proportion of items from that class in the 100 most active items is low (low precision). See appendix Sec. C for discussion of how this occurs and Fig. A5(a) for an illustrative example. ",
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+    {
+        "type": "table",
+        "img_path": "images/8b9e832ed98595bb4897c329665ff4b63d66b9a62b1f9643e36837fa4d5d8906.jpg",
+        "table_caption": [
+            "Table A1: The units with the highest CCMAS and precision scores in AlexNet. Unit fc6.1199 was displayed in Fig. 3. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>LAYER.UNIT CCMAS</td><td>Precision</td></tr><tr><td>Top CCMAS units</td><td></td></tr><tr><td>output.322 0.991</td><td>100%</td></tr><tr><td>fc7.31 0.94</td><td>11%</td></tr><tr><td>fc6.582 0.93</td><td>1%</td></tr><tr><td>conv5.78 0.75</td><td>5%</td></tr><tr><td></td><td></td></tr><tr><td>Top precision units</td><td></td></tr><tr><td>output.0 0.99</td><td>100%</td></tr><tr><td>fc7.255 0.90</td><td>97%</td></tr><tr><td>fc6.1199 0.92</td><td>95%</td></tr><tr><td>conv5.0 0.55</td><td>77%</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table A2 shows positive correlations between four of the selectivity measures used. There are moderate positive correlations between precision and CCMAS; and precision and Recall at $9 5 \\%$ precision. The other correlations between selectivity measures have weak positive correlations. All four selectivity measures are negatively correlated with the number of classes present in the $1 0 0 \\mathrm { m o s t }$ active items, that is, the more selective the unit, the fewer classes will be represented in the most active 100 items. ",
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+    {
+        "type": "table",
+        "img_path": "images/1967e595b7a92ec761360c4264d6c0c6f3b8db95d8d4b2a686035ab45448aa39.jpg",
+        "table_caption": [
+            "Table A2: The correlations between the different measures. (All $p$ ’s $< . 0 0 1$ ) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CCMAS</td><td>recalll0.95</td><td>Max. Inf.</td><td>No.classes in top100</td></tr><tr><td>precision</td><td>0.38</td><td>0.30</td><td>0.15</td><td>-0.68</td></tr><tr><td>CCMAS</td><td></td><td>0.09</td><td>0.14</td><td>-0.47</td></tr><tr><td>recall0.95</td><td></td><td></td><td>0.10</td><td>-0.19</td></tr><tr><td>Max. Inf.</td><td></td><td></td><td></td><td>-0.22</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "C FURTHER ISSUES WITH THE CCMAS MEASURE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The CCMAS measure is based on comparing the mean activation of a category with the mean activation for all other items, and this is problematic for a few reasons. First, in many units a large proportion of images do not activate a unit at all. For instance, our butterfly ‘detector’ unit fc6.1199 has a high proportion of images with an activation of 0.0 (see figure 3). Indeed, the inset on the middle figure shows that the distribution can be better described by exponential-derived fits rather than a Gaussian. This means that the CCMAS selectivity is heavily influenced by the the proportion of images that have an activation value of zero (or close to zero). This can lead to very different estimates of selectivity for CCMAS and precision or localist selectivity, which are driven by the most highly activated items. In A5 we generate example data to highlight ways in which CCMAS score may be non-intuitive. In subplot (a) we demonstrate that a unit can have a CCMAS score of of 1.0 despite only a single item activating the unit. The point that we wish to emphasise is that a high CCMAS score does not necessarily imply selectivity for a given class, but might in fact relate to selectivity for a small subset of items from a given class, and this is especially true when a unit’s activation is sparse (many items do not activate the unit). However, the reverse can also be true. In subplot (c) we demonstrate that a unit can have a very low CCMAS score of .06 despite all of the most active items being from the same class. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "In addition, if the CCMAS provided a good measure of a unit’s class selectivity, then one should expect that a high measure of selectivity for one class would imply that the unit is not highly selective for other classes. However, the CCMAS score for the most selective category and the second most selective category (CCMAS2) were similar across the conv5, fc6 and fc7. layers, with the mean CCMAS scores .491, .844, and .848, and the CCMAS2 scores .464, .821, .831. For example, unit fc7.0 has a CCMAS of .813 for the class ‘maypole’, and a CCMAS2 score of .808 for ‘chainsaw’ (with neither of these categories corresponding ‘orangutan’ that had the highest precision of score of $14 \\%$ ). ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/77025fa9e4dcfbee2b67b179b8eba0ebd1ff4a2f835efa476052be3c934048af.jpg",
+        "image_caption": [
+            "Figure A5: Example of where the CCMAS does not match intuitive understandings of selectivity. Generated example data: (a) If a unit is off to all but a single image from a large class of objects, the CCMAS for that class is 1 (maximum possible selectivity). (b) An archetypal ‘grandmother’ cell (GMC), where the unit is strongly activated to all members of one class and off to everything else. The CCMAS is the same for (b) as for (a) although the precision is vastly different. (c): If a unit has high activations for all classes, but one class (black squares) is 0.1 more than all others (coloured circles), the CCMAS is very low (0.06) despite being $\\% 1 0 0$ precision. The generated examples are for 10 classes of 100 items "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "D TESTING UNITS IN OTHER MODELS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To investigate units claimed by Zhou et al. (2018a) to be object detectors, we focus on units from a single layer that are reported to be ‘bus detectors’, that is, units with an $\\mathrm { I o U } \\geq . 0 4$ . We used the first 100 images per class from the ImageNet 2012 dataset as our test data. There are three classes of bus in this dataset: ‘n04146614 school bus’, ‘n04487081 trolleybus, trolley coach, trackless trolley’, ‘n03769881 minibus’, and this corresponded to 300 items out of 100000 images. Data for all bus unit detectors for VGG trained on places 365 are shown in table A3; for GoogLeNet trained on places 365 in table A4; and for GoogLeNet trained on ImageNet are shown in table A5. Note that for all units there are very few busses with activation at zero and that the mean activation for busses is higher than the mean activation for non-busses. However, all precision scores are all below .6, meaning that of the 100 items that most strongly activated the unit, at least 40 of them were not busses. Together these results suggests that whilst these units demonstrate some sensitivity to busses, they show poor specificity for busses (e.g., high false-alarm rate). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2863cccd24c44521d9b6b09d5e52e29b690b66d187ddb4d1d1a0c442281f9a70.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>unit</td><td>IoU</td><td>no.ax&gt;0 x∈A</td><td>no.ax&gt;0 x∈-A</td><td>μA</td><td>μ-A precision</td><td>CCMAS</td></tr><tr><td>conv5_3191</td><td>.15</td><td>99.0%</td><td>63.9%</td><td>131.9 16.1</td><td>.45</td><td>.78</td></tr><tr><td>conv5_320</td><td>.15</td><td>99.0%</td><td>49.1%</td><td>157.6</td><td>15.2 .53</td><td>.82</td></tr><tr><td>conv5_3333</td><td>.08</td><td>99.0%</td><td>71.4%</td><td>101.7</td><td>17.5 .24</td><td>.71</td></tr><tr><td>conv5_3145</td><td>.07</td><td>97.3%</td><td>61.7%</td><td>75.5</td><td>12.5 .19</td><td>.72</td></tr><tr><td>conv5_3113</td><td>.06</td><td>97.4%</td><td>41.0%</td><td>62.8</td><td>9.1 .12</td><td>.75</td></tr><tr><td>conv5_3443</td><td>.04</td><td>95.3%</td><td>38.2%</td><td>59.3</td><td>8.1 .12</td><td>.76</td></tr><tr><td>conv5_3131</td><td>.04</td><td>93.7%</td><td>22.3%</td><td>54.0</td><td>5.86 .08</td><td>.80</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table A3: Selectivity measures for VGG-16, trained on Places-365, top convolutional layer units identified by Zhou et al. (2018a) as object detectors. Standard errors not shown for space, but were below $\\pm 5$ . The IoU is from Zhou et al. (2018a)’s network dissection method. no. $a _ { x } { > } 0$ and no. $a _ { x } { > } 0$ $x \\in A$ refer to the proportion of activations that were greater than zero for busses and non-busses respectively. $\\mu _ { A }$ and $\\mu { \\neg } A$ are the class means for busses and non busses respectively. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/16805a82c41c0a8f8ef237357d76ebe77150b8e7e621c671d37a940c75e3662c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>unit</td><td>IoU</td><td>correct</td><td>no.ax&gt;0 x∈A</td><td>no.ax&gt;0 x∈-A</td><td>μA</td><td>μ-A</td><td>precision</td><td>CCMAS</td></tr><tr><td>incep4e824</td><td>.17</td><td>T</td><td>100.0</td><td>91.4</td><td>41.0</td><td>11.8</td><td>.27</td><td>.55</td></tr><tr><td>incep4e745</td><td>.13</td><td>T</td><td>98.3</td><td>74.8</td><td>34.8</td><td>11.4</td><td>.06</td><td>.51</td></tr><tr><td>incep4e791</td><td>.11</td><td>T</td><td>98.3</td><td>71.4</td><td>32.7</td><td>5.3</td><td>.41</td><td>.72</td></tr><tr><td>incep4e194</td><td>.11</td><td>F</td><td>100.0</td><td>85.3</td><td>26.6</td><td>8.8</td><td>.02</td><td>.51</td></tr><tr><td>incep4e82</td><td>.11</td><td>T</td><td>100.0</td><td>97.3</td><td>26.7</td><td>10.9</td><td>.14</td><td>.42</td></tr><tr><td>incep4e736</td><td>.11</td><td>T</td><td>100.0</td><td>78.8</td><td>38.7</td><td>9.9</td><td>.05</td><td>.59</td></tr><tr><td>incep4e663</td><td>.10</td><td>F</td><td>96.0</td><td>38.0</td><td>33.4</td><td>3.7</td><td>.15</td><td>.80</td></tr><tr><td>incep4eg4</td><td>.10</td><td>T</td><td>100.0</td><td>91.6</td><td>38.3</td><td>9.5</td><td>.35</td><td>.60</td></tr><tr><td>incep4e772</td><td>.08</td><td>F</td><td>97.3</td><td>54.6</td><td>21.7</td><td>5.2</td><td>.00</td><td>.61</td></tr><tr><td>incep4e113</td><td>.08</td><td>F</td><td>100.0</td><td>88.0</td><td>24.9</td><td>9.2</td><td>.02</td><td>.46</td></tr><tr><td>incep4e708</td><td>.06</td><td>F</td><td>100.0</td><td>85.1</td><td>29.7</td><td>9.1</td><td>.02</td><td>.53</td></tr><tr><td>incep4eg01</td><td>.06</td><td>F</td><td>100.0</td><td>64.5</td><td>35.2</td><td>6.4</td><td>.14</td><td>.69</td></tr><tr><td>incep4e199</td><td>.06</td><td>F</td><td>99.7</td><td>92.2</td><td>21.5</td><td>7.7</td><td>.09</td><td>.47</td></tr><tr><td>incep4eg</td><td>.05</td><td>F</td><td>99.7</td><td>83.5</td><td>18.5</td><td>7.3</td><td>.01</td><td>.43</td></tr><tr><td>incep4e121</td><td>.05</td><td>F</td><td>100.0</td><td>90.4</td><td>17.9</td><td>8.9</td><td>.01</td><td>.34</td></tr><tr><td>incep4e622</td><td>.05</td><td>T</td><td>96.0</td><td>65.0</td><td>27.5</td><td>6.4</td><td>.20</td><td>.62</td></tr><tr><td>incep4e97</td><td>.04</td><td>T</td><td>99.3</td><td>86.4</td><td>21.1</td><td>9.3</td><td>.04</td><td>.39</td></tr></table>",
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+            795,
+            734
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Table A4: Selectivity measures for GoogLeNet, trained on Places-365, layer inception4e units identified by Zhou et al. (2018a) as object detectors. Standard errors not shown for space, but were below $\\pm 2$ . The IoU is from Zhou et al. (2018a)’s network dissection method. A units is marked as correct if there was a single bus in the 4 example pictures on the website (http://netdissect. csail.mit.edu/dissect/googlenet_places365/), and false if not. Where units are False, this might suggest that the units were responding to ’bus like’ features in none bus objects. no. $a _ { x } { > } 0$ and no $. a _ { x } { > } 0 \\ x \\in A$ refer to the proportion of activations that were greater than zero for busses and non-busses respectively. $\\mu _ { A }$ and $\\mu { \\_ } A$ are the class means for busses and non busses respectively. ",
+        "bbox": [
+            173,
+            743,
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+            869
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+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/20a3100f985e518d41d794ad00908d377f9f670e3a7890f6af10e3ed803d484f.jpg",
+        "table_caption": [
+            "Table A5: Selectivity measures for GoogLeNet, trained on ImageNet, layer inception4e units units identified by Zhou et al. (2018a) as object detectors. Standard errors not shown for space, but were below $\\pm 2$ . A units is marked as correct if there was a single bus in the 4 example pictures on the website (http://netdissect.csail.mit.edu/dissect/googlenet_imagenet/), and false if not. This might suggest that the units were responding to ’bus like’ features in none bus objects. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>unit</td><td>IoU</td><td>correct</td><td>no.ax&gt;0 x∈A</td><td>no.ax&gt;0 x∈-A</td><td>μA</td><td>μ-A</td><td>precision</td><td>CCMAS</td></tr><tr><td>incep4e494</td><td>.11</td><td>F</td><td>99.0</td><td>82.4</td><td>72.5</td><td>22.8</td><td>.00</td><td>.52</td></tr><tr><td>incep4e828</td><td>.10</td><td>T</td><td>100.0</td><td>72.6</td><td>109.4</td><td>17.6</td><td>.45</td><td>.72</td></tr><tr><td>incep4e569</td><td>.10</td><td>T</td><td>99.7</td><td>85.9</td><td>74.9</td><td>20.0</td><td>.05</td><td>.58</td></tr><tr><td>incep4e384</td><td>.10</td><td>T</td><td>100.0</td><td>71.6</td><td>67.0</td><td>18.5</td><td>.00</td><td>.57</td></tr><tr><td>incep4e455</td><td>.09</td><td>T</td><td>99.7</td><td>89.6</td><td>69.1</td><td>14.3</td><td>.3</td><td>.66</td></tr><tr><td>incep4e579</td><td>.09</td><td>T</td><td>100.0</td><td>97.0</td><td>91.5</td><td>26.0</td><td>.23</td><td>.56</td></tr><tr><td>incep4e331</td><td>.08</td><td>T</td><td>98.0</td><td>75.5</td><td>51.0</td><td>11.8</td><td>.12</td><td>.62</td></tr><tr><td>incep4e582</td><td>.08</td><td>T</td><td>100.0</td><td>83.4</td><td>125.7</td><td>21.95</td><td>.58</td><td>.70</td></tr><tr><td>incep4e498</td><td>.07</td><td>T</td><td>97.7</td><td>77.2</td><td>73.5</td><td>15.0</td><td>.52</td><td>.66</td></tr><tr><td>incep4e534</td><td>.07</td><td>F</td><td>99.3</td><td>81.2</td><td>62.7</td><td>19.</td><td>.02</td><td>.53</td></tr><tr><td>incep4e693</td><td>.07</td><td>T</td><td>98.7</td><td>91.2</td><td>75.4</td><td>22.3</td><td>.15</td><td>.54</td></tr><tr><td>incep4e673</td><td>.07</td><td>T</td><td>99.7</td><td>88.4</td><td>88.6</td><td>23.0</td><td>.33</td><td>.59</td></tr><tr><td>incep4e469</td><td>.06</td><td>T</td><td>98.7</td><td>78.1</td><td>34.7</td><td>14.6</td><td>.00</td><td>.41</td></tr><tr><td>incep4e207</td><td>.06</td><td>T</td><td>100.0</td><td>93.5</td><td>76.1</td><td>21.3</td><td>.07</td><td>.56</td></tr><tr><td>incep4e491</td><td>.06</td><td>F</td><td>99.0</td><td>74.5</td><td>41.1</td><td>13.7</td><td>.01</td><td>.50</td></tr><tr><td>incep4e645</td><td>.06</td><td>T</td><td>98.0</td><td>83.9</td><td>59.9</td><td>18.1</td><td>.20</td><td>.54</td></tr><tr><td>incep4e527</td><td>.06</td><td>F</td><td>100.0</td><td>91.5</td><td>58.0</td><td>21.7</td><td>.00</td><td>.46</td></tr><tr><td>incep4e511</td><td>.05</td><td>F</td><td>100.0</td><td>89.4</td><td>53.5</td><td>21.7</td><td>.00</td><td>.42</td></tr><tr><td>incep4e308</td><td>.05</td><td>F</td><td>100.0</td><td>89.4</td><td>53.5</td><td>21.7</td><td>.00</td><td>.42</td></tr><tr><td>incep4e541</td><td>.05</td><td>F</td><td>99.67</td><td>88.7</td><td>44.9</td><td>13.7</td><td>.00</td><td>.53</td></tr><tr><td>incep4e367</td><td>.05</td><td>T</td><td>97.3</td><td>80.3</td><td>37.7</td><td>15.4</td><td>.02</td><td>.42</td></tr><tr><td>incep4e665</td><td>.05</td><td>T</td><td>100.0</td><td>82.45</td><td>107.2</td><td>21.0</td><td>.33</td><td>.67</td></tr><tr><td>incep4e532</td><td>.05</td><td>T</td><td>100.0</td><td>91.5</td><td>52.9</td><td>22.4</td><td>.05</td><td>.41</td></tr><tr><td>incep4e297</td><td>.04</td><td>T</td><td>99.7</td><td>90.2</td><td>48.2</td><td>17.9</td><td>.00</td><td>.46</td></tr><tr><td>incep4e480</td><td>.04</td><td>T</td><td>100.0</td><td>92.9</td><td>69.4</td><td>21.4</td><td>.02</td><td>.53</td></tr></table>",
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+    }
+]

parse/train/r1gelyrtwH/r1gelyrtwH.md

@@ -0,0 +1,376 @@
+# PHYSICS-AWARE DIFFERENCE GRAPH NETWORKS FOR SPARSELY-OBSERVED DYNAMICS
+
+Sungyong Seo∗, Chuizheng Meng∗, Yan Liu
+
+Department of Computer Science   
+University of Southern California   
+{sungyons,chuizhem,yanliu.cs}@usc.edu
+
+# ABSTRACT
+
+Sparsely available data points cause numerical error on finite differences which hinders us from modeling the dynamics of physical systems. The discretization error becomes even larger when the sparse data are irregularly distributed or defined on an unstructured grid, making it hard to build deep learning models to handle physics-governing observations on the unstructured grid. In this paper, we propose a novel architecture, Physics-aware Difference Graph Networks (PA-DGN), which exploits neighboring information to learn finite differences inspired by physics equations. PA-DGN leverages data-driven end-to-end learning to discover underlying dynamical relations between the spatial and temporal differences in given sequential observations. We demonstrate the superiority of PA-DGN in the approximation of directional derivatives and the prediction of graph signals on the synthetic data and the real-world climate observations from weather stations.
+
+# 1 INTRODUCTION
+
+Modeling real world phenomena, such as climate observations, traffic flow, physics and chemistry simulation (Li et al., 2018; Geng et al., 2019; Long et al., 2018; de Bezenac et al., 2018; SanchezGonzalez et al., 2018; Gilmer et al., 2017), is important but extremely challenging. While deep learning has achieved remarkable successes in prediction tasks by learning latent representations from data-rich applications such as image recognition (Krizhevsky et al., 2012), text understanding (Wu et al., 2016), and speech recognition (Hinton et al., 2012), we confront many challenging scenarios in modeling natural phenomena with deep neural networks when only a limited number of observations are available. Particularly, the sparsely available data points cause substantial numerical error when we utilize existing finite difference operators and the limitation requires a more principled way to redesign deep learning models.
+
+While many methods have been proposed to model physics-simulated observations using deep learning, many of them are designed under the assumption that input is on a continuous domain. For example, Raissi et al. (2017a;b) proposed physics-informed neural networks (PINNs) to learn nonlinear relations between input (spatial- and temporal-coordinates $( { \pmb x } , t ) )$ and output simulated with a given partial differential equation (PDE). Since Raissi et al. (2017a;b) use the coordinates as input and compute derivatives based on the coordinates to represent the equation, the setting is only valid when the data are densely observed over spatial and temporal space.
+
+Prior knowledge related to physics equations has been combined with data-driven models for various purposes. Chen et al. (2015) proposed a nonlinear diffusion process for image restoration and de Bezenac et al. (2018) incorporated the transport physics (advection-diffusion equation) with deep neural networks for forecasting sea surface temperature by extracting the motion field. Lutter et al. (2019) introduced deep Lagrangian networks specialized to learn Lagrangian mechanics with learnable parameters. Seo & Liu (2019) proposed a physics-informed regularizer to impose dataspecific physics equations. In common, the methods in Chen et al. (2015); de Bezenac et al. (2018); Lutter et al. (2019) are not efficiently applicable to sparsely discretized input as only a small number of data points are available and continuous properties on given space are not easily recovered. It is unsuitable to directly use continuous differential operators to provide local behaviors because it is hard to approximate the continuous derivatives precisely with the sparse points (Shewchuk, 2002; Amenta & Kil, 2004; Luo et al., 2009). Furthermore, they are only applicable when the specific physics equations are explicitly given and still hard to be generalized to incorporate other types of equations.
+
+As another direction to modeling physics-simulated data, Long et al. (2018) proposed PDE-Net which uncovers the underlying hidden PDEs and predicts the dynamics of complex systems. Ruthotto & Haber (2018) derived new CNNs: parabolic and hyperbolic CNNs based on ResNet (He et al., 2016) architecture motivated by PDE theory. While Long et al. (2018); Ruthotto & Haber (2018) are flexible to uncover hidden physics from the constrained kernels, it is still restrictive to a regular grid where the proposed constraints on the learnable filters are easily defined.
+
+The topic of reasoning physical dynamics of discrete objects has been actively studied (SanchezGonzalez et al., 2018; Battaglia et al., 2016; Chang et al., 2016) as the appearance of graph-based neural networks (Kipf & Welling, 2017; Santoro et al., 2017; Gilmer et al., 2017). Although these models can handle sparsely located data points without explicitly given physics equations, they are purely data-driven so that the physics-inspired inductive bias for exploiting finite differences is not considered at all. In contrast, our method consists of physics-aware modules allowing efficiently leveraging the inductive bias to learn spatiotemporal data from the physics system.
+
+In this paper, we propose physics-aware difference graph networks (PA-DGN) whose architecture is motivated to leverage differences of sparsely available data from the physical systems. The differences are particularly important since most of the physics-related dynamic equations (e.g., Navier–Stokes equations) handle differences of physical quantities in spatial and temporal space instead of using the quantities directly. Inspired by the property, we first propose spatial difference layer (SDL) to efficiently learn the local representations by aggregating neighboring information in the sparse data points. The layer is based on graph networks (GN) as it easily leverages structural features to learn the localized representations and share the parameters for computing the localized features. Then, the layer is followed by recurrent graph networks (RGN) to predict the temporal difference which is another core component of physics-related dynamic equations. PA-DGN is applicable to various tasks and we provide two representative tasks; the approximation of directional derivatives and the prediction of graph signals.
+
+Our contributions are:
+
+• We tackle a limitation of the sparsely discretized data which cause numerical error when we model the physical system by proposing spatial difference layer (SDL) for efficiently exploiting neighboring information under the limitation of sparsely observable points.   
+• We combine SDL with recurrent graph networks to build PA-DGN which automatically learns the underlying spatiotemporal dynamics in graph signals.   
+• We verify that PA-DGN is effective in approximating directional derivatives and predicting graph signals in synthetic data. Then, we conduct exhaustive experiments to predict climate observations from land-based weather stations and demonstrate that PA-DGN outperforms other baselines.
+
+# 2 PHYSICS-AWARE DIFFERENCE GRAPH NETWORK
+
+In this section, we introduce the building module used to learn spatial differences of graph signals and describe how the module is used to predict signals in the physics system.
+
+# 2.1 DIFFERENCE OPERATORS ON GRAPH
+
+As approximation of derivatives in continuous domain, difference operators have been used as a core role to compute numerical solutions of (continuous) differential equations. Since it is hard to derive closed-form expressions of derivatives in real-world data, the difference operators have been considered as alternative tools to describe and solve PDEs in practice. The operators are especially important for physics-related data (e.g., meteorological observations) because the governing rules behind the observations are mostly differential equations.
+
+![](images/c729f07d57a78bf799f70079991ad77dbfac4589faed989f6fb73a67b408728f.jpg)  
+Figure 1: Examples of difference operators applied to graph signal. Filters used for the processing are (b) $\textstyle \sum _ { j } ( f _ { i } - f _ { j } )$ (c) $\textstyle \sum _ { j } ( 1 . 1 f _ { i } { \bar { - } } f _ { j } )$ , (d) $f _ { j } - 0 . 5 f _ { i }$ .
+
+Graph signals Given a graph $\mathcal { G } = ( \mathbb { V } , \mathbb { E } )$ where $\mathbb { V }$ is a set of vertices $\mathbb { V } = \{ 1 , \ldots , N _ { v } \}$ and $\mathbb { E }$ is a set of edges $\mathbb { E } \subseteq \bar { \{ ( i , j ) | i , j \in \mathbb { V } \} }$ $\mathbf { \left| \mathbb { E } \right| } = N _ { e } )$ , graph signals on all nodes at time $t$ are $f ( t ) = \{ f _ { i } ( t ) \mid \bar { i } \in \mathbb { V } \}$ where $f _ { i } : \mathbb { V } \to \mathbb { R }$ . Graph signals on edges can also be defined similarly, $F ( t ) = \{ F _ { i j } ( t ) \mid ( i , j ) \in \mathbb { E } \}$ where $F _ { i j } : \mathbb { E }  \mathbb { R }$ . Both signals can be multidimensional.
+
+Gradient on graph The gradient $( \nabla )$ of a function on nodes of a graph is represented by finite difference
+
+$$
+\begin{array} { r } { \nabla : L ^ { 2 } ( \mathbb { V } ) \to L ^ { 2 } ( \mathbb { E } ) , \qquad ( \nabla f ) _ { i j } = \left( f _ { j } - f _ { i } \right) \quad \mathrm { i f } \ ( i , j ) \in \mathbb { E } \mathrm { ~ a n d ~ } 0 \mathrm { ~ o t h e r w i s e } , } \end{array}
+$$
+
+where $L ^ { 2 } ( \mathbb { V } )$ and $L ^ { 2 } ( \mathbb { E } )$ denote Hilbert spaces for node/edge functions, respectively. The gradients on a graph provide finite differences of graph signals and they become edge $( i , j )$ features.
+
+Laplace-Beltrami operator Laplace-Beltrami operator (or Laplacian, $\Delta$ ) in graph domain is defined as
+
+$$
+\Delta : L ^ { 2 } ( \mathbb { V } ) \to L ^ { 2 } ( \mathbb { V } ) , \qquad ( \Delta f ) _ { i } = \sum _ { j : ( i , j ) \in \mathbb { E } } ( f _ { i } - f _ { j } ) \quad \forall i , j \in \mathbb { V }
+$$
+
+This operator is usually regarded as a matrix form in other literature, $\pmb { { \cal L } } = \pmb { { \cal D } } - \pmb { { \cal A } }$ where $\pmb { A }$ is an adjacency matrix and $\begin{array} { r } { \dot { D } = \mathrm { d i a g } ( \sum _ { j : j \neq i } A _ { i j } ) } \end{array}$ is a degree matrix.
+
+# 2.2 DIFFERENCE OPERATORS ON TRIANGULATED MESH
+
+According to Crane (2018), the gradient and Laplacian operators on the triangulated mesh can be discretized by incorporating the coordinates of nodes. To obtain the gradient operator, the per-face gradient of each triangular face is calculated first. Then, the gradient on each node is the area-weighted average of all its neighboring faces, and the gradient on edge $( i , j )$ is defined as the dot product between the per-node gradient and the direction vector $e _ { i j }$ . The Laplacian operator can be discretized with Finite Element Method (FEM):
+
+$$
+( \Delta f ) _ { i } = \frac { 1 } { 2 } \sum _ { j : ( i , j ) \in \mathbb { E } } \left( \cot \alpha _ { j } + \cot \beta _ { j } \right) \left( f _ { j } - f _ { i } \right) ,
+$$
+
+where node $j$ belongs to node $i$ ’s immediate neighbors $( j \in \mathbb { N } _ { i } )$ and $( \alpha _ { j } , \beta _ { j } )$ are two opposing angles of the edge $( i , j )$ .
+
+# 2.3 SPATIAL DIFFERENCE LAYER
+
+While the difference operators are generalized in Riemannian manifolds (Lai et al., 2013; Lim, 2015), there exist numerical error compared to those in continuous space and the error can be larger when the nodes are spatially far from neighboring nodes because the connected nodes $( j \in \mathbb { N } _ { i } )$ ) of $i \cdot$ -th node fail to represent local features around the node. Furthermore, the error is even larger if available data points are sparsely distributed (e.g., sensor-based observations). In other words, the difference operators are unlikely to discover meaningful spatial variations behind the sparse observations since they are highly limited to immediate neighboring information only. To mitigate the limitation, we propose spatial difference layer (SDL) which consists of a set of parameters to define learnable difference operators as a form of gradient and Laplacian to fully utilize neighboring information:
+
+![](images/877d4826df123901cdca612fd7715d80895076ebabfe7e3ae1d04828fbea9bce.jpg)  
+Figure 2: Physics-aware Difference Graph Networks for graph signal prediction. Blue boxes have learnable parameters and all parameters are trained through end-to-end learning. The nodes/edges can be multidimensional.
+
+$$
+( ^ { w } \nabla f ) _ { i j } = w _ { i j } ^ { ( g _ { 1 } ) } ( f _ { j } - w _ { i j } ^ { ( g _ { 2 } ) } f _ { i } ) , \qquad ( ^ { w } \Delta f ) _ { i } = \sum _ { j : ( i , j ) \in \mathbb { R } } w _ { i j } ^ { ( l _ { 1 } ) } ( f _ { i } - w _ { i j } ^ { ( l _ { 2 } ) } f _ { j } )
+$$
+
+where $w _ { i j }$ are the parameters tuning the difference operators along with the corresponding edge direction $e _ { i j }$ . The two forms (Eq 1) are associated with edge and node features, respectively. The superscript in $^ w \nabla$ and $w _ { \Delta }$ denotes that the difference operators are functions of the learnable parameters $w$ . $w _ { i j } ^ { ( g ) }$ and w(l)ij are obtained by integrating local information as follow:
+
+$$
+w _ { i j } = g ( \{ f _ { k } , F _ { m n } | k , ( m , n ) \in h \mathrm { - h o p ~ n e i g h b o r h o o d ~ o f ~ e d g e ~ } ( i , j ) \} )
+$$
+
+While the standard difference operators consider two connected nodes only $\mathit { i }$ and $j$ ) for each edge $( i , j )$ , Eq 2 uses a larger view $h$ -hop) to represent the differences between $i$ and $j$ nodes. Since graph networks (GN) (Battaglia et al., 2018) are efficient networks to aggregate neighboring information, we use GN for $g ( \cdot )$ function and $w _ { i j }$ are edge features of output of GN. Eq 2 can be viewed as a higher-order difference equation because nodes/edges which are multi-hop apart are considered.
+
+$w _ { i j }$ has a similar role of parameters in convolution kernels of CNNs. For example, while the standard gradient operator can be regarded as an example of simple edge-detecting filters, the operator can be a sharpening filter if w(g1)ij and $\begin{array} { r } { w _ { i j } ^ { ( g _ { 2 } ) } = \bar { \frac { | \mathbb { N } _ { i } | + 1 } { | \mathbb { N } _ { i } | } } } \end{array}$ for $i$ node and the operators over each edge are summed. In other words, by modulating $w _ { i j }$ , it is readily extended to conventional kernels including edge detection or sharpening filters and even further complicated kernels. On top of $w _ { i j }$ , the difference forms in Eq 1 make an optimizing process for learnable parameters based on the differences instead of the values intentionally. $\mathrm { E q } 1$ thus naturally provides the physics-inspired inductive bias which is particularly effective for modeling physics-related observations. Furthermore, it is easily possible to increase the number of channels for w(g)ij and w(l)ij to be more expressive. Figure 1 illustrates how the exemplary filters convolve the given graph signals.
+
+# 2.4 RECURRENT GRAPH NETWORKS
+
+Difference graph Once the modulated spatial differences $( ^ { w } \nabla f ( t ) , ^ { w } \Delta f ( t ) )$ are obtained, they are concatenated with the current signals $f ( t )$ to construct node-wise $\left( z _ { i } \right)$ and edge-wise $( z _ { i j } )$ features and the graph is called a difference graph. The difference graph includes all information to describe spatial variations.
+
+Recurrent graph networks Given a snapshot $( f ( t ) , F ( t ) )$ of a sequence of graph signals, one difference graph is obtained and is used to predict next graph signals. While a non-linear layer can be used to combine the learned spatial differences to predict the next signals, it is limited to discover spatial relations only among the features in the difference graph. Since many equations describing physics-related phenomena are non-static (e.g., Navier–Stokes equations), we adopt recurrent graph networks (RGN) (Sanchez-Gonzalez et al., 2018) with a graph state $\mathcal { G } _ { h }$ as input to combine the spatial differences with temporal variations. RGN returns a graph state $( \mathcal { G } _ { h } ^ { * } = ( h ^ { * ( v ) } , h ^ { * ( e ) } ) )$ and next graph signal $z _ { i } ^ { * }$ and $z _ { i j } ^ { * }$ . The update rule is described as follow:
+
+1. $( \boldsymbol { z } _ { i j } ^ { * } , \pmb { h } ^ { * ( e ) } )  \phi ^ { e } ( \boldsymbol { z } _ { i j } , \boldsymbol { z } _ { i } , \boldsymbol { z } _ { j } , \boldsymbol { h } ^ { ( e ) } )$ for all $( i , j ) \in \mathbb { E }$ pairs, 2. $( z _ { i } ^ { * } , h ^ { * ( v ) } ) \gets \phi ^ { v } ( z _ { i } , \bar { z } _ { i } ^ { \prime } , h ^ { ( v ) } )$ for all $i \in \mathbb { V }$ , $\bar { z } _ { i } ^ { \prime }$ is an aggregated edge attribute related to the node $i$ ,
+
+where $\phi ^ { e } , \phi ^ { v }$ are edge and node update functions, respectively, and they can be any recurrent unit.   
+Finally, the prediction is made through a decoder by feeding the graph signal, $z _ { i } ^ { * }$ and $z _ { i j } ^ { * }$ .
+
+Learning objective Let $\hat { f }$ and $\hat { F }$ denote predictions of the target node/edge signals. PA-DGN is trained by minimizing the following objective:
+
+$$
+\mathcal { L } = \sum _ { i \in \mathbb { V } } \vert \vert f _ { i } - \hat { f } _ { i } \vert \vert ^ { 2 } + \sum _ { ( i , j ) \in \mathbb { E } } \vert \vert F _ { i j } - \hat { F } _ { i j } \vert \vert ^ { 2 } .
+$$
+
+For multistep predictions, $\mathcal { L }$ is summed over all predicting steps. If only one type (node or edge) of signal is given, the corresponding term in Eq 3 is used to optimize the parameters in SDL and RGN simultaneously.
+
+# 3 EFFECTIVENESS OF SPATIAL DIFFERENCE LAYER
+
+To investigate if the proposed spatial difference forms (Eq 1) can be beneficial to learning physicsrelated patterns, we use SDL on two different tasks: (1) approximate directional derivatives and (2) predict synthetic graph signals.
+
+# 3.1 APPROXIMATION OF DIRECTIONAL DERIVATIVES
+
+As we claimed in Section 2.3, the standard difference forms (gradient and Laplacian) on a graph can cause significant numerical error easily because they are susceptible to a distance of two points and variations of a given function. To evaluate the applicability of the proposed SDL, we train SDL to approximate directional derivatives on a graph. First, we define a synthetic function and its gradients on 2D space and sample 200 points $( x _ { i } , y _ { i } )$ . Then, we construct a graph on the sampled points by using $k$ -NN algorithm $( k = 4$ ). With the known gradient $\begin{array} { r } { \bigg ( \nabla f = ( \frac { \partial f } { \partial x } , \frac { \partial f } { \partial y } ) \bigg ) } \end{array}$ at each point (a node in the graph), we can compute directional derivatives by projecting $\nabla f$ to a connected edge $e _ { i j }$ (See Figure 3). We compare against four baselines: (1) the finite gradient (FinGrad) (2) multilayer perceptron (MLP) (3) graph networks (GN) (4) a different form of $\mathrm { E q } 1$ (One-w). For the finite gradient $( ( f _ { j } - f _ { i } ) / | | x _ { j } - x _ { i } | | )$ , there is no learnable parameter and it only uses two points. For MLP, we feed $( f _ { i } , f _ { j } , \pmb { x } _ { i } , \pmb { x } _ { j } )$ as input to see whether learnable parameters can benefit the approximation or not. For GN, we use distances of two connected points as edge features and function values on the points as node features. The edge feature output of GN is used as a prediction for the directional derivative on the edge. Finally, we modify the proposed form as $( \overset { w } { \nabla { f } } ) _ { i j } = w _ { i j } f _ { j } - f _ { i }$ . GN and the modified form are used to verify the effectiveness of Eq 1. Note that we define two synthetic functions (Figure 4) which have different property; (1) monotonically increasing from a center and (2) periodically varying.
+
+![](images/067e3e5946fdec6e962b58900cdfdb1a2cc8877584c427fc82ba89e26a60dbda.jpg)  
+Figure 3: Directional derivative on graph
+
+Table 1: Mean squared error $( 1 0 ^ { - 2 } )$ for approximation of directional derivatives.   
+
+<table><tr><td>Functions</td><td>FinGrad</td><td>MLP</td><td>GN</td><td>One-w</td><td>SDL</td></tr><tr><td>fi(x,y)=0.1x² +0.5y²</td><td>6.42±0.47</td><td>2.12±0.32</td><td>1.05±0.42</td><td>1.41±0.44</td><td>0.97±0.39</td></tr><tr><td>f2(x,y)= sin(x) +cos(y)</td><td>5.90±0.04</td><td>2.29±0.77</td><td>2.17±0.34</td><td>6.73±1.17</td><td>1.26±0.05</td></tr></table>
+
+![](images/9fac30ff5be81be8490024894820601036be32ed3ceb7491f1ab6ed14988804d.jpg)  
+Figure 4: Gradients and graph structure of sampled points. Left: the synthetic function is $f _ { 1 } ( x , y ) =$ $0 . \check { 1 } x ^ { 2 } + 0 . 5 y ^ { 2 }$ . Right: the synthetic function is $f _ { 2 } ( x , y ) = \sin ( x ) + \cos ( y )$ .
+
+Approximation accuracy As shown in Table 1, the proposed spatial difference layer outperforms others by a large margin. As expected, FinGrad provides the largest error since it only considers two points without learnable parameters. It is found that the learnable parameters can significantly benefit to approximate the directional derivatives even if input is the same (FinGrad vs. MLP). Note that utilizing neighboring information (GN, One-w, SDL) is generally helpful to learn spatial variations properly. However, simply training parameters in GN is not sufficient and explicitly defining difference, which is important to understand spatial variations, provides more robust inductive bias. One important thing we found is that One-w is not effective as much as GN and it can be even worse than FinGrad. It is because of its limited degree of freedom. As implied in the form $( \nabla _ { w } f ) _ { i j } = w _ { i j } * f _ { j } - f _ { i }$ , only one $w _ { i j }$ adjusts the relative difference between $f _ { i }$ and $f _ { j }$ , and this is not enough to learn whole possible linear combinations of $f _ { i }$ and $f _ { j }$ . The unstable performance supports that the form of SDL is not ad-hoc but more rigorously designed.
+
+# 3.2 GRAPH SIGNAL PREDICTION
+
+We evaluate PA-DGN on the synthetic data sampled from the simulation of specific convectiondiffusion equations, to provide if the proposed model can predict next signals of the simulated dynamics from observations on discrete nodes only. For the simulated dynamics, we use an equation slightly modified based on the one in Long et al. (2018).
+
+$$
+\frac { d f _ { i } ( t ) } { d t } = a ( i ) ( \nabla f ) _ { \hat { x } } + b ( i ) ( \nabla f ) _ { \hat { y } } + c ( i ) \Delta f , \qquad f _ { i } ( 0 ) = f _ { o } ( i ) ,
+$$
+
+where the index $i$ points the $i$ -th node whose coordinate is $( x _ { i } , y _ { i } )$ in the 2D space $( [ 0 , 2 \pi ] \times [ 0 , 2 \pi ] )$ and $\hat { x }$ and $\hat { y }$ indicate $x$ - and $y$ -direction in the space. $a ( i ) = 0 . 5 ( \cos ( y _ { i } ) + x _ { i } ( 2 \pi - \bar { x _ { i } } ) \sin ( x _ { i } ) \bar { ) } + 0 . \bar { 6 }$ $b ( i ) = 2 ( \cos ( y _ { i } ) + \sin ( x _ { i } ) ) + 0 . 8 _ { }$ , and $\begin{array} { r } { c ( i ) = 0 . 5 \left( 1 - \frac { \sqrt { ( x _ { i } - \pi ) ^ { 2 } + ( y _ { i } - \pi ) ^ { 2 } } } { \sqrt { 2 } \pi } \right) } \end{array}$ . Then, we uniformly sample 250 points in the above 2D space. The task is to predict signal values of all points in the future $M$ steps given observed values of the first $N$ steps. For our experiments, we choose $N = 5$ and $M = 1 5$ . Since there is no a priori graph structure on sampled points, we construct a graph with $k$ -NN algorithm $ { [ k = 4 ] }$ ) using the Euclidean distance. Figure 5 shows the dynamics and the graph structure of sampled points.
+
+To evaluate the effect of the proposed SDL on the above prediction task, we cascade SDL and a linear regression model as our prediction model since the dynamics follows a linear partial differential equation. We compare its performance with four baselines: (1) vector auto-regressor (VAR); (2) multilayer perceptron (MLP); (3) StandardOP: the standard approximation of differential operators in Section 2.1 followed by a linear regressor; (4) MeshOP: similar to StandardOP but use the discretization on triangulated mesh in Section 2.2 for differential operators.
+
+Table 2: Mean absolute error $( 1 0 ^ { - 2 } )$ for graph signal prediction.   
+
+<table><tr><td>VAR</td><td>MLP</td><td>StandardOP</td><td>MeshOP</td><td>SDL</td></tr><tr><td>16.84±0.41</td><td>15.75±0.53</td><td>11.99±0.29</td><td>12.82±0.06</td><td>10.87±0.98</td></tr></table>
+
+Prediction Performance Table 2 shows the prediction performance of different models measured with mean absolute error. The prediction model with our proposed spatial differential layer outperforms other baselines. All models incorporating any form of spatial difference operators (StandardOP,
+
+![](images/44d4bb29f77c07af9c69d1563184127ca0bf654c6929bccc6caaf555c37c827b.jpg)  
+Figure 5: Synthetic dynamics and graph structure of sampled points.
+
+MeshOP and SDL) outperform those without spatial difference operators (VAR and MLP), showing that introducing spatial differences information inspired by the intrinsic dynamics helps prediction. However, in cases where points with observable signal are sparse in the space, spatial difference operators derived with fixed rules can be inaccurate and sub-optimal for prediction since the locally linear assumption which they are based on no longer holds. Our proposed SDL, to the contrary, is capable of bridging the gap between approximated difference operators and accurate ones by introducing learnable coefficients utilizing neighboring information, and thus improves the prediction performance.
+
+# 4 PREDICTION: GRAPH SIGNALS ON LAND-BASED WEATHER SENSORS
+
+We evaluate the proposed model on the task of predicting climate observations (Temperature) from the land-based weather stations located in the United States.
+
+# 4.1 EXPERIMENTAL SET-UP
+
+Data and task We sample the weather stations located in the United States from the Online Climate Data Directory of the National Oceanic and Atmospheric Administration (NOAA) and choose the stations which have actively measured meteorological observations during 2015. We choose two geographically close but meteorologically diverse groups of stations: the Western and Southeastern states. We use $k$ -Nearest Neighbor (NN) algorithm $k = 4$ ) to generate graph structures and the final adjacency matrix is $A = \bar { ( } A _ { k } + A _ { k } ^ { \top } ) / 2$ to make it symmetric where $A _ { k }$ is the output adjacency matrix from $k$ -NN algorithm.
+
+Figure 6 shows the distributions of the land-based weather stations and their connectivity. Since the stations are not synchronized and have different timestamps for the observations, we aggregate the time series hourly. The 1-year sequential data are split into the train set (8 months), the validation set (2 months), and the test set (2 months), respectively.
+
+Our main task is to predict the next graph signals based on the current and past graph signals. All methods we evaluate are trained through the objective (Eq 3) with the Adam optimizer and we use scheduled sampling (Bengio et al., 2015) for the models with recurrent modules. We evaluate PA-DGN and other baselines on two prediction tasks, (1) 1-step and (2) multistep-ahead predictions. Furthermore, we demonstrate the ablation study that provides how much the spatial derivatives from our proposed SDL are important signals to predict the graph dynamics.
+
+![](images/a27f4d0228aea93218bb3e8d15df642d05e54640be82b0c45bea4364b2457316.jpg)  
+Figure 6: Weather stations in (left) western (right) southeastern states in the United States and $k$ -NN graph.
+
+# 4.2 GRAPH SIGNAL PREDICTIONS
+
+We compare against the widely used baselines (VAR, MLP, and GRU) for 1-step and multistep prediction. Then, we use RGN (Sanchez-Gonzalez et al., 2018) to examine how much the graph structure is beneficial. Finally, we evaluate PA-DGN to verify if the proposed architecture (Eq 1) is able to reduce prediction loss. Experiment results for the prediction task are summarized in Table 3.
+
+Overall, RGN and PA-DGN are better than other baselines and it implies that the graph structure provides useful inductive bias for the task. It is intuitive as the meteorological observations are continuously changing over the space and time and thus, the observations at the $i$ -th station are strongly related to those of its neighboring stations.
+
+PA-DGN outperforms RGN and the discrepancy comes from the fact that the spatial derivatives (Eq 1) we feed in PA-DGN are beneficial and this finding is expected because the meteorological signals at a certain point are a function of not only its previous signal but also the relative differences between neighbor signals and itself. Knowing the relative differences among local observations is particularly essential to understand physics-related dynamics. For example, Diffusion equation, which describes how physical quantities (e.g., heat) are transported through space over time, is also a function of relative differences of the quantities ( $\begin{array} { r } { \frac { d f } { d t } = D \bar { \Delta } f ) } \end{array}$ rather than values of the neighbor signals. In other words, spatial differences are physics-aware features and it is desired to leverage the features as input to learn dynamics related to physical phenomena.
+
+Table 3: Graph signal prediction results (MAE) on multistep predictions. In each row, we report the average with standard deviations from all baselines and PA-DGN. One step is 1-hour time interval.   
+
+<table><tr><td>Region</td><td>Method</td><td>1-step</td><td>6-step</td><td>12-step</td></tr><tr><td rowspan="7">West</td><td>VAR</td><td>0.1241 ± 0.0234</td><td>0.4295 ± 0.1004</td><td>0.4820 ± 0.1298</td></tr><tr><td>MLP</td><td>0.1040 ± 0.0003</td><td>0.3742 ± 0.0238</td><td>0.4998 ± 0.0637</td></tr><tr><td>GRU</td><td>0.0913 ± 0.0047</td><td>0.1871 ± 0.0102</td><td>0.2707 ± 0.0006</td></tr><tr><td>RGN</td><td>0.0871 ± 0.0033</td><td>0.1708 ± 0.0024</td><td>0.2666 ± 0.0252</td></tr><tr><td>RGN(StandardOP)</td><td>0.0860 ± 0.0018</td><td>0.1674 ± 0.0019</td><td>0.2504 ± 0.0107</td></tr><tr><td>RGN(MeshOP)</td><td>0.0840 ± 0.0015</td><td>0.2119 ± 0.0018</td><td>0.4305 ± 0.0177</td></tr><tr><td>PA-DGN</td><td>0.0840 ± 0.0004</td><td>0.1614 ± 0.0042</td><td>0.2439 ± 0.0163</td></tr><tr><td rowspan="7">SouthEast</td><td>VAR</td><td>0.0889 ± 0.0025</td><td>0.2250 ± 0.0013</td><td>0.3062 ± 0.0032</td></tr><tr><td>MLP</td><td>0.0722 ± 0.0012</td><td>0.1797 ± 0.0086</td><td>0.2514 ± 0.0154</td></tr><tr><td>GRU</td><td>0.0751 ± 0.0037</td><td>0.1724 ± 0.0130</td><td>0.2446 ± 0.0241</td></tr><tr><td>RGN</td><td>0.0790 ± 0.0113</td><td>0.1815 ± 0.0239</td><td>0.2548 ± 0.0210</td></tr><tr><td>RGN(StandardOP)</td><td>0.0942 ± 0.0121</td><td>0.2135 ± 0.0187</td><td>0.2902 ± 0.0348</td></tr><tr><td>RGN(MeshOP)</td><td>0.0905 ± 0.0012</td><td>0.2052 ± 0.0012</td><td>0.2602 ± 0.0062</td></tr><tr><td>PA-DGN</td><td>0.0721 ± 0.0002</td><td>0.1664 ± 0.0011</td><td>0.2408 ± 0.0056</td></tr></table>
+
+# 4.3 CONTRIBUTION OF SPATIAL DERIVATIVES
+
+We further investigate if the modulated spatial derivatives $( \mathrm { E q ~ 1 } )$ ) are effectively advantageous compared to the spatial derivatives defined in Riemannian manifolds. First, RGN without any spatial derivatives is assessed for the prediction tasks on Western and Southeastern states graph signals. Note that this model does not use any extra features but the graph signal, $f ( t )$ . Secondly, we add (1) StandardOP, the discrete spatial differences (Gradient and Laplacian) in Section 2.1 and (2) MeshOP, the triangular mesh approximation of differential operators in Section 2.2 separately as additional signals to RGN. Finally, we incorporate with RGN our proposed Spatial Difference Layer.
+
+Table 3 shows the contribution of each component. As expected, PA-DGN provides much higher drops in MAE $3 . 5 6 \%$ , $5 . 5 0 \%$ , $8 . 5 1 \%$ and $8 . 7 3 \%$ , $8 . 3 2 \%$ , $5 . 4 9 \%$ on two datasets, respectively) compared to RGN without derivatives and the results demonstrate that the derivatives, namely, relative differences from neighbor signals are effectively useful. However, neither RGN with StandardOP nor with MeshOP can consistently outperform RGN. We also found that PA-DGN consistently shows positive effects on the prediction error compared to the fixed derivatives. This finding is a piece of evidence to support that the parameters modulating spatial derivatives in our proposed Spacial Difference Layer are properly inferred to optimize the networks and to improve the prediction performance.
+
+# 5 CONCLUSION
+
+In this paper, we introduce a novel architecture (PA-DGN) that approximates spatial derivatives to use them to represent PDEs which have a prominent role for physics-aware modeling. PA-DGN
+
+effectively learns the modulated derivatives for predictions and the derivatives can be used to discover hidden physics describing interactions between temporal and spatial derivatives.
+
+# ACKNOWLEDGEMENTS
+
+This work is supported in part by NSF Research Grant IIS-1254206 and MINERVA grant N00014-17- 1-2281, granted to co-author Yan Liu in her academic role at the University of Southern California. The views and conclusions are those of the authors and should not be interpreted as representing the official policies of the funding agency, or the U.S. Government. Last but not least, we appreciate anonymous reviewers for your thorough comments and suggestions.
+
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+
+# A APPENDIX
+
+# A.1 SIMULATED DATA
+
+For the simulated dynamics, we discretize the following partial differential equation similar to the one in Long et al. (2018) to simulate the corresponding linear variable-coefficient convection-diffusion equation on graphs.
+
+In a continuous space, we define the linear variable-coefficient convection-diffusion equation as:
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \frac { \partial f } { \partial t } } & { = a ( x , y ) f _ { x } + b ( x , y ) f _ { y } + c ( x , y ) \Delta f } \\ { \dot { f } \rvert _ { t = 0 } } & { = f _ { 0 } ( x , y ) } \\ { \textrm { \scriptsize 1 } \Omega = [ 0 , 2 \pi ] \times [ 0 , 2 \pi ] , ( t , x , y ) \in [ 0 , 0 . 2 ] \times \Omega , a ( x , y ) = 0 . 5 \big ( \cos ( y ) + x ( 2 \pi - x ) \sin ( x ) \big ) + } \\ { b ( x , y ) = 2 ( \cos ( y ) + \sin ( x ) ) + 0 . 8 , c ( x , y ) = 0 . 5 \bigg ( 1 - \frac { \sqrt { ( x _ { i } - \pi ) ^ { 2 } + ( y _ { i } - \pi ) ^ { 2 } } } { \sqrt { 2 \pi } } \bigg ) . } \end{array} \right. } \end{array}
+$$
+
+We follow the setting of initialization in Long et al. (2018):
+
+$$
+f _ { 0 } ( x , y ) = \sum _ { | k | , | l | \leq N } \lambda _ { k , l } \cos ( k x + l y ) + \gamma _ { k , l } \sin ( k x + l y )
+$$
+
+, where $\begin{array} { r } { N = 9 , \lambda _ { k , l } , \gamma _ { k , l } \sim \mathcal { N } \left( 0 , \frac { 1 } { 5 0 } \right) } \end{array}$ , and $k$ and $l$ are chosen randomly.
+
+We use spatial difference operators to approximate spatial derivatives:
+
+$$
+\left\{ \begin{array} { r l } { f _ { x } ( x _ { i } , y _ { i } ) } & { { } = \displaystyle \frac { 1 } { 2 s } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } - s , y _ { i } ) ) - \displaystyle \frac { 1 } { 2 s } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } + s , y _ { i } ) ) } \\ { f _ { y } ( x _ { i } , y _ { i } ) } & { { } = \displaystyle \frac { 1 } { 2 s } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } , y _ { i } - s ) ) - \displaystyle \frac { 1 } { 2 s } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } , y _ { i } + s ) ) } \\ { f _ { x x } ( x _ { i } , y _ { i } ) } & { { } = \displaystyle \frac { 1 } { s ^ { 2 } } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } - s , y _ { i } ) ) + \displaystyle \frac { 1 } { s ^ { 2 } } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } + s , y _ { i } ) ) } \\ { f _ { y y } ( x _ { i } , y _ { i } ) } & { { } = \displaystyle \frac { 1 } { s ^ { 2 } } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } , y _ { i } - s ) ) + \displaystyle \frac { 1 } { s ^ { 2 } } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } , y _ { i } + s ) ) } \end{array} \right.
+$$
+
+, where $s$ is the spatial grid size for discretization.
+
+Then we rewrite (5) with difference operators defined on graphs:
+
+$$
+\left\{ \begin{array} { l l } { \frac { \partial f } { \partial t } = a ( i ) ( \nabla f ) _ { \hat { x } } + b ( i ) ( \nabla f ) _ { \hat { y } } + c ( i ) ( ( \Delta f ) _ { \hat { x } \hat { x } } + ( \Delta f ) _ { \hat { y } \hat { y } } ) } \\ { f _ { i } ( 0 ) = f _ { o } ( i ) } \end{array} \right.
+$$
+
+, where
+
+$$
+\begin{array} { r } { a ( i ) ( x _ { j } , y _ { j } ) = \left\{ \begin{array} { l l } { \displaystyle \frac { a ( x _ { i } , y _ { i } ) } { 2 s } } & { \mathrm { i f ~ } x _ { i } = x _ { j } + s , y _ { i } = y _ { j } } \\ { - \displaystyle \frac { a ( x _ { i } , y _ { i } ) } { 2 s } } & { \mathrm { i f ~ } x _ { i } = x _ { j } - s , y _ { i } = y _ { j } } \\ { \displaystyle \frac { b ( x _ { i } , y _ { i } ) } { 2 s } } & { \mathrm { i f ~ } x _ { i } = x _ { j } , y _ { i } = y _ { j } + s } \\ { - \displaystyle \frac { b ( x _ { i } , y _ { i } ) } { 2 s } } & { \mathrm { i f ~ } x _ { i } = x _ { j } , y _ { i } = y _ { j } - s } \end{array} \right. } \\ { c ( i ) ( x _ { j } , y _ { j } ) = \frac { c } { s ^ { 2 } } } \end{array}
+$$
+
+Then we replace the gradient w.r.t time in (8) with temporal discretization:
+
+$$
+\left\{ \begin{array} { l l } { f ( t + 1 ) = \Delta t ( a ( i ) ( \nabla f ) _ { \hat { x } } + b ( i ) ( \nabla f ) _ { \hat { y } } + c ( i ) ( ( \Delta f ) _ { \hat { x } \hat { x } } + ( \Delta f ) _ { \hat { y } \hat { y } } ) ) + f ( t ) } \\ { f _ { i } ( 0 ) = f _ { o } ( i ) } \end{array} \right.
+$$
+
+, where $\Delta t$ is the time step in temporal discretization.
+
+Equation (12) is used for simulating the dynamics described by the equation (5). Then, we uniformly sample 250 points in the above 2D space and choose their corresponding time series of $u$ as the dataset used in our synthetic experiments. We generate 1000 sessions on a $5 0 \times 5 0$ regular mesh with time step size $\Delta t = 0 . 0 1$ . 700 sessions are used for training, 150 for validation and 150 for test.
+
+# A.2 EXPERIMENT SETTINGS
+
+Here we provide additional details for models we used in this work, including model architecture settings and hyper-parameter settings.
+
+# A.2.1 MODEL SETTINGS
+
+Unless mentioned otherwise, all models use a hidden dimension of size 64.
+
+• VAR: A vector autoregression model with 2 lags. Input is the concatenated features of previous 2 frames. The weights are shared among all nodes in the graph.   
+• MLP: A multilayer perceptron model with 2 hidden layers. Input is the concatenated features of previous 2 frames. The weights are shared among all nodes in the graph. GRU: A Gated Recurrent Unit network with 2 hidden layers. Input is the concatenated features of previous 2 frames. The weights are shared among all nodes in the graph. RGN: A recurrent graph neural network model with 2 GN blocks. Each GN block has an edge update block and a node update block, both of which use a 2-layer GRU cell as the update function. We set its hidden dimension to 73 so that it has the same number of learnable parameters as our proposed model PA-DGN. RGN(StandardOP): Similar to RGN, but use the output of difference operators in Section 2.1 as extra input features. We set its hidden dimension to 73. RGN(MeshOP): Similar to RGN(StandardOP), but the extra input features are calculated using opeartors in Section 2.2. We set its hidden dimension to 73. PA-DGN: Our proposed model. The spatial derivative layer uses a message passing neural network (MPNN) with 2 GN blocks using 2-layer MLPs as update functions. The forward network part uses a recurrent graph neural network with 2 recurrent GN blocks using 2-layer GRU cells as update functions.
+
+The numbers of learnable parameters of all models are listed as follows:
+
+Table 4: Numbers of learnable parameters.   
+
+<table><tr><td>Model</td><td>VAR</td><td>MLP</td><td>GRU</td><td>RGN</td><td>RGN(StandardOP)</td><td>RGN(MeshOP)</td><td>PA-DGN</td></tr><tr><td> #Params</td><td>3</td><td>4,417</td><td>37,889</td><td>345,876</td><td>341,057</td><td>342,152</td><td>340,001</td></tr></table>
+
+# A.2.2 TRAINING SETTINGS
+
+The number of evaluation runs We performed 3 times for every experiment in this paper to report the mean and standard deviations.
+
+Length of prediction For experiments on synthetic data, all models take first 5 frames as input and predict the following 15 frames. For experiments on NOAA datasets, all models take first 12 frames as input and predict the following 12 frames.
+
+Training hyper-parameters We use Adam optimizer with learning rate 1e-3, batch size 8, and weight decay of 5e-4. All experiments are trained for a maximum of 2000 epochs with early stopping. All experiments are trained using inverse sigmoid scheduled sampling with the coefficient $k = 1 0 7$
+
+Environments All experiments are implemented with Python3.6 and PyTorch 1.1.0, and are run with NVIDIA GTX 1080 Ti GPUs.
+
+# A.3 EFFECT OF DIFFERENT GRAPH STRUCTURES
+
+In this section, we evaluate the effect of 2 different graph structures on baselines and our models: (1) k-NN: a graph constructed with $k$ -NN algorithm $k = 4 ,$ ; (2) TriMesh: a graph generated with Delaunay Triangulation. All graphs use the Euclidean distance.
+
+Table 5: Mean absolute error $( 1 0 ^ { - 2 } )$ ) for graph signal prediction on the synthetic dataset.   
+
+<table><tr><td rowspan="2">VAR</td><td rowspan="2">MLP</td><td colspan="2">StandardOP</td><td colspan="2">MeshOP</td><td colspan="2">SDL</td></tr><tr><td>k-NN</td><td>TriMesh</td><td>k-NN</td><td>TriMesh</td><td>k-NN</td><td>TriMesh</td></tr><tr><td>17.30</td><td>16.27</td><td>12.00</td><td>12.29</td><td>12.87</td><td>12.82</td><td>11.04</td><td>12.40</td></tr></table>
+
+Table 6: Graph signal prediction results (MAE) on multistep predictions. In each row, we report the average with standard deviations from all baselines and PA-DGN. One step is 1 hour time interval.   
+
+<table><tr><td>Region</td><td>Method</td><td>Graph</td><td>1-step</td><td>6-step</td><td>12-step</td></tr><tr><td rowspan="7">West</td><td>VAR MLP</td><td></td><td>0.1241 ± 0.0234</td><td>0.4295 ± 0.1004</td><td>0.4820 ± 0.1298</td></tr><tr><td>GRU</td><td>1</td><td>0.1040 ± 0.0003 0.0913 ± 0.0047</td><td>0.3742 ± 0.0238 0.1871 ± 0.0102</td><td>0.4998 ± 0.0637 0.2707 ± 0.0006</td></tr><tr><td>RGN</td><td>k-NN TriMesh</td><td>0.0871 ± 0.0033</td><td>0.1708 ± 0.0024</td><td>0.2666 ± 0.0252</td></tr><tr><td>RGN (StandardOP)</td><td>k-NN TriMesh</td><td>0.0897 ± 0.0030 0.0860 ± 0.0018</td><td>0.1723 ± 0.0116 0.1674 ± 0.0019</td><td>0.2800 ± 0.0414 0.2504 ± 0.0107</td></tr><tr><td>RGN</td><td>k-NN</td><td>0.0842 ± 0.0011 0.0840 ± 0.0015</td><td>0.1715 ± 0.0027 0.2119 ± 0.0018</td><td>0.2517 ± 0.0369 0.4305 ± 0.0177</td></tr><tr><td>(MeshOP)</td><td>TriMesh k-NN</td><td>0.0846 ± 0.0017 0.0840 ± 0.0004</td><td>0.2090 ± 0.0077</td><td>0.4051 ± 0.0457</td></tr><tr><td>PA-DGN VAR</td><td>TriMesh</td><td>0.0849 ± 0.0012</td><td>0.1614 ± 0.0042 0.1610 ± 0.0029</td><td>0.2439 ± 0.0163 0.2473 ± 0.0162</td></tr><tr><td rowspan="7">SouthEast</td><td>MLP GRU</td><td></td><td>0.0889 ± 0.0025 0.0722 ± 0.0012</td><td>0.2250 ± 0.0013 0.1797 ± 0.0086</td><td>0.3062 ± 0.0032 0.2514 ± 0.0154</td></tr><tr><td>RGN</td><td>= k-NN</td><td>0.0751 ± 0.0037 0.0790 ± 0.0113</td><td>0.1724 ± 0.0130 0.1815 ± 0.0239</td><td>0.2446 ± 0.0241 0.2548 ± 0.0210</td></tr><tr><td>RGN (StandardOP)</td><td>TriMesh k-NN</td><td>0.0932 ± 0.0105 0.0942 ± 0.0121</td><td>0.2076 ± 0.0200 0.2135 ± 0.0187</td><td>0.2854 ± 0.0211 0.2902 ± 0.0348</td></tr><tr><td rowspan="2">RGN (MeshOP)</td><td>TriMesh</td><td>0.0868 ± 0.0132</td><td>0.1885 ± 0.0305</td><td>0.2568 ± 0.0328</td></tr><tr><td>k-NN</td><td>0.0913 ± 0.0016</td><td>0.2069 ± 0.0031</td><td>0.2649 ± 0.0092</td></tr><tr><td></td><td>TriMesh k-NN</td><td>0.0877 ± 0.0020 0.0721 ± 0.0002</td><td>0.2043 ± 0.0026</td><td>0.2579 ± 0.0057</td></tr><tr><td>PA-DGN</td><td>TriMesh</td><td>0.0876 ± 0.0096</td><td>0.1664 ± 0.0011 0.2002 ± 0.0163</td><td>0.2408 ± 0.0056 0.2623 ± 0.0180</td></tr></table>
+
+Table 5 and Table 6 show the effect of different graph structures on the synthetic dataset used in Section 3.2 and the real-world dataset in Section 4.2 separately. We find that for different models the effect of graph structures is not homogeneous. For RGN and PA-DGN, $k$ -NN graph is more beneficial to the prediction performance than TriMesh graph, because these two models rely more on neighboring information and a $k$ -NN graph incorporates it better than a Delaunay Triangulation graph. However, switching from TriMesh graph to $k$ -NN graph is harmful to the prediction accuracy of RGN(MeshOP) since Delaunay Triangulation is a well-defined method for generating triangulated mesh in contrast to $k$ -NN graphs. Given the various effect of graph structures on different models, our proposed PA-DGN under $k$ -NN graphs always outperforms other baselines using any graph structure.
+
+![](images/00af0c99e444548cf3f703e615283468f9daffee0b6b915718fed2d1cc52a107.jpg)  
+Figure 7: MAE across the nodes.
+
+# A.4 THE DISTRIBUTION OF PREDICTION ERROR ACROSS NODES
+
+Figure 7 provides the distribution of MAEs across the nodes of PA-DGN applied to the graph signal prediction task of the west coast region of the real-world dataset in Section 4.2. As shown in the figure, nodes with the highest prediction error for short-term prediction are gathered in the inner part where the observable nodes are sparse, while for long-term prediction nodes in the area with a limited number of observable points no longer have the largest MAE. This implies that PA-DGN can utilize neighboring information efficiently even under the limitation of sparsely observable points.
+
+# A.5 EVALUATION ON NEMO SEA SURFACE TEMPERATURE (SST) DATASET
+
+We tested our proposed method and baselines on the NEMO sea surface temperature (SST) dataset1. We first download the data in the area between $5 0 \mathrm { N } ^ { \circ }  – 6 5 \mathrm { N } ^ { \circ }$ and $7 5 \mathrm { W } ^ { \circ } \ – 1 0 \mathrm { W } ^ { \circ }$ starting from 2016-01- 01 to 2017-12-31, then we crop the $[ 0 , 5 5 0 ] \times [ 1 0 0 , 6 5 0 ]$ square from the area and sample 250 points from the square as our chosen dataset. We divide the data into 24 sequences, each lasting 30 days, and truncate the tail. All models use the first 5-day SST as input and predict the SST in the following 15 and 25 days. We use the data in 2016 for training all models and the left for testing.
+
+For StandardOP, MeshOP and SDL, we test both options using linear regression and using RGN for the prediction part and report the best result. The results in Table 7 show that all methods incorporating spatial differences gain improvement on prediction and that our proposed learnable SDL outperforms all other baselines.
+
+Table 7: Mean absolute error $( 1 0 ^ { - 2 } )$ for SST graph signal prediction.   
+
+<table><tr><td></td><td>VAR</td><td>MLP</td><td>GRU</td><td>RGU</td><td>StandardOP</td><td>MeshOP</td><td>SDL</td></tr><tr><td>15-step</td><td>15.123</td><td>15.058</td><td>15.101</td><td>15.172</td><td>14.756</td><td>14.607</td><td>14.382</td></tr><tr><td>25-step</td><td>19.533</td><td>19.473</td><td>19.522</td><td>19.705</td><td>18.983</td><td>18.977</td><td>18.434</td></tr></table>
+
+# A.6 EVALUATION ON DATASETS WITH DIFFERENT SPARSITY
+
+We changed the number of nodes to control the sparsity of data. As shown in Table 8, our proposed model outperforms others under various settings of sparsity on the synthetic experiment in Section 3.2.
+
+Table 8: Mean absolute error $( 1 0 ^ { - 2 } )$ ) for graph signal prediction with different sparsity.   
+
+<table><tr><td>#Nodes</td><td>VAR</td><td>MLP</td><td>StandardOP</td><td>MeshOP</td><td>SDL</td></tr><tr><td>250</td><td>0.1730</td><td>0.1627</td><td>0.1200</td><td>0.1287</td><td>0.1104</td></tr><tr><td>150</td><td>0.1868</td><td>0.1729</td><td>0.1495</td><td>0.1576</td><td>0.1482</td></tr><tr><td>100</td><td>0.1723</td><td>0.1589</td><td>0.1629</td><td>0.1696</td><td>0.1465</td></tr></table>
+
+Furthermore, we sampled 400 points and trained SDL as described in Section 3.1, and resampled fewer points (350,300,250,200) to evaluate if SDL generalizes less sparse setting. As Table 9 shows, MSE increases when fewer sample points are used. However, SDL is able to provide much more accurate gradients even if it is trained under a new graph with different properties. Thus, the results support that SDL is able to generalize the c setting.
+
+Table 9: Mean squared error $( 1 0 ^ { - 2 } )$ for approximations of directional derivatives of function $f _ { 2 } ( x , y ) = \sin \left( x \right) + \cos \left( y \right)$ with different sparsity.   
+
+<table><tr><td>Method</td><td>350 Nodes</td><td>300 Nodes</td><td>250 Nodes</td><td>200 Nodes</td></tr><tr><td>FinGrad</td><td>2.88 ± 0.11</td><td>3.42 ± 0.14</td><td>3.96 ± 0.17</td><td>4.99 ± 0.31</td></tr><tr><td>SDL</td><td>1.03 ± 0.09</td><td>1.14 ± 0.12</td><td>1.40 ± 0.10</td><td>1.76 ± 0.10</td></tr></table>

parse/train/r1gelyrtwH/r1gelyrtwH_content_list.json

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+    {
+        "type": "text",
+        "text": "PHYSICS-AWARE DIFFERENCE GRAPH NETWORKS FOR SPARSELY-OBSERVED DYNAMICS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            820,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sungyong Seo∗, Chuizheng Meng∗, Yan Liu ",
+        "bbox": [
+            186,
+            170,
+            483,
+            185
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Computer Science   \nUniversity of Southern California   \n{sungyons,chuizhem,yanliu.cs}@usc.edu ",
+        "bbox": [
+            184,
+            185,
+            547,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sparsely available data points cause numerical error on finite differences which hinders us from modeling the dynamics of physical systems. The discretization error becomes even larger when the sparse data are irregularly distributed or defined on an unstructured grid, making it hard to build deep learning models to handle physics-governing observations on the unstructured grid. In this paper, we propose a novel architecture, Physics-aware Difference Graph Networks (PA-DGN), which exploits neighboring information to learn finite differences inspired by physics equations. PA-DGN leverages data-driven end-to-end learning to discover underlying dynamical relations between the spatial and temporal differences in given sequential observations. We demonstrate the superiority of PA-DGN in the approximation of directional derivatives and the prediction of graph signals on the synthetic data and the real-world climate observations from weather stations. ",
+        "bbox": [
+            233,
+            295,
+            766,
+            462
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            491,
+            336,
+            507
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Modeling real world phenomena, such as climate observations, traffic flow, physics and chemistry simulation (Li et al., 2018; Geng et al., 2019; Long et al., 2018; de Bezenac et al., 2018; SanchezGonzalez et al., 2018; Gilmer et al., 2017), is important but extremely challenging. While deep learning has achieved remarkable successes in prediction tasks by learning latent representations from data-rich applications such as image recognition (Krizhevsky et al., 2012), text understanding (Wu et al., 2016), and speech recognition (Hinton et al., 2012), we confront many challenging scenarios in modeling natural phenomena with deep neural networks when only a limited number of observations are available. Particularly, the sparsely available data points cause substantial numerical error when we utilize existing finite difference operators and the limitation requires a more principled way to redesign deep learning models. ",
+        "bbox": [
+            174,
+            522,
+            825,
+            661
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While many methods have been proposed to model physics-simulated observations using deep learning, many of them are designed under the assumption that input is on a continuous domain. For example, Raissi et al. (2017a;b) proposed physics-informed neural networks (PINNs) to learn nonlinear relations between input (spatial- and temporal-coordinates $( { \\pmb x } , t ) )$ and output simulated with a given partial differential equation (PDE). Since Raissi et al. (2017a;b) use the coordinates as input and compute derivatives based on the coordinates to represent the equation, the setting is only valid when the data are densely observed over spatial and temporal space. ",
+        "bbox": [
+            174,
+            669,
+            825,
+            767
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Prior knowledge related to physics equations has been combined with data-driven models for various purposes. Chen et al. (2015) proposed a nonlinear diffusion process for image restoration and de Bezenac et al. (2018) incorporated the transport physics (advection-diffusion equation) with deep neural networks for forecasting sea surface temperature by extracting the motion field. Lutter et al. (2019) introduced deep Lagrangian networks specialized to learn Lagrangian mechanics with learnable parameters. Seo & Liu (2019) proposed a physics-informed regularizer to impose dataspecific physics equations. In common, the methods in Chen et al. (2015); de Bezenac et al. (2018); Lutter et al. (2019) are not efficiently applicable to sparsely discretized input as only a small number of data points are available and continuous properties on given space are not easily recovered. It is unsuitable to directly use continuous differential operators to provide local behaviors because it is hard to approximate the continuous derivatives precisely with the sparse points (Shewchuk, 2002; Amenta & Kil, 2004; Luo et al., 2009). Furthermore, they are only applicable when the specific physics equations are explicitly given and still hard to be generalized to incorporate other types of equations. ",
+        "bbox": [
+            174,
+            773,
+            825,
+            898
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            172
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As another direction to modeling physics-simulated data, Long et al. (2018) proposed PDE-Net which uncovers the underlying hidden PDEs and predicts the dynamics of complex systems. Ruthotto & Haber (2018) derived new CNNs: parabolic and hyperbolic CNNs based on ResNet (He et al., 2016) architecture motivated by PDE theory. While Long et al. (2018); Ruthotto & Haber (2018) are flexible to uncover hidden physics from the constrained kernels, it is still restrictive to a regular grid where the proposed constraints on the learnable filters are easily defined. ",
+        "bbox": [
+            174,
+            180,
+            825,
+            263
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The topic of reasoning physical dynamics of discrete objects has been actively studied (SanchezGonzalez et al., 2018; Battaglia et al., 2016; Chang et al., 2016) as the appearance of graph-based neural networks (Kipf & Welling, 2017; Santoro et al., 2017; Gilmer et al., 2017). Although these models can handle sparsely located data points without explicitly given physics equations, they are purely data-driven so that the physics-inspired inductive bias for exploiting finite differences is not considered at all. In contrast, our method consists of physics-aware modules allowing efficiently leveraging the inductive bias to learn spatiotemporal data from the physics system. ",
+        "bbox": [
+            174,
+            270,
+            825,
+            368
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose physics-aware difference graph networks (PA-DGN) whose architecture is motivated to leverage differences of sparsely available data from the physical systems. The differences are particularly important since most of the physics-related dynamic equations (e.g., Navier–Stokes equations) handle differences of physical quantities in spatial and temporal space instead of using the quantities directly. Inspired by the property, we first propose spatial difference layer (SDL) to efficiently learn the local representations by aggregating neighboring information in the sparse data points. The layer is based on graph networks (GN) as it easily leverages structural features to learn the localized representations and share the parameters for computing the localized features. Then, the layer is followed by recurrent graph networks (RGN) to predict the temporal difference which is another core component of physics-related dynamic equations. PA-DGN is applicable to various tasks and we provide two representative tasks; the approximation of directional derivatives and the prediction of graph signals. ",
+        "bbox": [
+            174,
+            376,
+            825,
+            541
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are: ",
+        "bbox": [
+            174,
+            549,
+            318,
+            563
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We tackle a limitation of the sparsely discretized data which cause numerical error when we model the physical system by proposing spatial difference layer (SDL) for efficiently exploiting neighboring information under the limitation of sparsely observable points.   \n• We combine SDL with recurrent graph networks to build PA-DGN which automatically learns the underlying spatiotemporal dynamics in graph signals.   \n• We verify that PA-DGN is effective in approximating directional derivatives and predicting graph signals in synthetic data. Then, we conduct exhaustive experiments to predict climate observations from land-based weather stations and demonstrate that PA-DGN outperforms other baselines. ",
+        "bbox": [
+            215,
+            575,
+            825,
+            713
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PHYSICS-AWARE DIFFERENCE GRAPH NETWORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            734,
+            614,
+            751
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce the building module used to learn spatial differences of graph signals and describe how the module is used to predict signals in the physics system. ",
+        "bbox": [
+            173,
+            767,
+            823,
+            796
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 DIFFERENCE OPERATORS ON GRAPH",
+        "text_level": 1,
+        "bbox": [
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+            813,
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "As approximation of derivatives in continuous domain, difference operators have been used as a core role to compute numerical solutions of (continuous) differential equations. Since it is hard to derive closed-form expressions of derivatives in real-world data, the difference operators have been considered as alternative tools to describe and solve PDEs in practice. The operators are especially important for physics-related data (e.g., meteorological observations) because the governing rules behind the observations are mostly differential equations. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c729f07d57a78bf799f70079991ad77dbfac4589faed989f6fb73a67b408728f.jpg",
+        "image_caption": [
+            "Figure 1: Examples of difference operators applied to graph signal. Filters used for the processing are (b) $\\textstyle \\sum _ { j } ( f _ { i } - f _ { j } )$ (c) $\\textstyle \\sum _ { j } ( 1 . 1 f _ { i } { \\bar { - } } f _ { j } )$ , (d) $f _ { j } - 0 . 5 f _ { i }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Graph signals Given a graph $\\mathcal { G } = ( \\mathbb { V } , \\mathbb { E } )$ where $\\mathbb { V }$ is a set of vertices $\\mathbb { V } = \\{ 1 , \\ldots , N _ { v } \\}$ and $\\mathbb { E }$ is a set of edges $\\mathbb { E } \\subseteq \\bar { \\{ ( i , j ) | i , j \\in \\mathbb { V } \\} }$ $\\mathbf { \\left| \\mathbb { E } \\right| } = N _ { e } )$ , graph signals on all nodes at time $t$ are $f ( t ) = \\{ f _ { i } ( t ) \\mid \\bar { i } \\in \\mathbb { V } \\}$ where $f _ { i } : \\mathbb { V } \\to \\mathbb { R }$ . Graph signals on edges can also be defined similarly, $F ( t ) = \\{ F _ { i j } ( t ) \\mid ( i , j ) \\in \\mathbb { E } \\}$ where $F _ { i j } : \\mathbb { E }  \\mathbb { R }$ . Both signals can be multidimensional. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Gradient on graph The gradient $( \\nabla )$ of a function on nodes of a graph is represented by finite difference ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca3bfff6311655da1ef17835f7a431e1404a51b7fab14257060c01125b662d01.jpg",
+        "text": "$$\n\\begin{array} { r } { \\nabla : L ^ { 2 } ( \\mathbb { V } ) \\to L ^ { 2 } ( \\mathbb { E } ) , \\qquad ( \\nabla f ) _ { i j } = \\left( f _ { j } - f _ { i } \\right) \\quad \\mathrm { i f } \\ ( i , j ) \\in \\mathbb { E } \\mathrm { ~ a n d ~ } 0 \\mathrm { ~ o t h e r w i s e } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            375,
+            759,
+            393
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $L ^ { 2 } ( \\mathbb { V } )$ and $L ^ { 2 } ( \\mathbb { E } )$ denote Hilbert spaces for node/edge functions, respectively. The gradients on a graph provide finite differences of graph signals and they become edge $( i , j )$ features. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Laplace-Beltrami operator Laplace-Beltrami operator (or Laplacian, $\\Delta$ ) in graph domain is defined as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0b73b4a139d7aa55275c1d006fb7cbdc47ba23284ce8eb3dc6abf55767d207b7.jpg",
+        "text": "$$\n\\Delta : L ^ { 2 } ( \\mathbb { V } ) \\to L ^ { 2 } ( \\mathbb { V } ) , \\qquad ( \\Delta f ) _ { i } = \\sum _ { j : ( i , j ) \\in \\mathbb { E } } ( f _ { i } - f _ { j } ) \\quad \\forall i , j \\in \\mathbb { V }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            481,
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+            517
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This operator is usually regarded as a matrix form in other literature, $\\pmb { { \\cal L } } = \\pmb { { \\cal D } } - \\pmb { { \\cal A } }$ where $\\pmb { A }$ is an adjacency matrix and $\\begin{array} { r } { \\dot { D } = \\mathrm { d i a g } ( \\sum _ { j : j \\neq i } A _ { i j } ) } \\end{array}$ is a degree matrix. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 DIFFERENCE OPERATORS ON TRIANGULATED MESH",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "According to Crane (2018), the gradient and Laplacian operators on the triangulated mesh can be discretized by incorporating the coordinates of nodes. To obtain the gradient operator, the per-face gradient of each triangular face is calculated first. Then, the gradient on each node is the area-weighted average of all its neighboring faces, and the gradient on edge $( i , j )$ is defined as the dot product between the per-node gradient and the direction vector $e _ { i j }$ . The Laplacian operator can be discretized with Finite Element Method (FEM): ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4eef4691b4f5bf40bb610cc048bf82e57f66bd0db60860e3020ea7f9feb72e89.jpg",
+        "text": "$$\n( \\Delta f ) _ { i } = \\frac { 1 } { 2 } \\sum _ { j : ( i , j ) \\in \\mathbb { E } } \\left( \\cot \\alpha _ { j } + \\cot \\beta _ { j } \\right) \\left( f _ { j } - f _ { i } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            690,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where node $j$ belongs to node $i$ ’s immediate neighbors $( j \\in \\mathbb { N } _ { i } )$ and $( \\alpha _ { j } , \\beta _ { j } )$ are two opposing angles of the edge $( i , j )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 SPATIAL DIFFERENCE LAYER ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While the difference operators are generalized in Riemannian manifolds (Lai et al., 2013; Lim, 2015), there exist numerical error compared to those in continuous space and the error can be larger when the nodes are spatially far from neighboring nodes because the connected nodes $( j \\in \\mathbb { N } _ { i } )$ ) of $i \\cdot$ -th node fail to represent local features around the node. Furthermore, the error is even larger if available data points are sparsely distributed (e.g., sensor-based observations). In other words, the difference operators are unlikely to discover meaningful spatial variations behind the sparse observations since they are highly limited to immediate neighboring information only. To mitigate the limitation, we propose spatial difference layer (SDL) which consists of a set of parameters to define learnable difference operators as a form of gradient and Laplacian to fully utilize neighboring information: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/877d4826df123901cdca612fd7715d80895076ebabfe7e3ae1d04828fbea9bce.jpg",
+        "image_caption": [
+            "Figure 2: Physics-aware Difference Graph Networks for graph signal prediction. Blue boxes have learnable parameters and all parameters are trained through end-to-end learning. The nodes/edges can be multidimensional. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c045ba5d328f3f7fbc93959675b092ddf2f0e2dcb63ef72a2d30450f9fb78c2f.jpg",
+        "text": "$$\n( ^ { w } \\nabla f ) _ { i j } = w _ { i j } ^ { ( g _ { 1 } ) } ( f _ { j } - w _ { i j } ^ { ( g _ { 2 } ) } f _ { i } ) , \\qquad ( ^ { w } \\Delta f ) _ { i } = \\sum _ { j : ( i , j ) \\in \\mathbb { R } } w _ { i j } ^ { ( l _ { 1 } ) } ( f _ { i } - w _ { i j } ^ { ( l _ { 2 } ) } f _ { j } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            367,
+            754,
+            405
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { i j }$ are the parameters tuning the difference operators along with the corresponding edge direction $e _ { i j }$ . The two forms (Eq 1) are associated with edge and node features, respectively. The superscript in $^ w \\nabla$ and $w _ { \\Delta }$ denotes that the difference operators are functions of the learnable parameters $w$ . $w _ { i j } ^ { ( g ) }$ and w(l)ij are obtained by integrating local information as follow: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/64841cbcc584c9fe30cf972c19a83eb6fd3e95423671774739a1e88701ffa918.jpg",
+        "text": "$$\nw _ { i j } = g ( \\{ f _ { k } , F _ { m n } | k , ( m , n ) \\in h \\mathrm { - h o p ~ n e i g h b o r h o o d ~ o f ~ e d g e ~ } ( i , j ) \\} )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            477,
+            727,
+            496
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "While the standard difference operators consider two connected nodes only $\\mathit { i }$ and $j$ ) for each edge $( i , j )$ , Eq 2 uses a larger view $h$ -hop) to represent the differences between $i$ and $j$ nodes. Since graph networks (GN) (Battaglia et al., 2018) are efficient networks to aggregate neighboring information, we use GN for $g ( \\cdot )$ function and $w _ { i j }$ are edge features of output of GN. Eq 2 can be viewed as a higher-order difference equation because nodes/edges which are multi-hop apart are considered. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$w _ { i j }$ has a similar role of parameters in convolution kernels of CNNs. For example, while the standard gradient operator can be regarded as an example of simple edge-detecting filters, the operator can be a sharpening filter if w(g1)ij and $\\begin{array} { r } { w _ { i j } ^ { ( g _ { 2 } ) } = \\bar { \\frac { | \\mathbb { N } _ { i } | + 1 } { | \\mathbb { N } _ { i } | } } } \\end{array}$ for $i$ node and the operators over each edge are summed. In other words, by modulating $w _ { i j }$ , it is readily extended to conventional kernels including edge detection or sharpening filters and even further complicated kernels. On top of $w _ { i j }$ , the difference forms in Eq 1 make an optimizing process for learnable parameters based on the differences instead of the values intentionally. $\\mathrm { E q } 1$ thus naturally provides the physics-inspired inductive bias which is particularly effective for modeling physics-related observations. Furthermore, it is easily possible to increase the number of channels for w(g)ij and w(l)ij to be more expressive. Figure 1 illustrates how the exemplary filters convolve the given graph signals. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 RECURRENT GRAPH NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Difference graph Once the modulated spatial differences $( ^ { w } \\nabla f ( t ) , ^ { w } \\Delta f ( t ) )$ are obtained, they are concatenated with the current signals $f ( t )$ to construct node-wise $\\left( z _ { i } \\right)$ and edge-wise $( z _ { i j } )$ features and the graph is called a difference graph. The difference graph includes all information to describe spatial variations. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Recurrent graph networks Given a snapshot $( f ( t ) , F ( t ) )$ of a sequence of graph signals, one difference graph is obtained and is used to predict next graph signals. While a non-linear layer can be used to combine the learned spatial differences to predict the next signals, it is limited to discover spatial relations only among the features in the difference graph. Since many equations describing physics-related phenomena are non-static (e.g., Navier–Stokes equations), we adopt recurrent graph networks (RGN) (Sanchez-Gonzalez et al., 2018) with a graph state $\\mathcal { G } _ { h }$ as input to combine the spatial differences with temporal variations. RGN returns a graph state $( \\mathcal { G } _ { h } ^ { * } = ( h ^ { * ( v ) } , h ^ { * ( e ) } ) )$ and next graph signal $z _ { i } ^ { * }$ and $z _ { i j } ^ { * }$ . The update rule is described as follow: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1. $( \\boldsymbol { z } _ { i j } ^ { * } , \\pmb { h } ^ { * ( e ) } )  \\phi ^ { e } ( \\boldsymbol { z } _ { i j } , \\boldsymbol { z } _ { i } , \\boldsymbol { z } _ { j } , \\boldsymbol { h } ^ { ( e ) } )$ for all $( i , j ) \\in \\mathbb { E }$ pairs, 2. $( z _ { i } ^ { * } , h ^ { * ( v ) } ) \\gets \\phi ^ { v } ( z _ { i } , \\bar { z } _ { i } ^ { \\prime } , h ^ { ( v ) } )$ for all $i \\in \\mathbb { V }$ , $\\bar { z } _ { i } ^ { \\prime }$ is an aggregated edge attribute related to the node $i$ , ",
+        "bbox": [
+            210,
+            145,
+            617,
+            202
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\phi ^ { e } , \\phi ^ { v }$ are edge and node update functions, respectively, and they can be any recurrent unit.   \nFinally, the prediction is made through a decoder by feeding the graph signal, $z _ { i } ^ { * }$ and $z _ { i j } ^ { * }$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Learning objective Let $\\hat { f }$ and $\\hat { F }$ denote predictions of the target node/edge signals. PA-DGN is trained by minimizing the following objective: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/714d079c21931c1731a8b08eefb7f05f74776678f6529f7f340d73812aded3e5.jpg",
+        "text": "$$\n\\mathcal { L } = \\sum _ { i \\in \\mathbb { V } } \\vert \\vert f _ { i } - \\hat { f } _ { i } \\vert \\vert ^ { 2 } + \\sum _ { ( i , j ) \\in \\mathbb { E } } \\vert \\vert F _ { i j } - \\hat { F } _ { i j } \\vert \\vert ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            297,
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+            335
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For multistep predictions, $\\mathcal { L }$ is summed over all predicting steps. If only one type (node or edge) of signal is given, the corresponding term in Eq 3 is used to optimize the parameters in SDL and RGN simultaneously. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 EFFECTIVENESS OF SPATIAL DIFFERENCE LAYER ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            406,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To investigate if the proposed spatial difference forms (Eq 1) can be beneficial to learning physicsrelated patterns, we use SDL on two different tasks: (1) approximate directional derivatives and (2) predict synthetic graph signals. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 APPROXIMATION OF DIRECTIONAL DERIVATIVES",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As we claimed in Section 2.3, the standard difference forms (gradient and Laplacian) on a graph can cause significant numerical error easily because they are susceptible to a distance of two points and variations of a given function. To evaluate the applicability of the proposed SDL, we train SDL to approximate directional derivatives on a graph. First, we define a synthetic function and its gradients on 2D space and sample 200 points $( x _ { i } , y _ { i } )$ . Then, we construct a graph on the sampled points by using $k$ -NN algorithm $( k = 4$ ). With the known gradient $\\begin{array} { r } { \\bigg ( \\nabla f = ( \\frac { \\partial f } { \\partial x } , \\frac { \\partial f } { \\partial y } ) \\bigg ) } \\end{array}$ at each point (a node in the graph), we can compute directional derivatives by projecting $\\nabla f$ to a connected edge $e _ { i j }$ (See Figure 3). We compare against four baselines: (1) the finite gradient (FinGrad) (2) multilayer perceptron (MLP) (3) graph networks (GN) (4) a different form of $\\mathrm { E q } 1$ (One-w). For the finite gradient $( ( f _ { j } - f _ { i } ) / | | x _ { j } - x _ { i } | | )$ , there is no learnable parameter and it only uses two points. For MLP, we feed $( f _ { i } , f _ { j } , \\pmb { x } _ { i } , \\pmb { x } _ { j } )$ as input to see whether learnable parameters can benefit the approximation or not. For GN, we use distances of two connected points as edge features and function values on the points as node features. The edge feature output of GN is used as a prediction for the directional derivative on the edge. Finally, we modify the proposed form as $( \\overset { w } { \\nabla { f } } ) _ { i j } = w _ { i j } f _ { j } - f _ { i }$ . GN and the modified form are used to verify the effectiveness of Eq 1. Note that we define two synthetic functions (Figure 4) which have different property; (1) monotonically increasing from a center and (2) periodically varying. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/067e3e5946fdec6e962b58900cdfdb1a2cc8877584c427fc82ba89e26a60dbda.jpg",
+        "image_caption": [
+            "Figure 3: Directional derivative on graph "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            661,
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/d434284a186b629e1732483f588ce3282ab063b57cccfc6ab6c336a0cd255e9a.jpg",
+        "table_caption": [
+            "Table 1: Mean squared error $( 1 0 ^ { - 2 } )$ for approximation of directional derivatives. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Functions</td><td>FinGrad</td><td>MLP</td><td>GN</td><td>One-w</td><td>SDL</td></tr><tr><td>fi(x,y)=0.1x² +0.5y²</td><td>6.42±0.47</td><td>2.12±0.32</td><td>1.05±0.42</td><td>1.41±0.44</td><td>0.97±0.39</td></tr><tr><td>f2(x,y)= sin(x) +cos(y)</td><td>5.90±0.04</td><td>2.29±0.77</td><td>2.17±0.34</td><td>6.73±1.17</td><td>1.26±0.05</td></tr></table>",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/9fac30ff5be81be8490024894820601036be32ed3ceb7491f1ab6ed14988804d.jpg",
+        "image_caption": [
+            "Figure 4: Gradients and graph structure of sampled points. Left: the synthetic function is $f _ { 1 } ( x , y ) =$ $0 . \\check { 1 } x ^ { 2 } + 0 . 5 y ^ { 2 }$ . Right: the synthetic function is $f _ { 2 } ( x , y ) = \\sin ( x ) + \\cos ( y )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Approximation accuracy As shown in Table 1, the proposed spatial difference layer outperforms others by a large margin. As expected, FinGrad provides the largest error since it only considers two points without learnable parameters. It is found that the learnable parameters can significantly benefit to approximate the directional derivatives even if input is the same (FinGrad vs. MLP). Note that utilizing neighboring information (GN, One-w, SDL) is generally helpful to learn spatial variations properly. However, simply training parameters in GN is not sufficient and explicitly defining difference, which is important to understand spatial variations, provides more robust inductive bias. One important thing we found is that One-w is not effective as much as GN and it can be even worse than FinGrad. It is because of its limited degree of freedom. As implied in the form $( \\nabla _ { w } f ) _ { i j } = w _ { i j } * f _ { j } - f _ { i }$ , only one $w _ { i j }$ adjusts the relative difference between $f _ { i }$ and $f _ { j }$ , and this is not enough to learn whole possible linear combinations of $f _ { i }$ and $f _ { j }$ . The unstable performance supports that the form of SDL is not ad-hoc but more rigorously designed. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 GRAPH SIGNAL PREDICTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We evaluate PA-DGN on the synthetic data sampled from the simulation of specific convectiondiffusion equations, to provide if the proposed model can predict next signals of the simulated dynamics from observations on discrete nodes only. For the simulated dynamics, we use an equation slightly modified based on the one in Long et al. (2018). ",
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+        "type": "equation",
+        "img_path": "images/f947fc4c459361ea1c20b6ce3fc7c14da2474d50db7bc6ca4e7f8b151d48e759.jpg",
+        "text": "$$\n\\frac { d f _ { i } ( t ) } { d t } = a ( i ) ( \\nabla f ) _ { \\hat { x } } + b ( i ) ( \\nabla f ) _ { \\hat { y } } + c ( i ) \\Delta f , \\qquad f _ { i } ( 0 ) = f _ { o } ( i ) ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where the index $i$ points the $i$ -th node whose coordinate is $( x _ { i } , y _ { i } )$ in the 2D space $( [ 0 , 2 \\pi ] \\times [ 0 , 2 \\pi ] )$ and $\\hat { x }$ and $\\hat { y }$ indicate $x$ - and $y$ -direction in the space. $a ( i ) = 0 . 5 ( \\cos ( y _ { i } ) + x _ { i } ( 2 \\pi - \\bar { x _ { i } } ) \\sin ( x _ { i } ) \\bar { ) } + 0 . \\bar { 6 }$ $b ( i ) = 2 ( \\cos ( y _ { i } ) + \\sin ( x _ { i } ) ) + 0 . 8 _ { }$ , and $\\begin{array} { r } { c ( i ) = 0 . 5 \\left( 1 - \\frac { \\sqrt { ( x _ { i } - \\pi ) ^ { 2 } + ( y _ { i } - \\pi ) ^ { 2 } } } { \\sqrt { 2 } \\pi } \\right) } \\end{array}$ . Then, we uniformly sample 250 points in the above 2D space. The task is to predict signal values of all points in the future $M$ steps given observed values of the first $N$ steps. For our experiments, we choose $N = 5$ and $M = 1 5$ . Since there is no a priori graph structure on sampled points, we construct a graph with $k$ -NN algorithm $ { [ k = 4 ] }$ ) using the Euclidean distance. Figure 5 shows the dynamics and the graph structure of sampled points. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To evaluate the effect of the proposed SDL on the above prediction task, we cascade SDL and a linear regression model as our prediction model since the dynamics follows a linear partial differential equation. We compare its performance with four baselines: (1) vector auto-regressor (VAR); (2) multilayer perceptron (MLP); (3) StandardOP: the standard approximation of differential operators in Section 2.1 followed by a linear regressor; (4) MeshOP: similar to StandardOP but use the discretization on triangulated mesh in Section 2.2 for differential operators. ",
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+        "table_caption": [
+            "Table 2: Mean absolute error $( 1 0 ^ { - 2 } )$ for graph signal prediction. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>VAR</td><td>MLP</td><td>StandardOP</td><td>MeshOP</td><td>SDL</td></tr><tr><td>16.84±0.41</td><td>15.75±0.53</td><td>11.99±0.29</td><td>12.82±0.06</td><td>10.87±0.98</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Prediction Performance Table 2 shows the prediction performance of different models measured with mean absolute error. The prediction model with our proposed spatial differential layer outperforms other baselines. All models incorporating any form of spatial difference operators (StandardOP, ",
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+        "img_path": "images/44d4bb29f77c07af9c69d1563184127ca0bf654c6929bccc6caaf555c37c827b.jpg",
+        "image_caption": [
+            "Figure 5: Synthetic dynamics and graph structure of sampled points. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "MeshOP and SDL) outperform those without spatial difference operators (VAR and MLP), showing that introducing spatial differences information inspired by the intrinsic dynamics helps prediction. However, in cases where points with observable signal are sparse in the space, spatial difference operators derived with fixed rules can be inaccurate and sub-optimal for prediction since the locally linear assumption which they are based on no longer holds. Our proposed SDL, to the contrary, is capable of bridging the gap between approximated difference operators and accurate ones by introducing learnable coefficients utilizing neighboring information, and thus improves the prediction performance. ",
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+    {
+        "type": "text",
+        "text": "4 PREDICTION: GRAPH SIGNALS ON LAND-BASED WEATHER SENSORS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate the proposed model on the task of predicting climate observations (Temperature) from the land-based weather stations located in the United States. ",
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SET-UP ",
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+    {
+        "type": "text",
+        "text": "Data and task We sample the weather stations located in the United States from the Online Climate Data Directory of the National Oceanic and Atmospheric Administration (NOAA) and choose the stations which have actively measured meteorological observations during 2015. We choose two geographically close but meteorologically diverse groups of stations: the Western and Southeastern states. We use $k$ -Nearest Neighbor (NN) algorithm $k = 4$ ) to generate graph structures and the final adjacency matrix is $A = \\bar { ( } A _ { k } + A _ { k } ^ { \\top } ) / 2$ to make it symmetric where $A _ { k }$ is the output adjacency matrix from $k$ -NN algorithm. ",
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+    {
+        "type": "text",
+        "text": "Figure 6 shows the distributions of the land-based weather stations and their connectivity. Since the stations are not synchronized and have different timestamps for the observations, we aggregate the time series hourly. The 1-year sequential data are split into the train set (8 months), the validation set (2 months), and the test set (2 months), respectively. ",
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+    {
+        "type": "text",
+        "text": "Our main task is to predict the next graph signals based on the current and past graph signals. All methods we evaluate are trained through the objective (Eq 3) with the Adam optimizer and we use scheduled sampling (Bengio et al., 2015) for the models with recurrent modules. We evaluate PA-DGN and other baselines on two prediction tasks, (1) 1-step and (2) multistep-ahead predictions. Furthermore, we demonstrate the ablation study that provides how much the spatial derivatives from our proposed SDL are important signals to predict the graph dynamics. ",
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+        "img_path": "images/a27f4d0228aea93218bb3e8d15df642d05e54640be82b0c45bea4364b2457316.jpg",
+        "image_caption": [
+            "Figure 6: Weather stations in (left) western (right) southeastern states in the United States and $k$ -NN graph. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.2 GRAPH SIGNAL PREDICTIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare against the widely used baselines (VAR, MLP, and GRU) for 1-step and multistep prediction. Then, we use RGN (Sanchez-Gonzalez et al., 2018) to examine how much the graph structure is beneficial. Finally, we evaluate PA-DGN to verify if the proposed architecture (Eq 1) is able to reduce prediction loss. Experiment results for the prediction task are summarized in Table 3. ",
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+    {
+        "type": "text",
+        "text": "Overall, RGN and PA-DGN are better than other baselines and it implies that the graph structure provides useful inductive bias for the task. It is intuitive as the meteorological observations are continuously changing over the space and time and thus, the observations at the $i$ -th station are strongly related to those of its neighboring stations. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "PA-DGN outperforms RGN and the discrepancy comes from the fact that the spatial derivatives (Eq 1) we feed in PA-DGN are beneficial and this finding is expected because the meteorological signals at a certain point are a function of not only its previous signal but also the relative differences between neighbor signals and itself. Knowing the relative differences among local observations is particularly essential to understand physics-related dynamics. For example, Diffusion equation, which describes how physical quantities (e.g., heat) are transported through space over time, is also a function of relative differences of the quantities ( $\\begin{array} { r } { \\frac { d f } { d t } = D \\bar { \\Delta } f ) } \\end{array}$ rather than values of the neighbor signals. In other words, spatial differences are physics-aware features and it is desired to leverage the features as input to learn dynamics related to physical phenomena. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/d23acca0f95b947d7b87da60b568e78ac86dcc1cb1843cdc4da434ff05b83676.jpg",
+        "table_caption": [
+            "Table 3: Graph signal prediction results (MAE) on multistep predictions. In each row, we report the average with standard deviations from all baselines and PA-DGN. One step is 1-hour time interval. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Region</td><td>Method</td><td>1-step</td><td>6-step</td><td>12-step</td></tr><tr><td rowspan=\"7\">West</td><td>VAR</td><td>0.1241 ± 0.0234</td><td>0.4295 ± 0.1004</td><td>0.4820 ± 0.1298</td></tr><tr><td>MLP</td><td>0.1040 ± 0.0003</td><td>0.3742 ± 0.0238</td><td>0.4998 ± 0.0637</td></tr><tr><td>GRU</td><td>0.0913 ± 0.0047</td><td>0.1871 ± 0.0102</td><td>0.2707 ± 0.0006</td></tr><tr><td>RGN</td><td>0.0871 ± 0.0033</td><td>0.1708 ± 0.0024</td><td>0.2666 ± 0.0252</td></tr><tr><td>RGN(StandardOP)</td><td>0.0860 ± 0.0018</td><td>0.1674 ± 0.0019</td><td>0.2504 ± 0.0107</td></tr><tr><td>RGN(MeshOP)</td><td>0.0840 ± 0.0015</td><td>0.2119 ± 0.0018</td><td>0.4305 ± 0.0177</td></tr><tr><td>PA-DGN</td><td>0.0840 ± 0.0004</td><td>0.1614 ± 0.0042</td><td>0.2439 ± 0.0163</td></tr><tr><td rowspan=\"7\">SouthEast</td><td>VAR</td><td>0.0889 ± 0.0025</td><td>0.2250 ± 0.0013</td><td>0.3062 ± 0.0032</td></tr><tr><td>MLP</td><td>0.0722 ± 0.0012</td><td>0.1797 ± 0.0086</td><td>0.2514 ± 0.0154</td></tr><tr><td>GRU</td><td>0.0751 ± 0.0037</td><td>0.1724 ± 0.0130</td><td>0.2446 ± 0.0241</td></tr><tr><td>RGN</td><td>0.0790 ± 0.0113</td><td>0.1815 ± 0.0239</td><td>0.2548 ± 0.0210</td></tr><tr><td>RGN(StandardOP)</td><td>0.0942 ± 0.0121</td><td>0.2135 ± 0.0187</td><td>0.2902 ± 0.0348</td></tr><tr><td>RGN(MeshOP)</td><td>0.0905 ± 0.0012</td><td>0.2052 ± 0.0012</td><td>0.2602 ± 0.0062</td></tr><tr><td>PA-DGN</td><td>0.0721 ± 0.0002</td><td>0.1664 ± 0.0011</td><td>0.2408 ± 0.0056</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 CONTRIBUTION OF SPATIAL DERIVATIVES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We further investigate if the modulated spatial derivatives $( \\mathrm { E q ~ 1 } )$ ) are effectively advantageous compared to the spatial derivatives defined in Riemannian manifolds. First, RGN without any spatial derivatives is assessed for the prediction tasks on Western and Southeastern states graph signals. Note that this model does not use any extra features but the graph signal, $f ( t )$ . Secondly, we add (1) StandardOP, the discrete spatial differences (Gradient and Laplacian) in Section 2.1 and (2) MeshOP, the triangular mesh approximation of differential operators in Section 2.2 separately as additional signals to RGN. Finally, we incorporate with RGN our proposed Spatial Difference Layer. ",
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+    {
+        "type": "text",
+        "text": "Table 3 shows the contribution of each component. As expected, PA-DGN provides much higher drops in MAE $3 . 5 6 \\%$ , $5 . 5 0 \\%$ , $8 . 5 1 \\%$ and $8 . 7 3 \\%$ , $8 . 3 2 \\%$ , $5 . 4 9 \\%$ on two datasets, respectively) compared to RGN without derivatives and the results demonstrate that the derivatives, namely, relative differences from neighbor signals are effectively useful. However, neither RGN with StandardOP nor with MeshOP can consistently outperform RGN. We also found that PA-DGN consistently shows positive effects on the prediction error compared to the fixed derivatives. This finding is a piece of evidence to support that the parameters modulating spatial derivatives in our proposed Spacial Difference Layer are properly inferred to optimize the networks and to improve the prediction performance. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduce a novel architecture (PA-DGN) that approximates spatial derivatives to use them to represent PDEs which have a prominent role for physics-aware modeling. PA-DGN ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "effectively learns the modulated derivatives for predictions and the derivatives can be used to discover hidden physics describing interactions between temporal and spatial derivatives. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work is supported in part by NSF Research Grant IIS-1254206 and MINERVA grant N00014-17- 1-2281, granted to co-author Yan Liu in her academic role at the University of Southern California. The views and conclusions are those of the authors and should not be interpreted as representing the official policies of the funding agency, or the U.S. Government. Last but not least, we appreciate anonymous reviewers for your thorough comments and suggestions. ",
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+        "type": "text",
+        "text": "Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al. Google’s neural machine translation system: Bridging the gap between human and machine translation. arXiv preprint arXiv:1609.08144, 2016. ",
+        "bbox": [
+            174,
+            728,
+            826,
+            785
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 SIMULATED DATA ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            132,
+            344,
+            147
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For the simulated dynamics, we discretize the following partial differential equation similar to the one in Long et al. (2018) to simulate the corresponding linear variable-coefficient convection-diffusion equation on graphs. ",
+        "bbox": [
+            173,
+            159,
+            825,
+            202
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In a continuous space, we define the linear variable-coefficient convection-diffusion equation as: ",
+        "bbox": [
+            173,
+            207,
+            802,
+            222
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bbd751caaa0b6a73fc2f198310df255a80bc71a2b52a40405887bd310dfe3a94.jpg",
+        "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { \\frac { \\partial f } { \\partial t } } & { = a ( x , y ) f _ { x } + b ( x , y ) f _ { y } + c ( x , y ) \\Delta f } \\\\ { \\dot { f } \\rvert _ { t = 0 } } & { = f _ { 0 } ( x , y ) } \\\\ { \\textrm { \\scriptsize 1 } \\Omega = [ 0 , 2 \\pi ] \\times [ 0 , 2 \\pi ] , ( t , x , y ) \\in [ 0 , 0 . 2 ] \\times \\Omega , a ( x , y ) = 0 . 5 \\big ( \\cos ( y ) + x ( 2 \\pi - x ) \\sin ( x ) \\big ) + } \\\\ { b ( x , y ) = 2 ( \\cos ( y ) + \\sin ( x ) ) + 0 . 8 , c ( x , y ) = 0 . 5 \\bigg ( 1 - \\frac { \\sqrt { ( x _ { i } - \\pi ) ^ { 2 } + ( y _ { i } - \\pi ) ^ { 2 } } } { \\sqrt { 2 \\pi } } \\bigg ) . } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            173,
+            223,
+            828,
+            309
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We follow the setting of initialization in Long et al. (2018): ",
+        "bbox": [
+            174,
+            311,
+            566,
+            327
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e8ab11baf1a72986d95c00672518f66a4e67ca31758e554b22a35ce0372b82c8.jpg",
+        "text": "$$\nf _ { 0 } ( x , y ) = \\sum _ { | k | , | l | \\leq N } \\lambda _ { k , l } \\cos ( k x + l y ) + \\gamma _ { k , l } \\sin ( k x + l y )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            328,
+            689,
+            364
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": ", where $\\begin{array} { r } { N = 9 , \\lambda _ { k , l } , \\gamma _ { k , l } \\sim \\mathcal { N } \\left( 0 , \\frac { 1 } { 5 0 } \\right) } \\end{array}$ , and $k$ and $l$ are chosen randomly. ",
+        "bbox": [
+            173,
+            366,
+            650,
+            385
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We use spatial difference operators to approximate spatial derivatives: ",
+        "bbox": [
+            173,
+            388,
+            633,
+            404
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0855fa6a689486c56414b01b45c11d28e70cd01169c07b350cd4caa5439c0966.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { r l } { f _ { x } ( x _ { i } , y _ { i } ) } & { { } = \\displaystyle \\frac { 1 } { 2 s } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } - s , y _ { i } ) ) - \\displaystyle \\frac { 1 } { 2 s } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } + s , y _ { i } ) ) } \\\\ { f _ { y } ( x _ { i } , y _ { i } ) } & { { } = \\displaystyle \\frac { 1 } { 2 s } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } , y _ { i } - s ) ) - \\displaystyle \\frac { 1 } { 2 s } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } , y _ { i } + s ) ) } \\\\ { f _ { x x } ( x _ { i } , y _ { i } ) } & { { } = \\displaystyle \\frac { 1 } { s ^ { 2 } } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } - s , y _ { i } ) ) + \\displaystyle \\frac { 1 } { s ^ { 2 } } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } + s , y _ { i } ) ) } \\\\ { f _ { y y } ( x _ { i } , y _ { i } ) } & { { } = \\displaystyle \\frac { 1 } { s ^ { 2 } } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } , y _ { i } - s ) ) + \\displaystyle \\frac { 1 } { s ^ { 2 } } ( f ( x _ { i } , y _ { i } ) - f ( x _ { i } , y _ { i } + s ) ) } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            222,
+            405,
+            766,
+            554
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": ", where $s$ is the spatial grid size for discretization. ",
+        "bbox": [
+            176,
+            554,
+            498,
+            568
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Then we rewrite (5) with difference operators defined on graphs: ",
+        "bbox": [
+            178,
+            569,
+            596,
+            582
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1442a47a2c5a51a9c15d39f981d099655e61a48526b260a57a10b102418e7700.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { l l } { \\frac { \\partial f } { \\partial t } = a ( i ) ( \\nabla f ) _ { \\hat { x } } + b ( i ) ( \\nabla f ) _ { \\hat { y } } + c ( i ) ( ( \\Delta f ) _ { \\hat { x } \\hat { x } } + ( \\Delta f ) _ { \\hat { y } \\hat { y } } ) } \\\\ { f _ { i } ( 0 ) = f _ { o } ( i ) } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            583,
+            696,
+            618
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": ", where ",
+        "bbox": [
+            173,
+            619,
+            225,
+            633
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6094c1bfb468410b6f1c88e9395c4ad9d716f6b02cc1e9eefa089fed9ac196a.jpg",
+        "text": "$$\n\\begin{array} { r } { a ( i ) ( x _ { j } , y _ { j } ) = \\left\\{ \\begin{array} { l l } { \\displaystyle \\frac { a ( x _ { i } , y _ { i } ) } { 2 s } } & { \\mathrm { i f ~ } x _ { i } = x _ { j } + s , y _ { i } = y _ { j } } \\\\ { - \\displaystyle \\frac { a ( x _ { i } , y _ { i } ) } { 2 s } } & { \\mathrm { i f ~ } x _ { i } = x _ { j } - s , y _ { i } = y _ { j } } \\\\ { \\displaystyle \\frac { b ( x _ { i } , y _ { i } ) } { 2 s } } & { \\mathrm { i f ~ } x _ { i } = x _ { j } , y _ { i } = y _ { j } + s } \\\\ { - \\displaystyle \\frac { b ( x _ { i } , y _ { i } ) } { 2 s } } & { \\mathrm { i f ~ } x _ { i } = x _ { j } , y _ { i } = y _ { j } - s } \\end{array} \\right. } \\\\ { c ( i ) ( x _ { j } , y _ { j } ) = \\frac { c } { s ^ { 2 } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            633,
+            681,
+            779
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Then we replace the gradient w.r.t time in (8) with temporal discretization: ",
+        "bbox": [
+            176,
+            795,
+            661,
+            810
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/220ff336444d5c2623c7b50c24f5bd82a20d0075fddb5e3b2cfefaf5a59ecbba.jpg",
+        "text": "$$\n\\left\\{ \\begin{array} { l l } { f ( t + 1 ) = \\Delta t ( a ( i ) ( \\nabla f ) _ { \\hat { x } } + b ( i ) ( \\nabla f ) _ { \\hat { y } } + c ( i ) ( ( \\Delta f ) _ { \\hat { x } \\hat { x } } + ( \\Delta f ) _ { \\hat { y } \\hat { y } } ) ) + f ( t ) } \\\\ { f _ { i } ( 0 ) = f _ { o } ( i ) } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            811,
+            756,
+            844
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": ", where $\\Delta t$ is the time step in temporal discretization. ",
+        "bbox": [
+            174,
+            847,
+            526,
+            861
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Equation (12) is used for simulating the dynamics described by the equation (5). Then, we uniformly sample 250 points in the above 2D space and choose their corresponding time series of $u$ as the dataset used in our synthetic experiments. We generate 1000 sessions on a $5 0 \\times 5 0$ regular mesh with time step size $\\Delta t = 0 . 0 1$ . 700 sessions are used for training, 150 for validation and 150 for test. ",
+        "bbox": [
+            173,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 EXPERIMENT SETTINGS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            385,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here we provide additional details for models we used in this work, including model architecture settings and hyper-parameter settings. ",
+        "bbox": [
+            174,
+            128,
+            823,
+            159
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2.1 MODEL SETTINGS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            172,
+            361,
+            188
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Unless mentioned otherwise, all models use a hidden dimension of size 64. ",
+        "bbox": [
+            174,
+            196,
+            663,
+            212
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• VAR: A vector autoregression model with 2 lags. Input is the concatenated features of previous 2 frames. The weights are shared among all nodes in the graph.   \n• MLP: A multilayer perceptron model with 2 hidden layers. Input is the concatenated features of previous 2 frames. The weights are shared among all nodes in the graph. GRU: A Gated Recurrent Unit network with 2 hidden layers. Input is the concatenated features of previous 2 frames. The weights are shared among all nodes in the graph. RGN: A recurrent graph neural network model with 2 GN blocks. Each GN block has an edge update block and a node update block, both of which use a 2-layer GRU cell as the update function. We set its hidden dimension to 73 so that it has the same number of learnable parameters as our proposed model PA-DGN. RGN(StandardOP): Similar to RGN, but use the output of difference operators in Section 2.1 as extra input features. We set its hidden dimension to 73. RGN(MeshOP): Similar to RGN(StandardOP), but the extra input features are calculated using opeartors in Section 2.2. We set its hidden dimension to 73. PA-DGN: Our proposed model. The spatial derivative layer uses a message passing neural network (MPNN) with 2 GN blocks using 2-layer MLPs as update functions. The forward network part uses a recurrent graph neural network with 2 recurrent GN blocks using 2-layer GRU cells as update functions. ",
+        "bbox": [
+            212,
+            219,
+            826,
+            497
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The numbers of learnable parameters of all models are listed as follows: ",
+        "bbox": [
+            174,
+            507,
+            643,
+            522
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/06aed820d6904a22742d63844b21f9fe7d9e9e2f1c5986c38181aa3219ffbcaf.jpg",
+        "table_caption": [
+            "Table 4: Numbers of learnable parameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>VAR</td><td>MLP</td><td>GRU</td><td>RGN</td><td>RGN(StandardOP)</td><td>RGN(MeshOP)</td><td>PA-DGN</td></tr><tr><td> #Params</td><td>3</td><td>4,417</td><td>37,889</td><td>345,876</td><td>341,057</td><td>342,152</td><td>340,001</td></tr></table>",
+        "bbox": [
+            176,
+            560,
+            821,
+            602
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2.2 TRAINING SETTINGS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            630,
+            379,
+            643
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The number of evaluation runs We performed 3 times for every experiment in this paper to report the mean and standard deviations. ",
+        "bbox": [
+            173,
+            654,
+            823,
+            683
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Length of prediction For experiments on synthetic data, all models take first 5 frames as input and predict the following 15 frames. For experiments on NOAA datasets, all models take first 12 frames as input and predict the following 12 frames. ",
+        "bbox": [
+            174,
+            696,
+            825,
+            739
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Training hyper-parameters We use Adam optimizer with learning rate 1e-3, batch size 8, and weight decay of 5e-4. All experiments are trained for a maximum of 2000 epochs with early stopping. All experiments are trained using inverse sigmoid scheduled sampling with the coefficient $k = 1 0 7$ ",
+        "bbox": [
+            173,
+            753,
+            823,
+            796
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Environments All experiments are implemented with Python3.6 and PyTorch 1.1.0, and are run with NVIDIA GTX 1080 Ti GPUs. ",
+        "bbox": [
+            173,
+            810,
+            823,
+            838
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 EFFECT OF DIFFERENT GRAPH STRUCTURES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            856,
+            522,
+            869
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we evaluate the effect of 2 different graph structures on baselines and our models: (1) k-NN: a graph constructed with $k$ -NN algorithm $k = 4 ,$ ; (2) TriMesh: a graph generated with Delaunay Triangulation. All graphs use the Euclidean distance. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/d468aec310655781b013e6464c1913c352bf846238e3c4b1b0ab896fb6ed9cbb.jpg",
+        "table_caption": [
+            "Table 5: Mean absolute error $( 1 0 ^ { - 2 } )$ ) for graph signal prediction on the synthetic dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">VAR</td><td rowspan=\"2\">MLP</td><td colspan=\"2\">StandardOP</td><td colspan=\"2\">MeshOP</td><td colspan=\"2\">SDL</td></tr><tr><td>k-NN</td><td>TriMesh</td><td>k-NN</td><td>TriMesh</td><td>k-NN</td><td>TriMesh</td></tr><tr><td>17.30</td><td>16.27</td><td>12.00</td><td>12.29</td><td>12.87</td><td>12.82</td><td>11.04</td><td>12.40</td></tr></table>",
+        "bbox": [
+            240,
+            127,
+            756,
+            191
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/a8cbc5b8d191321311d47f0e2358c58dc0e98e14b548cc199e7ea5146abd5bfc.jpg",
+        "table_caption": [
+            "Table 6: Graph signal prediction results (MAE) on multistep predictions. In each row, we report the average with standard deviations from all baselines and PA-DGN. One step is 1 hour time interval. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Region</td><td>Method</td><td>Graph</td><td>1-step</td><td>6-step</td><td>12-step</td></tr><tr><td rowspan=\"7\">West</td><td>VAR MLP</td><td></td><td>0.1241 ± 0.0234</td><td>0.4295 ± 0.1004</td><td>0.4820 ± 0.1298</td></tr><tr><td>GRU</td><td>1</td><td>0.1040 ± 0.0003 0.0913 ± 0.0047</td><td>0.3742 ± 0.0238 0.1871 ± 0.0102</td><td>0.4998 ± 0.0637 0.2707 ± 0.0006</td></tr><tr><td>RGN</td><td>k-NN TriMesh</td><td>0.0871 ± 0.0033</td><td>0.1708 ± 0.0024</td><td>0.2666 ± 0.0252</td></tr><tr><td>RGN (StandardOP)</td><td>k-NN TriMesh</td><td>0.0897 ± 0.0030 0.0860 ± 0.0018</td><td>0.1723 ± 0.0116 0.1674 ± 0.0019</td><td>0.2800 ± 0.0414 0.2504 ± 0.0107</td></tr><tr><td>RGN</td><td>k-NN</td><td>0.0842 ± 0.0011 0.0840 ± 0.0015</td><td>0.1715 ± 0.0027 0.2119 ± 0.0018</td><td>0.2517 ± 0.0369 0.4305 ± 0.0177</td></tr><tr><td>(MeshOP)</td><td>TriMesh k-NN</td><td>0.0846 ± 0.0017 0.0840 ± 0.0004</td><td>0.2090 ± 0.0077</td><td>0.4051 ± 0.0457</td></tr><tr><td>PA-DGN VAR</td><td>TriMesh</td><td>0.0849 ± 0.0012</td><td>0.1614 ± 0.0042 0.1610 ± 0.0029</td><td>0.2439 ± 0.0163 0.2473 ± 0.0162</td></tr><tr><td rowspan=\"7\">SouthEast</td><td>MLP GRU</td><td></td><td>0.0889 ± 0.0025 0.0722 ± 0.0012</td><td>0.2250 ± 0.0013 0.1797 ± 0.0086</td><td>0.3062 ± 0.0032 0.2514 ± 0.0154</td></tr><tr><td>RGN</td><td>= k-NN</td><td>0.0751 ± 0.0037 0.0790 ± 0.0113</td><td>0.1724 ± 0.0130 0.1815 ± 0.0239</td><td>0.2446 ± 0.0241 0.2548 ± 0.0210</td></tr><tr><td>RGN (StandardOP)</td><td>TriMesh k-NN</td><td>0.0932 ± 0.0105 0.0942 ± 0.0121</td><td>0.2076 ± 0.0200 0.2135 ± 0.0187</td><td>0.2854 ± 0.0211 0.2902 ± 0.0348</td></tr><tr><td rowspan=\"2\">RGN (MeshOP)</td><td>TriMesh</td><td>0.0868 ± 0.0132</td><td>0.1885 ± 0.0305</td><td>0.2568 ± 0.0328</td></tr><tr><td>k-NN</td><td>0.0913 ± 0.0016</td><td>0.2069 ± 0.0031</td><td>0.2649 ± 0.0092</td></tr><tr><td></td><td>TriMesh k-NN</td><td>0.0877 ± 0.0020 0.0721 ± 0.0002</td><td>0.2043 ± 0.0026</td><td>0.2579 ± 0.0057</td></tr><tr><td>PA-DGN</td><td>TriMesh</td><td>0.0876 ± 0.0096</td><td>0.1664 ± 0.0011 0.2002 ± 0.0163</td><td>0.2408 ± 0.0056 0.2623 ± 0.0180</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 5 and Table 6 show the effect of different graph structures on the synthetic dataset used in Section 3.2 and the real-world dataset in Section 4.2 separately. We find that for different models the effect of graph structures is not homogeneous. For RGN and PA-DGN, $k$ -NN graph is more beneficial to the prediction performance than TriMesh graph, because these two models rely more on neighboring information and a $k$ -NN graph incorporates it better than a Delaunay Triangulation graph. However, switching from TriMesh graph to $k$ -NN graph is harmful to the prediction accuracy of RGN(MeshOP) since Delaunay Triangulation is a well-defined method for generating triangulated mesh in contrast to $k$ -NN graphs. Given the various effect of graph structures on different models, our proposed PA-DGN under $k$ -NN graphs always outperforms other baselines using any graph structure. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/00af0c99e444548cf3f703e615283468f9daffee0b6b915718fed2d1cc52a107.jpg",
+        "image_caption": [
+            "Figure 7: MAE across the nodes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.4 THE DISTRIBUTION OF PREDICTION ERROR ACROSS NODES ",
+        "text_level": 1,
+        "bbox": [
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+            315,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 provides the distribution of MAEs across the nodes of PA-DGN applied to the graph signal prediction task of the west coast region of the real-world dataset in Section 4.2. As shown in the figure, nodes with the highest prediction error for short-term prediction are gathered in the inner part where the observable nodes are sparse, while for long-term prediction nodes in the area with a limited number of observable points no longer have the largest MAE. This implies that PA-DGN can utilize neighboring information efficiently even under the limitation of sparsely observable points. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.5 EVALUATION ON NEMO SEA SURFACE TEMPERATURE (SST) DATASET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            440,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We tested our proposed method and baselines on the NEMO sea surface temperature (SST) dataset1. We first download the data in the area between $5 0 \\mathrm { N } ^ { \\circ }  – 6 5 \\mathrm { N } ^ { \\circ }$ and $7 5 \\mathrm { W } ^ { \\circ } \\ – 1 0 \\mathrm { W } ^ { \\circ }$ starting from 2016-01- 01 to 2017-12-31, then we crop the $[ 0 , 5 5 0 ] \\times [ 1 0 0 , 6 5 0 ]$ square from the area and sample 250 points from the square as our chosen dataset. We divide the data into 24 sequences, each lasting 30 days, and truncate the tail. All models use the first 5-day SST as input and predict the SST in the following 15 and 25 days. We use the data in 2016 for training all models and the left for testing. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "For StandardOP, MeshOP and SDL, we test both options using linear regression and using RGN for the prediction part and report the best result. The results in Table 7 show that all methods incorporating spatial differences gain improvement on prediction and that our proposed learnable SDL outperforms all other baselines. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/3ec87fc8d7a50e51dc76264ed065598994515792584d47450bfdc5358d21b484.jpg",
+        "table_caption": [
+            "Table 7: Mean absolute error $( 1 0 ^ { - 2 } )$ for SST graph signal prediction. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>VAR</td><td>MLP</td><td>GRU</td><td>RGU</td><td>StandardOP</td><td>MeshOP</td><td>SDL</td></tr><tr><td>15-step</td><td>15.123</td><td>15.058</td><td>15.101</td><td>15.172</td><td>14.756</td><td>14.607</td><td>14.382</td></tr><tr><td>25-step</td><td>19.533</td><td>19.473</td><td>19.522</td><td>19.705</td><td>18.983</td><td>18.977</td><td>18.434</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.6 EVALUATION ON DATASETS WITH DIFFERENT SPARSITY ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            602,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We changed the number of nodes to control the sparsity of data. As shown in Table 8, our proposed model outperforms others under various settings of sparsity on the synthetic experiment in Section 3.2. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/725bfd598065f4d63f818ee4a4ac4058da8d5f089b55298827bba865be95dca3.jpg",
+        "table_caption": [
+            "Table 8: Mean absolute error $( 1 0 ^ { - 2 } )$ ) for graph signal prediction with different sparsity. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>#Nodes</td><td>VAR</td><td>MLP</td><td>StandardOP</td><td>MeshOP</td><td>SDL</td></tr><tr><td>250</td><td>0.1730</td><td>0.1627</td><td>0.1200</td><td>0.1287</td><td>0.1104</td></tr><tr><td>150</td><td>0.1868</td><td>0.1729</td><td>0.1495</td><td>0.1576</td><td>0.1482</td></tr><tr><td>100</td><td>0.1723</td><td>0.1589</td><td>0.1629</td><td>0.1696</td><td>0.1465</td></tr></table>",
+        "bbox": [
+            277,
+            208,
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+            281
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we sampled 400 points and trained SDL as described in Section 3.1, and resampled fewer points (350,300,250,200) to evaluate if SDL generalizes less sparse setting. As Table 9 shows, MSE increases when fewer sample points are used. However, SDL is able to provide much more accurate gradients even if it is trained under a new graph with different properties. Thus, the results support that SDL is able to generalize the c setting. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/3d05bd9334b69fc810e83cfd69731bae97aeb4602ce885d6dfb9074de9670aea.jpg",
+        "table_caption": [
+            "Table 9: Mean squared error $( 1 0 ^ { - 2 } )$ for approximations of directional derivatives of function $f _ { 2 } ( x , y ) = \\sin \\left( x \\right) + \\cos \\left( y \\right)$ with different sparsity. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>350 Nodes</td><td>300 Nodes</td><td>250 Nodes</td><td>200 Nodes</td></tr><tr><td>FinGrad</td><td>2.88 ± 0.11</td><td>3.42 ± 0.14</td><td>3.96 ± 0.17</td><td>4.99 ± 0.31</td></tr><tr><td>SDL</td><td>1.03 ± 0.09</td><td>1.14 ± 0.12</td><td>1.40 ± 0.10</td><td>1.76 ± 0.10</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    }
+]